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### Determine the acceleration of a freely-falling object (all boards)

This resource introduces uniform and non-uniform acceleration. Section 206-3 outlines a suitable practical. There are also supporting questions.

- Episode 206: Uniform and non-uniform acceleration
- Motion of a ball rolling down a plank
- Investigating free fall with a light gate
- Measurement of g using an electronic timer
- Episode 112: Resistivity
- Measuring resistance with a voltmeter and an ammeter
- Episode 121: EMF and internal resistance
- Internal resistance of a potato cell
- Internal resistance of a shoe box cell
- Episode 118: Potential dividers
- Episode 209: Drag, air resistance, terminal velocity
- Episode 228: The Young modulus
- Episode 311: Speed, frequency and wavelength
- Episode 324: Stationary or standing waves
- Melde's experiment
- Episode 321: Interference patterns
- Diffraction of laser light
- Episode 211: Newton’s second law of motion
- Investigating Newton's second law of motion
- Force used to kick a football
- Episode 220: Momentum and its conservation
- Investigating momentum during collisions
- Episode 124: Preparation for capacitors topic
- Episode 129: Discharge of a capacitor
- Episode 128: Energy stored by a capacitor
- Episode 122: Using an oscilloscope
- Episode 130: R-C circuits and other systems
- Episode 110: Resistance and temperature
- The effect of temperature on a thermistor
- Episode 608: Latent heat
- Episode 607: Specific heat capacity
- Specific thermal capacity of aluminium
- Specific thermal capacity of aluminium more accurately
- The specific thermal capacity of lead
- Episode 601: Brownian motion and ideal gases
- Boyle's law
- Pressure of air at constant volume
- Expansion of a gas at constant pressure
- Episode 510: Properties of radiations
- Episode 511: Absorption experiments
- Gamma radiation: inverse square law
- Gamma radiation: range and stopping
- Alpha radiation: range and stopping
- Episode 303: Mass-spring systems
- Episode 304: Simple pendulum
- Investigating a mass-on-spring oscillator
- Datalogging S.H.M. of a mass on a spring
- Examples of simple harmonic motion
- S.H.M. with a cantilever
- Episode 501: Spectra and energy levels
- Episode 502: The photoelectric effect
- Episode 414: Electromagnetic induction
- Episode 412: The force on a conductor in a magnetic field
- Force on a wire carrying a current in a magnetic field

Practical Activity for 16-19

There are two components to the assessment of practical work:

**Questions in the written exams:**these will count towards the final grade and provide the only differentiation on practical work; they will be designed to give an advantage to students who have had a thorough exposure to practical activities during the course**An endorsement of laboratory techniques:**this will be reported separately from the main grade (as a pass/fail). The requirement is that students complete at least 12 practical tasks over the two years. The tasks have to cover a range of skills that have been specified by Ofqual; teachers will have to ensure (and confirm) that students have mastered those skills

The awarding organisations are providing guidance and, in most cases, a pack of teacher and student notes for 12 activities. However, you should not feel restricted to that small set. And no one wants to see the pack being used as a kind of assessment pack – comprising 12 mini ISAs with vast matrices of check lists. There are plenty of additional activities from a multitude of sources that will support the development of the required skills and some of them can even be used to assess those skills.

Listed below are some suggested practical activities identified against the three English examination boards (AQA, OCR, Edexcel) for students to achieve the practical endorsement to their A-level physics award. To successfully achieve endorsement, candidates will be expected to have completed and recorded 12 activities within which they will have encountered and demonstrated mastery of 12 practical techniques. The activities outlined by the exam boards have been mapped to these techniques and so completing them will provide the minimum necessary coverage to achieve endorsement.

Schools are permitted to choose alternative activities but in that case they will need to undertake mapping of techniques to their chosen activity and ensure that these alternatives provide the same coverage of the techniques. In addition there will be a significant element of the written papers which will examine practical skills and may make reference to these activities but may be set in the context of other activities. Many of the activities are common to all boards but with slightly different wording. There are separate entries and slightly different references where appropriate.

This resource introduces uniform and non-uniform acceleration. Section 206-3 outlines a suitable practical. There are also supporting questions.

Forces and Motion

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

This episode continues to look at basic kinematics and introduces the equations of motion for uniform acceleration. This involves a little calculation practice.

Lesson Summary

- Demonstration (or student experiment): Non-uniform acceleration (20 minutes)
- Discussion: Developing equations of motion (10 minutes)
- Student experiment: Measuring acceleration due to gravity
*g*(20 minutes) - Student questions: Calculations (30 minutes)
- Worked example: Average velocity (10 minutes)

Uniform acceleration is compared with non-uniform acceleration.

Students will have already considered uniformly accelerated motion. This demonstration (or experiment) uses a similar method to consider non-uniform motion. You can conclude the demonstration by discussing the relationships shown on the graphs, stressing that these hold for both uniform and non-uniform motion.

Episode 206-1: An experimental velocity-time graph (Word, 35 KB)

Here you can develop the equations of motion (the SUVAT equations

).

Confident mathematicians will enjoy the mild challenge of developing SUVAT equations whereas weaker or non-mathematical students may find the activity surprisingly difficult. It is therefore best to proceed through the activity at a reasonable pace so that you concentrate on the results and using the SUVAT equations.

Episode 206-2: Deriving the equations of motion (Word, 41 KB)

Measuring the acceleration due to gravity g is a nice, simple experiment that also brings up the concepts of precision and accuracy. Of course, the students will know

the value of *g*, and may well have measured it. Nonetheless, it is a useful exercise to build good experimental practice.

Episode 206-3: Measuring the acceleration of free fall (Word, 21 KB)

These are a few simple questions that go over the ideas met in the unit. They include practice with interpreting motion graphs.

Episode 206-4: Motion under gravity (Word, 38 KB)

Episode 206-5: Thrust SSC (Word, 67 KB)

You might like to use the question below to highlight that the equations of motion (SUVAT equations) only apply to uniform acceleration.

A cyclist travels a displacement of 300 m due North at a velocity of 10 m s^{-1}. She travels the next 300 m in the same direction at a velocity of 15 m s^{-1}. Calculate the average velocity of the cyclist.

Answer: First 300 m takes:

300 m10 m s^{-1} = 30 s

Second 300 m takes:

300 m15 m s^{-1} = 20 s

average velocity = total displacementtotal time

average velocity = 600 m50 s

*v*_{average} = 12 m s^{-1}

Many weaker pupils will assume the answer is 12.5 m s^{-1} . You will have to explain why the equation:

average velocity = *v*+*u*2

cannot be used in this example. The equation only applies to uniformly accelerated motion. The cyclist spends longer travelling at 12 m s^{-1} than at 15 m s^{-1} .

The following practical investigates rolling a ball down a plank.

**Demonstration and Class practical**

This is a version of an experiment devised and carried out by Galileo. He used it to discover and demonstrate that motion was subject to quite simple, and beautiful, mathematical description.

Apparatus and Materials

- Plank, long, with one grooved edge
- Marbles, large, a supply of
- Hanging pennants, metal or light plastic, up to 6
- Clamps and stands for holding each pennant
- String
- Masses, 50 g, 5 or 6

Health & Safety and Technical Notes

The long plank should be handled by two persons.

Do not stand on a stool or bench to drop an object on a string. It could be dropped down a stairwell as long as precautions are taken to prevent the object falling on anyone.

Read our standard health & safety guidance

A length of 3 m of grooved plank is suitable. Alternatively, you could use a plain plank with grooved moulding fixed to it.

The pennants

can be small rectangles of metal sheet, just large enough to make a clear clink

as a marble hits one and rolls past. Christmas tree or similar decorations can add a touch of colour. They should make a clear sound without possibility of damage and without resisting the motion of the marble too much.

Horizontal rods held in clamps can be used to hang the pennants. Alternatively, you can fix wires from which to hang the pennants to curtain track fitted to the plank. Or you could make small wire goalposts

which could sit over the grooved track.

It must be possible to move the pennant holders to different positions on the track. And whichever system you use, the pennants should be hung before the lesson.

Procedure

- Set up the grooved plank inclined so that marbles can roll down the groove.
- Hang small tin pennants above the groove so that the marbles hit them and make' clinks'.
- Place the pennants at regular intervals: 25 cm, 50 cm, 75 cm, 100 cm from the beginning of the plank. Roll a marble down the track, and listen for the time intervals between each
clink

. - Try placing the pennants so that the
clinks

happen at equal time intervals.

Teaching Notes

- Galileo showed that the total distance travelled increases with the square of the time. The consequence here is that, for the desired regular clink of the pennants, they will be at distances from the start which have the ratio 1:4:9:16 and so on.
- As an extension activity, students could make graphs. They should first use values of time as 1, 2, 3, 4 units, plotted on the
*x*-axis. They should plot the values of distance on the*y*-axis. They then make a second graph plotting the square of time on the*x*-axis, to obtain a straight line (unless friction plays a significant complicating role). - The same ratio, 1:4:9:16, occurs for all accelerations and hence, provided that frictional effects are not too great, for all slopes of the runway. Students could try this. Where graphs of distance against the square of the time are plotted, lines obtained for different runway slopes will have different gradients.
- You can do a similar activity for a body in free fall, whose acceleration is
the acceleration due to gravity

. Tie a length of string to a mass. Tie knots in the string. The first one should be 10 centimetres from where the string is tied to the mass. The next one should be 40 centimetres away, the third one should be 90 centimetres away, and the fourth one 160 centimetres away. Let the string fall through your fingers from 160 cm above the floor with the mass falling first. - Feeling the knots is not as easy to discern as hearing masses hit the floor. Attach masses to the end of a string and at 10, 40, 90, 160, 250 cms from the original mass.
- Hold the mass at the 250 cm position and lower the string out of a window or down a stairwell until the end mass reaches the floor. Leave go of the string, and the masses will hit the floor at equal time intervals.

*This experiment was safety-tested in January 2005*

The following practical investigates free fall using light gates.

**Demonstration**

The acceleration of an object allowed to fall under the force of gravity is found by dropping a card vertically through a light gate. The emphasis of this datalogging experiment is on investigating the relationship between the velocity of the card and the distance it has fallen from rest.

Apparatus and Materials

- Light gate, interface and computer
- Weighted card
- Clamp and stand
- Metre or half metre rule

Health & Safety and Technical Notes

Read our standard health & safety guidance

Clamp the light gate about 20 cm above the bench. Clamp a ruler so that the vertical distance may be measured from above the level of the light gate.

Cut black card to the precise length of 10.0 cm. Draw a pencil line across the width of the card at exactly half its length.

Measurements of the height fallen by the card should be made to this line rather than the lower or upper edge of the card. Adding two small blobs of Blutack, at the lower corners, will improve the stability of the card as it falls.

Configure the data-logging software to measure the transit time and calculate the velocity as the card passes through the light gate. A series of results is accumulated in a table. This should also include a column for the manual entry of distance measurements taken from the ruler.

Procedure

**Data collection**

- Hold the card above the light gate and next to the ruler so that its height above the gate may be measured carefully. Release the card so that it cuts through the light beam; a velocity measurement should appear in the table on the screen. Repeat this measurement from the same height several times; enter the height value in the height column of the table in the computer program.
- Repeat this procedure for a new starting height 2 cm above the first.
- Collect a series of measurements, each time increasing the height by 2 cm.

**Analysis**

- Depending upon the software, the results may be displayed on a bar chart as the experiment proceeds. Note the relative increase in values of velocity as greater heights are chosen.
- The relationship between velocity and height fallen is more precisely investigated by plotting a XY graph of these two quantities. (Y axis: velocity; X axis: height fallen.) Use a curve matching tool to identify the algebraic form of the relationship. This is usually of the form 'velocity is proportional to the square root of height'.
- Use the program to calculate a new column of data representing the square of the velocity. Plot this against height on a new graph. A straight line is the usual result, showing that the velocity squared is proportional to the height fallen.

Teaching Notes

- One of the chief values of real-time data-logging, exemplified here, is the interaction between the collection and simultaneous display of results. These can be used to prompt students' thinking. You can orchestrate discussion as the experiment proceeds. If you do this as a class experiment, students should be prompted to ask questions about the results and the process involved in collecting them.
- Small variations between individual results should be observed and possible sources of error discussed. For example, how
clean

was the release of the card, how precise was the height measurement, was the fall wobble-free? and so on. The visual display of results on a bar chart helps to show the significance of these variations, compared with overall trends in the relationship between velocity and height. The chart also makes it very easy to spot anomalous results due to the card wobbling or snagging on the ruler as it falls. The software usually facilitates the deletion or hiding of such values. - Software provides several alternative approaches to analysis. Two examples are described here; a straightforward plot of the collected data, velocity against height, and a second plot involving calculated data.
- The first graph yields a curve which may be conveniently evaluated using a software curve-fitting tool. (It is best to give students tools which allow them to experiment with simple general forms such as quadratics and power laws rather than polynomial fits whose physical significance is hard to interpret.)
- The second graph yields a straight line which lends itself easily to conventional analysis using concepts such as gradient and intercept.
- Note: The discussion here has assumed that the software is capable of calculating and showing
velocity

as the primary measurement. Physically the system actually measures transit times for the card passing through the light gate. It is necessary for the user to enter into the program the length of the card (the distance travelled during the transit time) at the beginning of the session. Calculations may then be performed as a matter of routine during the experiment. Eliminating this calculation step from the experimental procedure allows thinking and discussion to focus on the relationship between velocity and height fallen.

*This experiment was safety-tested in May 2006*

The following practical investigates free fall using a timer.

**Class practical**

This experiment gives a direct measurement of the acceleration due to gravity.

Apparatus and Materials

- Release mechanism (may be electromagnetic)
- Trip switch (hinged flap)
- Power supply, low voltage, DC
- Switch, SPDT
- Ball bearing ball, steel
- Retort stand and boss
- Electronic timer
- Leads, 4 mm

Health & Safety and Technical Notes

It may be useful to stand the trip switch in an up-turned box lid to catch the ball after its fall.

Read our standard health & safety guidance

Science equipment suppliers offer slightly different versions of the mechanisms for the release that starts the timer and the trip switch that stops the timer.

i) In one version, there is no electromagnet in the release mechanism. Instead, the ball is held in position manually, so that it completes a circuit between two of three pegs. When you release the ball, the break

in circuit starts the timer.

If you are using this arrangement, you may prefer to release the ball by holding it from a thread passing through the top of the support. This ensures that you do not obstruct the motion of the ball with your hand: the ball will be in free fall immediately the circuit is broken.

At the bottom of its fall, depending on its design, the mechanism may either make

or break

a second circuit, stopping the timer.

ii) Another version employs an electromagnet.

Procedure

- Set up the apparatus as shown in the diagram. You may need to adjust the distance of fall and the point at which the ball strikes the flap.
- Arrange the timer so that it
starts

when the electromagnet is switched off andstops

when the hinged flap opens. - Check that the flap does open when the ball strikes it. You may need to make the distance of fall larger, or move the flap so that the ball strikes it further from the hinge.
- Measure the distance
*h*from the bottom of the ball to the hinged flap. Be careful to avoid parallax error in this measurement. - Measure the fall time three times and find the average.
- Repeat step 5 for a range of heights between 0.5 m and 2.0 m.
- Plot a graph of 2
*h*against*t*^{ 2}. - Use the graph to find
*g*.

Teaching Notes

- The value of
*g*is calculated from*s*=12*at*^{ 2}. In this case*h*=12*gt*^{ 2}. - With students at intermediate level, it will be sufficient to obtain an average value for
*t*at just one height,*h*. Omit steps 7 and 8. - Steps 7 and 8: With more advanced students, repeat the experiment at different heights and find the gradient of the graph. They will see that the graph does not pass through the origin. An intercept on the
*t*^{ 2}axis indicates that there is an apparent time of fall even when the ball falls no distance at all, a systematic error. This is the time that it takes the electromagnet to release the ball once the current through it is switched off, in other words, the time for the magnetism to fall sufficiently to release the ball. **How Science Works Extension:**This experiment provides an opportunity to discuss experimental design and how it can be used to reduce or eliminate errors. The experiment contains two sources of systematic error: the time-delay in releasing the ball (as discussed above in note 2), and a similar delay in switching the timer off, because the hinged flap may not switch the timer at exactly the instant when the ball strikes it. Both of these will give rise to measured times which are longer than the time during which the ball is in free-fall.- You could discuss with the class how to reduce or eliminate these errors. One approach is to improve the basic design of the experiment so that the time-delays are less or zero. You could compare this method of measuring
*g*with others and discuss their relative merits. - Another approach is to consider how this experiment can be analyzed to reduce the systematic errors. The ‘true time of fall’ is less than the measured value
*t*by a fixed amount*t*_{error}(equal to the sum of the two time delays discussed above). How can we discover or otherwise eliminate this error? Here are two ways: - Think about the situation where
*h*=0. The ball will take zero time to fall through this height, but the timer will still show a time equal to*t*_{error}. Plot a graph of*t*against*h*. Extrapolate back to*h*=0. The graph will reach the axis at*t*=*t*_{error}. Now this value can be used to correct the measured values of*t*. (Students will need extra columns in their results tables to allow for this.) Plot a graph of 2*h*against*t*^{ 2}as before. - Alternatively, think about the situation where
*h*is infinite, or at least very large). The value of*t*will be very large, so the error in it will be small or negligible. So large values of*h*will give more accurate values of*g*. In the results table, add a column for 2*h**t*^{ 2}; the values should approach an accurate value for*g*. Plot a graph of 2*h*against*t*^{ 2}as before; it will be curved, but at the high end it will approach a straight line through the origin whose gradient will be a good value of*g*.

*This experiment was safety-checked in May 2005*

This resource is specifically on resistivity. Section 112-3 outlines a suitable practical. Standard simple and cheap apparatus is all that is required.

Electricity and Magnetism

Lesson for 16-19

- Activity time 110 minutes
- Level Advanced

In this episode, students learn how and why the resistance of a wire depends on the wire’s dimensions. They learn the definition of resistivity and use it in calculations.

Lesson Summary

- Discussion: Variation of resistance with length and area (5 minutes)
- Student experiment: Variation of resistance with length and area (30 minutes)
- Discussion: Variation of resistance with length and area (10 minutes)
- Student experiment: Measurement of resistivity (30 minutes)
- Student questions: Using these ideas (30 minutes)

The analogy to water flow will be useful here – ask them how they think the flow rate will be affected if you increase the cross-sectional area or length of the pipe along which the water has to flow. This should lead to two predictions about the resistance of a wire:

- resistance increases with length
- resistance decreases with diameter or cross-sectional area

It will be worth reminding them that doubling the diameter quadruples the cross-sectional area; many students get confused about the distinction and expect a wire of double diameter to have half the resistance.

You could ask them to do one or both of the following experiments. Both reinforce the idea that resistance depends on material dimensions:

Episode 112-1: How the dimensions of a conductor affect its resistance (Word, 44 KB)

Episode 112-2: Introduction to resistivity using conducting paper (Word, 49 KB)

Follow up with some theory suggesting:

Resistance is proportional to length *l*

Resistance is inversely proportional to cross-sectional area *A*

resistance = constant × lengthcross-section area

The constant is a property of the material used – its resistivity ρ.

*R* = ρ × *l**A*

The units of resistivity can be derived from the equation: Ω m.

Emphasise that this is ohm metre

, not ohm per metre

.

Discuss the great range of resistivities amongst materials. Values for metals are very small. The resistivity of a material is numerically equal to the resistance between opposite faces of a one-metre-cube of the material; although this is not a good definition of resistivity, imagining such a block of metal does indicate why its value should be so low (~ 10^{-9} Ω m).

Complete this section by asking your students to measure the resistivity of several metal wires.

This experiment provides an opportunity for a detailed discussion of the treatment of experimental errors.

Episode 112-3: Measuring electrical resistivity (Word, 30 KB)

Problems involving resistivity.

Students often get confused between cross-section area and diameter.

Make sure they are able to convert mm^{2} to m^{2} for resistivity calculations.

Episode 112-4: Electrical properties (Word, 28 KB)

The following practical is of relevance. Standard simple and cheap apparatus is all that is required.

Electricity and Magnetism

Practical Activity for 14-16

**Class praticals**

Determining resistance from measurements of potential difference (p.d.) and current.

Apparatus and Materials

- Ammeter, 0 to 1 A, DC
- Voltmeter, (0-15 V), DC
- Power supply, low voltage, DC
- Lamp (12 V, 6 W) in holder
- Resistor (approx 15 ohms, 10 watt)
- Various other components

Health & Safety and Technical Notes

Read our standard health & safety guidance

Remind the class that the lamp will get hot, so it should only be moved by handling the lamp holder.

Procedure

- Set up the circuit shown. Turn the power supply up until the p.d. across the lamp is 12 V (the normal operating voltage).
- Take readings of the p.d. and current.
- Calculate the resistance of the lamp at its running temperature.
- Now, for several different values of p.d., measure the current through the lamp. Plot a graph of your results; this graph is known as the
**voltage-current characteristic**of the lamp. - Replace the lamp in the circuit with the resistor. Repeat the experiment and calculate its resistance. Take sufficient readings to allow you to plot the
**voltage-current characteristic.**

Teaching Notes

- This series of experiments should give students practice in taking a pair of current and potential difference readings for various components so that the resistance of the component can be calculated from V/I = R.
- It can also be extended so that students plot the current/potential difference characteristics for components such as a carbon resistor, a diode, a light-emitting diode (LED), a thermistor, motor armature, electric fire element (12 V supply only!) and so on. Students will need to be able to select appropriate meters, as the current through some of these devices may be very small. Each member of the class could tackle one component and present their results to the class, or produce a wall display.
- Some things which appear not to obey Ohm's law might, in fact, do so; for example, the tungsten filament of a lamp. Tungsten's resistance increases as the lamp gets hotter, but if it could be maintained at a constant temperature then its resistance would be constant.
- For suggested graphs, see below

*This experiment was safety-tested in January 2007*

Download the support sheet / student worksheet for this practical.

This resource focuses on e.m.f. and internal resistance, outlining relevant practicals.

Electricity and Magnetism

Lesson for 16-19

- Activity time 130 minutes
- Level Advanced

The starting point for the theory can be either Kirchhoff’s second law or conservation of energy in the circuit (the same thing really) but a general discussion based on the circuit diagram below should use a variety of approaches.

Lesson Summary

- Discussion: Deriving an equation (15 minutes)
- Discussion: Practical effects of internal resistance (10 minutes)
- Student questions: Internal resistance of a power supply (20 minutes)
- Student experiment: Measuring internal resistance and EMF (45 minutes)
- Student questions: Practice questions (30 minutes)
- Discussion: More about the practical importance of internal resistance (10 minutes)

There are three ways to arrive at the equation relating EMF, terminal PD, current and internal resistance. It is worth discussing all three, to show their equivalence. The order you take will depend on the approach used previously with the class:

Kirchhoff’s 2^{nd} Law: As charge goes around the circuit the sum of EMFs must equal the sum of voltage drops leading to:

*E* = *I**R* + *I**r*

The terminal voltage is equal to *I**R* so this can be rearranged to give:

*V* = *E* − *I**r*

and interpreted as
terminal voltage = EMF − lost volts

Energy is conserved. Imagine a unit of charge, *Q*, moving around the circuit:

*Q**E* = *Q**I**R* + *Q**I**r*

This leads to the same equations as above.

Use Ohm’s law with *E* driving

current through the combined resistance (*R* + *r*):

*I* = *E**R* + *r*

Multiplying throughout by (*R* + *r*) leads to the same equations and conclusions as in (1).

At this point it might be worth pausing to illustrate the effects. Take a car as an example. The headlamps are connected in parallel across a twelve-volt battery. The starter motor is also in parallel controlled by the ignition switch. Since the starter motor has a low resistance it demands a very high current (say 60 A). The battery itself has a low internal resistance (say 0.01 Ω ). The headlamps themselves draw a much lower current. Ask them what happens when the engine is started (switch to starter motor closed for a short time). Look for an answer in general terms:

- sudden demand for more current
- large lost volts (around 0.05 Ω × 60 A = 3 V)
- terminal voltage drops to 12 V – 3 V = 9 V
- headlamps dim

When the engine fires, the starter motor switch is opened and the current drops. The terminal voltage rises and the headlamps return to normal. It’s better to turn the headlamps off when starting the car.

As an aside, a lot of students seem to think the engine is powered by the battery! Point out that its main purpose while the engine is running is to provide the sparks for ignition and that while the car is driving the alternator continually recharges the battery, the energy for both headlamps and driving comes ultimately from the fuel that is burnt (since the car has to work a little bit harder to turn the alternator).

Some simple questions about the internal resistance of a power supply.

Episode 121-1: Internal resistance of power supplies (Word, 30 KB)

There are two experiments here, in which students determine the EMF *E* and internal resistance *r* of cells – one involving a potato cell (leading to a high internal resistance) and one involving a normal C cell (much lower internal resistance). You could get them to do both or ask some students to do one and some the other. Beware that, if you use an alkaline, high power C cell, it will run down quickly when there is a low load resistance, so you are advised to use cheap, low power cells which polarise quickly, they will depolarise over night. An alternative is to construct an artificial cell with a larger internal resistance by adding a higher series resistance (e.g. 100 Ω ) to a standard cell.

Episode 121-2: Internal resistance of a source of EMF (Word, 48 KB)

Episode 121-3: Internal resistance of a C cell (Word, 28 KB)

To determine *E* and *r* from the experimental results, there are various approaches. The simplest is to measure terminal voltage (*V*) and current (*I*) and to plot *V* against *I*. This gives an intercept at *V* = *E* on the y-axis and has a gradient of − *r*.

Questions on EMF and internal resistance.

Episode 121-4: Questions on EMF and internal resistance (Word, 29 KB)

Sometimes it is desirable to have a high internal resistance. Ask the class what happens if a 5 V cell is shorted – i.e. its terminals are connected together by a wire of zero resistance? Some might think *I* = *V**R* with *R* = 0 should mean that an infinite current would flow (limited by other physical factors!)

Remind them of the internal resistance *r*. This limits the cell to a maximum (short-circuit) current of:

*I* = *E**r*

We can use this to prevent EHT supplies giving the user an unpleasant shock. Take an EHT power supply off the shelf and show the connections for the series internal

resistance. It is usually 5 M Ω .

These supplies are designed to provide a high voltage to a high resistance load (e.g. cathode ray tube) but if the terminals or wires connected to them were accidentally touched this could provide a nasty shock (lower resistance in the load and higher current). One way to deal with this is to connect a large resistance in series with the output (positive) terminal. If the terminals are shorted (e.g. by contact through a person) the current drawn is limited to *I* = *E**r*. A typical EHT supply (up to 5000 V) is protected by a 5 M Ω resistor so the maximum current if shorted is just 1 mA. That shouldn’t kill you! Be aware however that HT supplies (0-300 V) have a much lower internal resistance, and could kill you, so special shrouded leads should be used.

EHT supplies often have a further safety resistor

(e.g. 10 M Ω ) to reduce the maximum current still further. This resistor can be by-passed when necessary. No school EHT supply is allowed to provide more than 5 mA.

A fun practical involving a potato, zinc and copper electrodes.

**Class practical**

An introduction to the concept of internal resistance, using a more interesting example than a battery.

Apparatus and Materials

*For each student group*

- Digital multimeters, 2
- Leads, 4 mm, 5
- Cells, 1.5 V type C, 4
- Resistors of a range of values from 10 ohms to 100 ohms
- Crocodile clips, 10 pairs

Health & Safety and Technical Notes

Biology teachers should note the potato cell

in this experiment refers to a whole potato not an individual potato cell.

Read our standard health & safety guidance

Procedure

- Make your potato cell. Insert the copper and zinc electrodes at either end of the potato. Attach a 4 mm lead to each electrode using a crocodile clip.
- Set up the circuit as shown. Set the resistance substitution box to 4.7 kΩ.. This is the load resistance. Record the current and potential difference values in a suitable table.
- Change the load resistance and record the values of current and potential difference. Repeat this process to gather data for a range of load resistances. You will have to change the range of your ammeter. Take care not to confuse amps with milliamps or microamps!
- Plot a graph of
*V*against*I*. Describe the trend.

Teaching Notes

- This experiment can be used for a number of purposes – as an introduction to the concept of internal resistance, an interesting example of internal resistance or an example of a simple cell. If standard resistors are available it is possible to vary the load resistance in smaller steps.
- The VI graph line will surprise students who have not been introduced to the concept of internal resistance. Those students familiar with the equation
*V*=*ε*–*I**r*should be able to interpret the data in terms of the internal resistance of the potato cell. However, many students find internal resistance a difficult concept and may find the Internal resistance of a shoe box cell experiment a useful support activity... - If readings are entered into a spreadsheet it is easy for interested students to plot further graphs, including load resistance/power dissipated in resistor. Such a graph will show a peak power output when the load resistance is equal to the internal resistance of the cell.

Internal resistance of a shoe box cell

*This experiment comes from AS/A2 Advancing Physics. It has been re-written for this website by Lawrence Herklots, King Edward VI School, Southampton.*
*This experiment was safety-tested in June 2007*

A practical where students can make up a cell with internal resistance in a shoe-box.

**Class practical**

Finding an internal resistance of a supply from the power dissipated by a load resistance.

Apparatus and Materials

*For each student group*

- Digital multimeters, 2
- Leads, 4 mm, 5
- Cells, 1.5 V type C, 4
- Resistors of a range of values from 10 ohms to 100 ohms
- Crocodile clips, 10 pairs

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

- The shoe box cell contains a 6 V supply with an unknown resistor in series. The unknown resistor acts as the internal resistance in the shoe box cell. Your challenge is to find the value of the internal resistance without opening the box!
- Choose a value for the load resistance
*R*and set up the circuit as shown. Record the current and potential difference values.*E*is the e.m.f. of the equivalent perfect cell of internal resistance*r*. - Change the load resistance and once again record the values of current and potential difference values.
- Repeat this process for about ten values of load resistance.

**Analysis**

- Calculate the power dissipated by the load resistance by using the equation
*P*=*I**V*. - Plot a graph of power dissipated against load resistance.
- The power dissipated by the load resistance is a maximum value when it equals the ‘internal resistance’ of the shoe box cell. Use this fact to estimate the resistance in the box.
- The cells in the box have a total internal resistance of about 4 ohms. How does this fact change your answer?

Teaching Notes

- The resistance in the box needs to be about 50 ohms if a clear peak is to be found when the load resistance varies between about 10 ohms and 100 ohms. This will give a minimum current of about 40 mA. using 4 x 1.5 V cells.
- Resistors can be labelled and clipped together to give a good range of total resistance values.
- Students familiar with
*V*=*ε*–*I**r*should be able to interpret the data in terms of the internal resistance of the shoe box cell. This experiment can also be interpreted as an example of a potential divider. The internal and external resistances are in series, so the e.m.f. is divided between them. - Some students will be able to complete the analysis more quickly using a spreadsheet.

*This experiment was safety-tested in April 2006*

This resource is specifically about potential dividers.

Electricity and Magnetism

Lesson for 16-19

- Activity time 105 minutes
- Level Advanced

This episode introduces the use of a potential divider as a source of variable pd. Students will also learn to use potential dividers to detect temperature or light levels.

The potential divider circuit is a particularly useful arrangement but many students find this difficult to grasp at first. It is worth spending time to make sure they understand what is going on. It may also be worth emphasizing, especially with more able groups, that the output voltage is affected by the load resistance (many, on first meeting the device, do not realize this).

Lesson Summary

- Discussion and demonstration: The potential divider formula (15 minutes)
- Student experiment: Using potential dividers (40 minutes)
- Student questions: Using the formula (20 minutes)
- Discussion: The effect of the load on output (10 minutes)
- Student questions: The effect of the load on output (30 minutes)

The potential divider is one of the most useful circuits your students will meet so it is particularly important that they understand how it works and how to calculate the voltage across each of its resistors. For an unloaded potential divider the current through each resistor is the same so the voltage is proportional to the resistance. This means that the pd across the pair of resistors is divided in the same ratio as the resistors themselves:

i.e.

*V*_{1}*V*_{2} = *I**R*_{1}*I**R*_{2}

*or*

*V*_{1}*V*_{2} = *R*_{1}*R*_{2}

It is worth emphasizing the practical implication of this – if *R*_{1} >> *R*_{2} then *V*_{1} is more or less the supply voltage and if *R*_{1} << *R*_{2} then *V*_{1} is close to 0 V. You could encourage them to see *V*_{source} as an input to the potential divider and *V*_{1} as an output. The circuit itself provides a way to tap off a voltage between 0 V and *V*_{source} .

This can, of course be done continuously using a rheostat or potentiometer and it is well worth demonstrating a variety of these including the rotary potentiometers used as volume controls in hi-fi systems.

The potential divider equation can be derived by rearranging the ratios above to give:

*V*_{1} = *R*_{1}*R*_{2} × *V*_{2}

Start with Parts 1-3 of these experiments.

Once they have experimented with several versions of the potential divider they can apply the ideas to a simple sensor circuit (Part 4). Set them the task of building and testing either an electronic temperature sensor or an electronic light meter (or both).

You may feel that the latter experiment as presented is sufficient to introduce the idea of a temperature or light intensity sensor, but this could be extended. The thermistor can be used as part of an electronic thermometer if it is connected into a suitable potential divider circuit and calibrated. One way to do this is to use a water bath to warm the thermistor (you can use ice with salt to get well below 0 ° C and hot water to get above 80 ° C giving a good range). However, to do this you need water-proofed thermistors (e.g. wrapped in polythene or embedded in an epoxy). If the water reaches the electrical connections then the readings will be unreliable. Beware of safety in the laboratory with both hot water and mercury thermometers (they need these for their calibration). Hot water from a kettle is safer than using a beaker on a tripod heated by a Bunsen burner. A similar approach can be used to make a light intensity meter (this can be calibrated using a lux meter).

Episode 118-1: Potential dividers (Word, 42 KB)

Practice in potential divider calculations.

Episode 118-2: Tapping off a potential difference (Word, 44 KB)

With strong groups you might discuss the effect of loading a potential divider on its output voltage. The ideas to get across are:

Connecting a load across *R*_{1} reduces the output voltage.

This is because the effective resistance in the lower arm of the potential divider is now a parallel combination of *R*_{1} and *R*_{load} (less than *R*_{1} ) so a smaller fraction of the voltage is tapped off

.

If *R*_{load} >> *R*_{1} then there is no significant effect on the output voltage.

It is worth going back to their experiences as younger pupils tackling simple circuits, and considering what was happening when a lit bulb went out when shorted out

by a piece of wire. Pupils will often regress to a current based explanation, particularly under the pressure of examination conditions. We need to encourage them now to think in terms of potential difference, and resistance. In the shorting out

case it was not that much that all the current wanted to take the easier parallel route

, but that the low resistance of the wire in parallel reduced the combination’s total resistance, compared to the rest of the circuit.

Episode 118-3: Loading a potential divider (Word, 36 KB)

Episode 118-4: Brightness of bulbs (Word, 29 KB)

This resource introduces drag (terminal velocity). Section 209-1 outlines a practical on 'falling cupcakes' and section 209-2 outlines a practical on ‘ball-bearings falling through a viscous medium'. Both are relevant.

Forces and Motion

Lesson for 16-19

- Activity time 50 minutes
- Level Advanced

This episode starts by considering the forces acting on a falling body in air, and moves on to related experimental work.

Lesson Summary

- Discussion and demonstration: Falling bodies (10 minutes)
- Student experiment: Falling cupcakes (20 minutes)
- Student investigation/demonstration: Balls falling through fluids (20 minutes)

As an introduction to the lesson the following questions can be posed to the class. During the discussion, you will be able to introduce and define the terms upthrust, drag, air resistance, lift and terminal velocity.

Why does a piece of paper fall more slowly under gravity than a piece of chalk if the acceleration due to gravity is the same for all objects? (Demonstrate this.)

If an object is falling through the air with constant velocity, what can you say about the net force on the object?

How do the forces on an object vary as the object accelerates from rest?

The discussion should lead to the concept of an object reaching terminal velocity when the drag force has the same magnitude as the accelerating force. This can be shown clearly in a diagram such as this:

A useful exercise is to ask a student to sketch the velocity-time graph of a skydiver who accelerates to terminal velocity and then opens her parachute.She will decelerate to a new, lower terminal velocity. You can use a toy parachutist to illustrate this.

The concept of upthrust

is most easily introduced using a toy helium balloon (see Balloons – Party in Yellow pages). It will accelerate as its upthrust is greater than its weight, but will reach an upward terminal velocity when upthrust = weight + drag. If you have a high stairwell you can try taking measurements with students standing at different heights and timing the passage of the balloon – fun if you have the time.

An aeroplane is flying at constant velocity (and therefore at constant height). The net force on the aeroplane must be zero. How are the forces balanced? The forward thrust provided by the engines balances the backward drag of air resistance; the downward weight of the plane is balanced by the upward lift on the wings.

This lift is often explained by the Bernoulli effect – faster moving air over the top of the wing produces a pressure difference and an upward force. There are many easy demonstrations of this effect – the simplest being to blow between two sheets of paper causing them to stick together

. However, you need to be a little careful with this explanation; remember that planes can fly upside down. The angle of the aerofoil is also important. You may prefer to avoid mentioning Bernoulli and simply say that the wing of an aircraft is shaped and angled to push air downwards; since there is a downward force of the wing on the air, there will be a corresponding upward force of the air on the wing (Newton’s third law); this is the force we call lift

.

Students will now be familiar with drag, thrust, lift, upthrust and weight.

This works well if you make sure that the students are thinking about the questions posed. The activity itself is trivial but a lot of good physics can be brought out. It follows on very well from the introductory discussion and some students will find it useful to sketch force diagrams to help their answers. A written discussion of what the investigation shows could usefully be left for homework.

A more quantitative investigation of drag and terminal velocity is to observe the motion of bodies falling through a viscous medium. This is a rich area for student investigations but can be carried out as a demonstration by the teacher with students making measurements.

Start by dropping a ball bearing through the fluid. Can your students describe its motion? (At first it accelerates; acceleration gradually decreases to zero, when the ball has reached terminal velocity.)

The physics behind the demonstration is Stokes’ Law and you can use this as an opportunity to discuss this or just remind students of a little kinematics as they analyse the motion of the falling ball bearing.

Episode 209-1: Falling cupcakes (Word, 30 KB)

Episode 209-2: Ball bearings falling through a viscous medium (Word, 35 KB)

Two good simple practicals: the standard stretching wire; and, bending a beam. You may also want to read this resource - Episode 227: Hooke's law - which has helpful guidance on teaching Hooke's Law leading on to Young modulus.

Properties of Matter

Lesson for 16-19

- Activity time 170 minutes
- Level Advanced

The Young modulus is often regarded as the quintessential material property, and students can learn to measure it. It is a measure of the stiffness of a material; however, in practice, other properties of materials, scientists and engineers are often interested in, such as yield stress, have more influence on the selection of materials for a particular purpose.

Lesson Summary

- Discussion: Defining the Young modulus (20 minutes)
- Student activity: Studying data (20 minutes)
- Student experiment: Measuring the Young modulus (60 minutes)
- Student experiment: An alternative approach using a cantilever (30 minutes)
- Discussion: Comparing experimental approaches (10 minutes)
- Student questions: Involving the Young modulus (30 minutes)

A typical value of *k* might be 60 N m^{-1}.

What does this mean? (60 N will stretch the sample 1 m.) What would happen in practice if you did stretch a sample by 1 m? (It will probably snap!)

A measure of stiffness that is *independent* of the particular sample of a substance is the Young modulus *E*.

Recall other examples you have already met of sample independent

properties that only depend upon the substance itself:

- density = massvolume
- electrical resistivity = resistance × arealength
- specific heating capacity = energy transferredmass × temperature change
- thermal conductivity = power × lengtharea × temperature difference

We need to correct

*k* for sample shape and size (i.e. length and surface area).

Episode 228-1: The Young modulus (Word, 53 KB)

Note the definitions, symbols and units used:

Quantity | Definition | Symbol | Units |
---|---|---|---|

Stress | tensionarea=FA | σ (sigma) | N m^{-2}={Pa |

Strain | extension per original length= Δ xx | ε (epsilon) | No units (because it’s a ratio of two lengths) |

Young modulus | stressstrain | E | N m^{-2} = Pa |

Strains can be quoted in several ways: as a %, or decimal. E.g. a 5% strain is 0.05.

Episode 228-2: Hooke's law and the Young modulus (Word, 75 KB)

It is helpful if students can learn to find their way around tables of material properties. Give your students a table and ask them to find values of the Young modulus. Note that values are often given in GPa (1 × 10^{9} Pa).

Some interesting values of *E* :

- DNA ~ 10
^{8}Pa - spaghetti (dry) ~ 10
^{9}Pa - cotton thread ~ 10
^{10}Pa - plant cell walls ~ 10
^{11}Pa - carbon fullerene nanotubes ~ 10
^{12}Pa

Episode 228-3: Materials database (Word, 115 KB)

You can make measuring the Young modulus *E* a more interesting lab exercise than one which simply follows a recipe. Ask students to identify the quantities to be measured, how they might be measured, and so on. At the end, you could show the standard version of this experiment (with Vernier scale etc.) and point out how the problems have been minimized.

What needs to be measured? Look at the definition: we need to measure load (easy), cross-sectional area *A* , original length *x*_{0} (so make it reasonably long), and extension Δ *x* .

Problems? Original length – what does this correspond to for a particular experimental set up? Cross-sectional area: introduce the use of micrometer and/or vernier callipers. Is the sample uniform? If sample gets longer, won’t it get thinner? Extension – won’t it be quite small?

Should the sample be arranged vertically or horizontally?

Divide the class up into pairs and brainstorm possible methods of measuring the quantities above, including the pros and cons of their methods.

Some possibilities for measuring Δ *x* :

Attach a pointers to the wire

- Pro: measures Δ
*x*directly - Con: may affect the sample; only moves a small distance

Attach a pointer to the load

- Pro: measures Δ
*x*directly, does not effect the sample - Con: only moves a small distance

Attach a pulley wheel

- Pro:
amplifies

the Δ*x* - Con: need to convert angular measure to linear measure, introduces friction

Attach a pointer to the pulley wheel

- Pro:
amplifies

the Δ*x*even more - Con: need to convert angular measure to linear measure, introduces friction

Exploit an optical level

- Pro: a
frictionless

pointer,amplifies

the Δ*x*even more - Con: need to convert angular measure to linear measure, more tricky to setup?

Illuminate the pointer etc to produce a magnified shadow of the movement

- Pro: easy to see movement
- Con: need to calculate magnification, can be knocked out of place

use a lever system to amplify or diminish the load and provide a pointer

- Pro: useful for more delicate or stiff samples; can use smaller loads
- Con: fixing the sample so it doesn’t
slip

, need to convert angular measure to linear measure

Different groups could try the different ideas they come up with. Depending upon the time available, it may be worth having some of the ideas already set up.

Give different groups different materials, cut to different sizes, for example: metal wires (copper, manganin, constantan etc), nylon (fishing line), human hair (attach in a loop using Sellotape), rubber. Note that in the set up above, the sample is at an angle to the ruler – a source of systematic error.

Students should wear eye protection, provide safe landing for the load should sample break, e.g. a box containing old cloth. For the horizontal set up: bridges

over the sample to trap the flying ends, should the sample snap.

Good experimental practice: measure extension when adding to the load and when unloading, to check for any plastic behaviour.

Episode 228-4: Measuring the stiffness of a material (Word, 59 KB)

Episode 228-5: Stress–strain graph for mild steel (Word, 68 KB)

Information about the use of precision instruments (micrometer screw gauge, Vernier callipers and Vernier microscope).

Episode 228-6: Measure for measure (Word, 82 KB)

An alternative approach to measuring the Young modulus is to bend a cantilever. (Potential engineering students will benefit greatly from this.)

For samples too stiff to extend easily (e.g. wooden or plastic rulers, spaghetti, glass fibres) the deflection *y* of a cantilever is often quite easy to measure and is directly related to its Young modulus *E* .

If the weight of the cantilever itself is *m**g*, and the added load is *M**g* and *L* is the length of the cantilever (the distance from where the cantilever is supported *to* where the load is applied):

- y = 4 (Mg + 5mg/16)
*L*^{ 3} - E b
*d*^{ 3}

(for square cross-section *d* = *b* )

- y = 4 (Mg + 5mg/16)
*L*^{ 3} - 3 π
*r*^{ 4}E

Finish with a short plenary session to compare the pros and cons of the different experimental approaches.

Questions involving stress, strain and the Young modulus, including data-handling.

Episode 228-7: Calculations on stress, strain and the Young modulus (Word, 59 KB)

Episode 228-8: Stress, strain and the Young modulus (Word, 26 KB)

This resource focuses on frequency, wavelength and speed, including a practical approach to sound measurement using CRO, speaker(s) and a microphone: section 311-3.

Light, Sound and Waves

Lesson for 16-19

- Activity time 150 - 240 minutes
- Level Advanced

This episode considers how these three quantities are linked by the wave equation *v* = *f* × *λ* , measuring *f* using an oscilloscope, and measuring the velocity of sound in free air.

Lesson Summary

- Discussion and worked examples: Deducing and using the wave equation (30 minutes)
- Student questions: Practice with
*v*=*f*×*λ*(30 minutes) - Demonstration: An exploration of sound waves (20 minutes)
- Student experiment: Measuring frequency using a CRO (30 minutes)
- Demonstration: Measuring the speed of sound using double beam oscilloscope (10 minutes)
- Student experiments: Exploring waveforms (30–120 minutes)

Use a simple approach to deduce the wave equation

Justification/deduction of the wave equation *v* = *f* × *λ* . For example: the coaches of a train are going past; you count how many coaches go by in a second and you know the length of one – so you multiply the two together to get the train’s speed. Apply this to waves: count the number of waves passing each second ( the frequency), and multiply by the length of each ( the wavelength) to find the speed.

If your syllabus says *deduce* then you will have to present the algebra of

speed = distancetime

*v* = *l**T*

*v* = *f* × *λ*

Work through three examples:

A simple example, perhaps for sound in air, with values in Hz and m.

An example involving electromagnetic waves with frequency units such as MHz or GHz, to show how to deal with powers of ten; emphasise that *c* = 3 × 10^{8} m s^{-1} for all em waves in free space.

An example in which the equation must be rearranged, to find *f* or *λ* .

You may wish to make a selection from these.

Episode 311-1: Speed, wavelength and frequency (Word, 47 KB)

Episode 311-2: Using the wave equation (Word, 42 KB)

An exploration of sound waves.

Turn the volume down each time you change the frequency range because of the differing sensitivity of the ear. Keep the overall volume low.

Emphasise that the oscilloscope trace represents displacement against time. You will have to hammer home that the peak-to-peak separation is not the wavelength.

It is advisable to perform this activity as a demonstration, as a room full of signal generators and loudspeakers in operation can be very noisy!

If you are going to use a stroboscope to show up the vibrations of a loudspeaker cone, you must check whether there is anyone in the class who may be affected by it. (You might wish to omit the use of the stroboscope.)

Episode 311-3: Exploring waveforms with an oscilloscope (Word, 49 KB)

Students can measure the period, *T*, and hence calculate *f*. Each group will need a signal generator and a single beam oscilloscope.

Remind students to make sure that the oscilloscope is on the calibrated setting.

With a clear trace, note the time base setting and determine *T* (over several cycles if possible, an important technique). Then calculate *f* and compare with the setting on signal generator.

This is a good opportunity to check they are confident with the controls on the CRO and they can also explore the different waveforms from the signal generator.

(If you really have time for fun, issue low-voltage AC power supplies, switch off the timebase and input the 50 Hz signal to the X plates instead (often at the back) and give them the treat of Lissajous figures.)

Connect two microphones to a double-beam oscilloscope. Set up a signal generator and loudspeaker to give sound waves of frequency 1 kHz. (Their wavelength is thus about 0.3 m.)

Place one microphone close to the loudspeaker, and observe its trace. Place the second microphone further from the loudspeaker, in the same straight line. Observe its trace. Move it back and forth, noting the changing phase difference between the two traces as you move through the sound waves.

Measure the wavelength (with a ruler) by finding how far the microphone is moved between adjacent positions where the signals are in phase. Calculate the speed of sound.

Note that, if you don’t have two microphones, you can link the signal generator and loudspeaker to one input. Then find two consecutive positions of the microphone which are in antiphase with the signal. Antiphase is easy to see when the traces are superimposed on the screen.

If you don’t have a double beam oscilloscope, wait until a lesson on standing waves and then use a single beam one.

These experiments make use of the CD-ROM Multimedia Sound

. You could make a booklet from the relevant pages, so that students can investigate unsupervised in their own time.

Episode 311-4: Exploring waveforms with multimedia sound (Word, 37 KB)

Episode 311-5: Multimedia Sound: studying a pre-recorded sample (Word, 131 KB)

Episode 311-6: Multimedia Sound: recording your own sound sample (Word, 54 KB)

Episode 311-7: Superposition (Word, 63 KB)

Episode 311-8: Multimedia Sound: combining sounds (Word, 64 KB)

**Also: Investigation into the variation of the frequency of stationary waves on a string with length, tension and mass per unit length of the string (AQA)**

Specifically sections 324-1 and 324-2.

Light, Sound and Waves

Lesson for 14-16

- Activity time 180 minutes
- Level Advanced

Most musical instruments depend upon standing waves, as does the operation of a laser. In a sense, a diffraction pattern is a standing wave pattern.

Lesson Summary

- Demonstration: Setting up waves on a rope (15 minutes)
- Discussion: How superposition results in standing waves (20 minutes)
- Demonstration: Melde’s experiment (20 minutes)
- Discussion: Stringed instruments (10 minutes)
- Demonstration: Standing sound waves (20 minutes)
- Student experiments: Measuring
*λ*and*c*(30 minutes) - Demonstration: Measuring
*c*using a microwave oven (15 minutes) - Student questions: On Melde’s experiment, and on waves in pipes (30 minutes)
- Demonstration: Standing waves in 2 and 3 dimensions (20 minutes)

Standing waves do not travel from point to point. They are formed from the superposition of two identical waves travelling in opposite directions. This is most easily achieved by reflecting a travelling wave back upon itself.

Reflection from a denser medium gives a phase change of 180 ° ( π radian) for the reflected wave. Show this using a rope or stick wave machine held between two people– pull down and let go to make pulse. See that there is a phase change if reflecting from a fixed end, but no effect if reflecting off a less dense medium (e.g. a free end).

The incoming reflected wave superposes with the outgoing wave. The result (if you vibrate the one end of the rope at the correct frequency) is a standing wave. The mid-point of the rope vibrates up and down with a large amplitude – this is an antinode. Other points vibrate with smaller amplitude.

Double the frequency of vibration and you get two antinodes, with a node in between. The amplitude at a node is zero.

Discuss how two travelling waves superpose to give a standing wave. A node (where there is *NO DisplacEment* is a special point, where a positive displacement from one wave is always cancelled by an equal, negative displacement from the other wave.

Episode 324-1: Standing waves (Word, 51 KB)

A stretched string or rubber cord can be made to vibrate using a vibrator (of the type illustrated) connected to a signal generator. This is known as Melde’s experiment.

Show that different numbers of loops

can be formed; identify the pattern of frequencies (e.g. one loop at 40 Hz; two loops at 80 Hz; three at 120 Hz, etc.)

Point out that each point on the string oscillates with simple harmonic motion – if the frequency is high you just see the envelope. Freeze the wave using a stroboscope. To do this, place an electronic strobe between the students and the string, so that it illuminates the string. Adjust the frequency of the strobe until the string appears stationary. ( *Safety precaution* : warn about the flashing light in case you have any students present who may suffer from photo-induced epilepsy.)

The distance between successive nodes (or antinodes) is *λ* 2. Deduce the wavelength.

Notice that only certain values of wavelength are possible. If the string has length *L* , the fundamental has
*λ* 2 = L
, first harmonic *λ* L, second harmonic
3 *λ* 2 = L
and so on. The allowed wavelengths are *quantised* .

(This experiment can be extended to look at how the pattern changes with length and tension of the string.)

Episode 324-2: Standing waves on a rubber cord (Word, 96 KB)

Relate what you have observed to the way in which stringed instruments work. (Wind instruments are covered later.)

Episode 324-3: Standing waves on a guitar (Word, 29 KB)

Episode 324-4: What factors affect the note produced by a string? (Word, 51 KB)

You can observe the same effect with sound waves. To generate two identical waves travelling in opposite directions, reflect one wave off a hard board.

Episode 324-5: Standing waves in sound (Word, 57 KB)

Standing waves make it easy to determine wavelengths (twice the distance between adjacent nodes), and hence wave speed *c* (since
*c* = *f* × * λ *). These experiments allow students to investigate wavelength and speed for sound waves, microwaves and radio waves.

Have a short plenary session for the different groups to report to the whole class.

Episode 324-6: Kundt's experiment (Word, 39 KB)

Episode 324-7: Standing waves with microwaves (Word, 30 KB)

Episode 324-8: A stationary 1 GHz wave pattern (Word, 32 KB)

Use a microwave oven to measure the speed of light. This makes a memorable demo if you have a microwave oven to hand in the lab. Without using the turntable, place marsh mallows or slices of processed cheese in the oven. Observe that the heating occurs in definite places – the cooking starts first at the anti-nodes, where the standing waves have the maximum amplitude. The purpose of the turntable it to even out this localised heating. Measure the distance between nodes, deduce *λ* , and multiply by *f* to find *c* .

NB It is a widely repeated misconception that microwave heating is a resonance effect, i.e. that the frequency of the microwaves is chosen to be one of the vibration frequencies of the water molecule. It is not. Water molecules resonate at rather higher frequencies than the 2.5 GHz of microwave ovens (22 GHz is one such frequency for free water molecules). The frequency used in ovens is a compromise between too low a value, when no microwaves would be absorbed by the food at all, and a frequency too close to the resonant frequency, when the microwaves would all be absorbed in the outside layer, instead of penetrating a few cm. Microwaves are attenuated exponentially by foodstuffs with a half-depth of about 12 mm.

Some questions based on Melde’s vibrating string experiment.

Episode 324-9: Stationary waves in a string (Word, 106 KB)

Questions on standing waves in air in a pipe; note that you will have to explain that there is a node at the closed end of a pipe, and an antinode at an open end. (The air molecules cannot vibrate at a closed end.)

Episode 324-10: Standing waves in pipes (Word, 46 KB)

Episode 324-11: Standing waves in pipes questions (Word, 37 KB)

You can demonstrate standing waves in two and three dimensions. Mount a wire loop or a metal plate (a Chladni plate) on top of a vibrator.

A wire loop shows a sequence of nodes and antinodes around its length, and is an analogue for electron standing waves in an atom.

Episode 324-12: More complicated standing waves (Word, 42 KB)

Chladni plate: Fine sand sprinkled on the plate gathers at the nodes.

Jelly (make up a large cubical shape) on a vibrating plate can look fantastic.

Rubber sheet stretched over a loud speaker: the low frequency standing waves are clearly visible.

**Also: Investigation into the variation of the frequency of stationary waves on a string with length, tension and mass per unit length of the string (AQA)**

Melde’s experiment for standing waves on a string.

**Demonstration**

Using a vibration generator to investigate standing waves on a string.

Apparatus and Materials

- Weight hanger with slotted weights
- Vibration generator
- Pulley, single, on clamp
- Thread, 3 metres

Health & Safety and Technical Notes

If a fractional horsepower motor is used, it is essential to connect both field coils and armature as shown, before switching on the power. No changes should be made while the motor is running.

Read our standard health & safety guidance

A commercially available vibrator and signal generator can be used for this experiment. Alternatively, the end of the thread can be attached to the vibrating strip in a ticker tape vibrator. A video showing how to use a signal (vibration} generator is available at the National STEM Centre eLibrary:

Another suitable arrangement is to attach a wheel to a small motor: the thread is then attached to an eccentric screw.

Students could observe the thread through hand stroboscopes. Alternatively, the thread can be illuminated intermittently by light.

Procedure

- Set up the vibrator so that one end of the long thread is excited, whilst the other end passes over the pulley to the weight hanger. Adjust the load until several loops are clearly seen. Alternatively, clamp the thread at both ends.
- Put the vibrator near one end, driving the string into resonance at a node.

Teaching Notes

- If a vibrator driven by a signal generator is used, you can gradually increase the frequency, showing how the string goes in and out of resonance with an increasing number of loops. Show the pattern of frequencies as the number of loops increases 1, 2, 3, etc.
- You could use this apparatus to test the relationship between the tension, mass per unit length, frequency, and wavelength. Or you could calculate the speed of the wave by measuring its wavelength and frequency.
**How Science Works Extension:**An electronic stroboscope can be used to illuminate the vibrating string. At resonance, adjust the frequency of the strobe until the string appears stationary. The frequency of the strobe should match the frequency of the signal generator, but this depends on how well they are calibrated. Which is to be believed?- This illustrates the need for a standard of measurement. The mains frequency (50 or 60 Hz, depending on where you are) is very reliable. Ask students how this could be used. (Try driving the vibrator with a low voltage mains supply; adjust the length or tension of the string until it resonates. Take a reading from the stroboscope. Set the signal generator to 50 Hz and see if the string still resonates.)
- Students could research the mains frequency. Why might it change? What variation in frequency is permitted?

*This experiment was safety-checked in February 2006*

Episode 321-5 for measuring wavelength of a laser. Difficult with ’two-slit’ because faint or with grating easier to see pattern.

**Also: Determination of the wavelength of light and sound by two source superposition with a double-slit and diffraction grating (OCR)**

**Also: Investigation of interference effects to include the Young's slit experiment and interference by a diffraction grating (AQA)**

Light, Sound and Waves

Lesson for 16-19

- Activity time 160 minutes
- Level Advanced

When two or more waves meet, we may observe interference effects. It is likely that your students will have already met the basic ideas of constructive and destructive interference.

Lesson Summary

- Demonstrations: Simple interference phenomena (20 minutes)
- Demonstration: Two sound sources (15 minutes)
- Demonstration: Young’s two-slit experiment (15 minutes)
- Discussion: Deriving and using the formula (20 minutes)
- Student experiments: Double slit analogues (30 minutes)
- Student questions: Using the Young’s slits formula (40 minutes)
- Demonstrations: For students to explain (20 minutes)

The following simple demonstrations could be used to introduce this section; don’t feel that you have to give detailed explanations at this stage.

**Laser speckle pattern**. Shine a laser onto a screen. Move your head side to side and observe the dark and light speckles, due to the different path lengths to the eye from different positions on the spot of laser light (If the beam is too small to show the speckles, try expanding it by passing it through a low-power lens, either converging or diverging.)**Observe the colours in soap bubbles or oil films**. Light is partly reflected by the upper surface of the film, partly by the lower surface. Depending on the thickness of the film, these two light rays will superpose constructively or destructively, depending on the wavelength. Thus two paths giving constructive superposition at one wavelength will not give constructive superposition for other wavelengths – hence only the colour with thecorrect

wavelength is seen.**Tuning forks**. Hold the vibrating fork with its prongs vertical and close to the ear. Twist the fingers so the fork slowly rotates about a vertical axis. The loudness of the sound will rise and fall, four times per complete rotation. (Each prong acts as a source of sound waves; twisting the fork alters the distance between each prong and the eardrum.)

Emphasise that, in each case, there are two or more sources

of light or sound reaching the eye or ear. You are going to look at an experiment designed to have two sets of light waves meeting in a very controlled way, i.e. Young’s two-slit experiment.

In any work with lasers, it is worth pointing out to the class the label in the laser. It should say Class 2: do not stare down the beam

. With such a laser, a momentary reflection of the beam into someone’s eye will not cause an injury.

Because the wavelength of light is very small, it is worth setting up an equivalent experiment with sound waves. Use two loudspeakers connected to a single signal generator. At this stage, it is not necessary to make detailed measurements.

Episode 321-1: Hearing superposition (Word, 84 KB)

Young’s two-slit experiment is perhaps one of the most famous experimental arrangements in physics. It was inspired by Young’s discovery of interference that he related in May 1801: Given a pond with a canal connected to it. At two places in the pond waves are excited. In the canal two waves superpose forming a resultant wave. The amplitude of the resultant wave is determined by the phase difference with which the two waves arrive at the canal.

Shine laser light through a double slit on to a screen. You should see a series of evenly-spaced bright spots (fringes

or maxima). Ask students to relate this to the sound experiment. (The bright fringes are the equivalent of the loud points in the sound field.)

Point to the central bright fringe. Emphasise that two light rays reach this point, one from each slit. They have travelled the same distance, so there is no path difference between them. They started off in step (in phase) with each other, and now they arrive at the screen in phase with each other. Hence their displacements add up to give a brighter ray.

The next bright fringes (on either side of the central one) represent points where one ray has travelled l further than the other, so they are back in phase. Why is there a dark fringe in between? (One ray has travelled l/2 further than the other, so they are out of phase and interfere destructively.)

If we could measure these distances approximately, we could determine *λ* .

Show the effects of:

- Using two slits with a smaller separation (the fringes are further apart).
- Moving the screen closer to the slits (the fringes are closer together).

It is clear that we might use this experiment to determine the wavelength of light, but how?

A modern version of the Young’s Two Slit experiment was voted the most beautiful experiment in physics

in a *Physics World* readers’ poll in 2002. It still forms the basis of ongoing research into the fundamental quantum nature of matter.

If you have already covered the photon model for light, you may want to refer back to this. As early as 1909, it was established that fringes were found even if the source was so faint that only one photon at a time was in the apparatus. Fringes can also be seen using de Broglie (or matter) waves. The most massive particles used to generate fringes to date (March 2005) are fluorinated buckyballs C_{60}F_{48} (i.e. 1632 mass units).

Now derive or quote the formula (depending upon your specification). A good way to start is to ask your students to identify the important variables (they are all lengths), and to give their approximate sizes:

*λ* , wavelength of the light (~500 nm)

*d*, separation of the two slits (~1 mm)

*s*, separation of the fringes (bright to bright or dark to dark) (~1 mm)

*L*, distance between slits and screen (~1 m)

How can we make a balanced equation from four quantities that are so different in magnitude? The simplest solution is that the product of the biggest and smallest is equal to the product of the two in-between quantities. Hence:

*λ* *L* = *s**d*

or

*λ* *d* = *s**L*

Episode 321-2: Calculating wavelength in two-slit interference (Word, 79 KB)

Episode 321-3: Two-slit interference (Word, 51 KB)

Set up a circus of Young’s two-slit arrangements (depending upon the available equipment to hand) using light, microwaves, 3GHz radio waves, ultra-sound (less noisy than audible sound!) and a ripple tank, and get students to determine the wavelength of the waves being used in each case.

You may prefer to set some up as demonstrations.

Episode 321-4: Interference patterns in a ripple tank (Word, 48 KB)

Episode 321-5: Measuring the wavelength of laser light (Word, 42 KB)

The first set of questions covers the principles of Young’s experiment.

The second set is questions for practice in using the equation.

Episode 321-6: Questions on the two-slit experiment (Word, 28 KB)

Episode 321-7: Two-source interference: some calculations (Word, 23 KB)

Here are two fun demonstrations to round off this episode. Demonstrate them, and ask your students to provide explanations.

- A nice demo using audible sound is to fix two loudspeakers at each end of a longish piece of wood. Mount the wood on a suitable pivot (large nail) mid-way between the two speakers. Drive both in parallel from signal generator. Slowly scan the class. As the interference fringes sweep across the audience, they hear the regular change in volume.
- Make a
sound trombone

. Mount a small loudspeaker in the wide end of a small plastic funnel. Tubing from the other end divides into two tubes: one takes a direct route, the other a route whose length can be varied by a U-shaped glass tube slidingtrombone

section. The two routes combine and are fed into another funnel that acts as an earpiece.
The loudness of the sound depends upon the position of the trombone slider. It is obvious that there are two paths by which the sounds reach the ear. There may be a path difference between them. If the path lengths differ by an exact number of wavelengths, constructive interference increases the volume; integral half wavelength path differences mute the sound due to destructive interference.
Do not confuse this with beats; here, only one frequency is involved, whereas

beatingis an effect due to

Episode 321-5 for measuring wavelength of a laser. Difficult with ’two-slit’ because faint or with grating easier to see pattern.

**Also: Determination of the wavelength of light and sound by two source superposition with a double-slit and diffraction grating (OCR)**

**Also: Investigation of interference effects to include the Young's slit experiment and interference by a diffraction grating (AQA)**

Episode 321-5 for measuring wavelength of a laser. Difficult with ’two-slit’ because faint or with grating easier to see pattern.

**Also: Determination of the wavelength of light and sound by two source superposition with a double-slit and diffraction grating (OCR)**

**Also: Investigation of interference effects to include the Young's slit experiment and interference by a diffraction grating (AQA)**

** Class demonstration **

This demonstration shows that a beam of light is diffracted as it passes around a wire, highlighting the wave nature of light.

Apparatus and Materials

- Laser source
- Thin, straight wire, approx 25 cm
- Stand with 2 clamps
- Screen

You will probably need to work in a darkened room

Health & Safety and Technical Notes

Read our standard health & safety guidance

Care should be taken to ensure that the laser beam does not shine directly into students’ eyes. This can be avoided by fixing it firmly in a clamp directed away from the students and towards the screen. Ensure that there are no shiny, reflective objects close to the path of the beam.

Procedure

This film shows how to demonstrate the diffraction of light using a laser source and a wire

- Mount the laser pointer horizontally in a clamp.
- Mount the wire vertically between two clamps.
- Direct the laser light onto the screen. You will see a bright dot.
- As suggested in the film, ask your students to predict what they will see when the wire partially blocks the laser beam.
- Move the wire into the beam. You should see a diffraction pattern of light and dark 'fringes' on the screen

Teaching Notes

- We may talk casually about ‘light waves’, but students need to be convinced that light travels as a wave. This demonstration shows it.
- Students will need to be familiar with two ideas: that waves diffract as they pass around an obstacle, and that waves interfere constructively and destructively when they overlap. These ideas can be shown using a ripple tank.
- You can show diffraction and interference of light using single, double or multiple slits. However, students may find these difficult to appreciate. Diffraction by a simple wire is a more straightforward situation to explain. Students can also be asked to predict what will be seen on the screen when the wire is placed in the path of the light beam. They will probably expect to see a vertical shadow. The appearance of a diffraction pattern spread across the screen is a surprise worth exploring.
- A laser is used because it is a convenient source of a narrow beam of light. It has the added advantage that it produces light of a single wavelength; white light would produce a similar effect but the diffraction pattern would not be as wide as different wavelengths (colours) would interfere at different points.
- It is worth emphasising the extent to which light is diffracted as it passes around the wire. The diffraction pattern may be 50 cm wide when the diffracting wire is one metre from the screen. So light is being diffracted (bent) through an appreciable angle – perhaps 20 degrees.
- You could investigate the effect of rotating the wire; can students predict what will happen? (A vertical wire produces a horizontal diffraction pattern; a horizontal wire will produce a vertical pattern.)

Related Guidance

Classroom management in semi-darkness

A sequence of experiments to show the diffraction of light and how this can be used to determine the wavelength of light

A number of options but basic set up with trolleys, runways and falling loads would work. Measure acceleration using light gates etc, or falling load to hit ground then measure final velocity (single light gate or ticker tape). For this need also to know time to fall

Forces and Motion

Lesson for 16-19

- Activity time 80 minutes
- Level Advanced

This episode concerns Newton’s second law. Your students will probably have met the second law in the form F = ma; many will have performed experiments to demonstrate the law. It is therefore useful to approach the experimental demonstration of the law as an exercise in data gathering and analysis. Using a simple set of apparatus should allow students to work individually or in pairs and critically consider the limits of the experiment as well as re-familiarizing themselves with the second law.

Lesson Summary

- Discussion: Revision of kinematics (10 minutes)
- Student investigation: Relationship between acceleration and force (30 minutes)
- Discussion: Looking at the results (10 minutes)
- Student questions: Using Newton’s second law (30 minutes)

Before embarking on the main activity it is useful to run through the equations of motion (the SUVAT

equations) once again so that students will understand the recipe

they use to calculate acceleration. You want to establish that by referring to the equation *s* = *u**t* + 12*a**t*^{ 2} the acceleration of a body travelling distance from rest is given by
*a* = 2 *s**t*^{ 2}

In this experiment, a trolley is accelerated by weights which are hanging on the end of a string which passes over a pulley.

It is important to note that the mass which is being accelerated includes the mass of the weights on the end of the string.

After the preliminary discussion the students should be able to tackle this without too many difficulties. The questions at the end of the section are best attempted after the apparatus is cleared away and the students have drawn the graphs. You can use their responses as a basis for a plenary session in which further discussion of sources of error (timing – more difficult for shorter time intervals, non-uniform acceleration etc).

Episode 211-1: Effect of force & mass on acceleration of an object (Word, 45 KB)

Discuss your students’ results:

Do they find that acceleration is proportional to force, and inversely proportional to mass?

Numerically, are their results consistent with the equation *F* = *m* × *a*?

You may wish to point out that the experiment can only show proportionality. In other words, we can only conclude that

*F* = *k × m* × *a*
, where *k* is a constant. In the SI system of units, we choose *k* = 1.
This defines the Newton: 1 N = 1 kg m s^{-2}.

Make a selection from these questions: cut out those you think may be too trivial for some, and others (using resolved forces) which may confuse weaker students even though the concepts have already been covered. You may wish to reserve some of the questions for later use.

Episode 211-2: Newton’s second law (Word, 25 KB)

**Demonstration**

A trolley experiences an acceleration when an external force is applied to it. The aim of this datalogging experiment is explore the relationship between the magnitudes of the external force and the resulting acceleration.

Apparatus and Materials

- Light gate, interface and computer
- Dynamics trolley
- Pulley and string
- Slotted masses, 400 g
- Mass, 1 g
- Clamp
- Ruler
- Double segment black card (see diagram)

Health & Safety and Technical Notes

Take care when masses fall to the floor. Use a box or tray lined with bubble wrap (or similar) under heavy objects being lifted. This will prevent toes or fingers from being in the danger zone.

Read our standard health & safety guidance

Pass a piece of string with a mass hanging on one end over a pulley. Attach the other end to the trolley so that, when the mass is released, it causes the trolley to accelerate. Choose a length of string such that the mass does not touch the ground until the trolley nearly reaches the pulley. Fix a 1 kg mass on the trolley with Blu-tack to make the total mass (trolley plus mass) of about 2 kg . This produces an acceleration which is not too aggressive when the maximum force (4 N) is applied.

The force is conveniently increased in 1 newton steps when slotted masses of 100 g are added. Place the unused slotted masses on the trolley. Transfer them to the slotted mass holder each time the accelerating force is increased. This ensures that the total mass experiencing acceleration remains constant throughout the experiment.

Fit a double segment black card on to the trolley. Clamp the light gate at a height which allows both segments of the card to interrupt the light beam when the trolley passes through the gate. Measure the width of each segment with a ruler, and enter the values into the software.

Connect the light gate via an interface to a computer running data-logging software. The program should be configured to obtain measurements of acceleration derived from the double interruptions of the light beam by the card.

The internal calculation within the program involves using the interruption times for the two segments to obtain two velocities. The difference between these, divided by the time between them, yields the acceleration.

A series of results is accumulated in a table. This should also include a column for the manual entry of values for force

in newtons. It is informative to display successive measurements on a simple bar chart.

Procedure

- Select the falling mass to be 100 g. Pull the trolley back so that the mass is raised to just below the pulley. Position the light gate so that it will detect the motion of the trolley soon after it has started moving.
- Set the software to record data, then release the trolley. Observe the measurement for the acceleration of the trolley.
- Repeat this measurement from the same starting position for the trolley several times. Enter from the keyboard '1' (1 newton) in the force column of the table (see below).
- Transfer 100 g from the trolley to the slotted mass, to increase it to 200 g. Release the trolley from the same starting point as before. Repeat this several times. Enter '2' (2 newtons) in the force column of the table.
- Repeat the above procedure for slotted masses of 300 g and 400 g.
- Depending upon the software, the results may be displayed on a bar chart as the experiment proceeds. Note the relative increase in values of acceleration as the slotted mass is increased.
- The relationship between acceleration and applied force is investigated more precisely by plotting an XY graph of these two quantities. (Y axis: acceleration; X axis: force.) Use a curve-matching tool to identify the algebraic form of the relationship. This is usually of the form 'acceleration is proportional to the applied force'.
- This relationship is indicative of Newton's second law of motion.

Teaching Notes

- This is a computer-assisted version of the classic experiment. The great advantage of this version is that the software presents acceleration values instantly. This avoids preoccupation with the calculation process, and greatly assists thinking about the relationship between acceleration and force. Each repetition with the same force gives a similar acceleration. If the force is doubled, this results in a doubling of the acceleration, and so on. The uniform increases in the acceleration can be confirmed by using cursors to read off corresponding values from the graph.
- The resulting straight line fit on the graph should be scrutinized for sources of error. The quality of the fit is reduced if the suggested procedure for maintaining the total mass constant is ignored. Also, a common outcome is a very small intercept near the graph origin. The most likely cause of this is neglect of the effect of friction on the motion of the trolley.
- The gradient of the line may be correlated with
*1/mass*of the system (trolley and slotted masses). - There is a variation of this experiment, in which the force is held constant but the mass of the trolley is altered by attaching further masses. This may be conducted to provide data for the complementary relationship indicated by Newton's second law: for a given applied force, the acceleration of the trolley is inversely proportional to its mass.

*This experiment was safety-tested in November 2006*

Download the support sheet / student worksheet for this practical.

This resource uses an electronic timer and a cardboard box on wheels to determine the force used to kick a football.

**Demonstration**

Using impact time and change of momentum of a football to measure the force needed to kick the ball.

Apparatus and Materials

- Scaler or electronic timer accurate to 0.001 s
- Round football (rugby type not suitable)
- 30 cm flexible leads with crocodile clips, 2
- Stopwatch or stopclock
- Balance (to measure mass of ball)
- Aluminium foil square, 15 cm by 15 cm
- Aluminium foil square, 7.5 cm by 7.5 cm
- Sellotape
- Plasticene

Health & Safety and Technical Notes

Take care that the football is aimed so that it does not cause damage, and there is no danger of the timer being knocked off the bench.

Read our standard health & safety guidance

The large foil is Sellotaped to the football: the small foil is taped to the toe of the kicker’s foot.

Connections to the foil are made with crocodile clips. The other ends of the leads should be a loose fit in the ‘timer input’ sockets so that they will come out easily in the event of an accident. It is sensible to have a student holding the timer on the bench.

Providing that the flexible leads are arranged so that the period of contact takes place before the ball pulls the foil away from the crocodile-clip contact, no difficulties should arise. You should obtain consistent results.

It is possible to get a value for the time of flight of a ball kicked with medium force down a 10 m corridor (or even a 5 m laboratory).

Procedure

- Place the ball on a laboratory table, using three small lumps of Plasticene to stabilize it.
- Kick the ball in a horizontal direction from a standing position, with only medium force. More vigorous kicks can be used out of doors to show the longer time of contact.
- Find the time of contact of the ball with the foot from the scaler or timer,
*t*, seconds. - Find the mass of the ball,
*m*kg, using a balance.

Teaching Notes

- Measure how far the ball travels horizontally before it hits the floor,
*s*, then*s = vT*. - The time of flight,
*T*, can be found from the height of the table,*h = 1/2 gT*^{ 2 }.
The acceleration due to gravity = 10 m s - Substituting in the equation
*v = s /T*gives a value for the initial velocity of the ball. - Therefore using
*Ft*= Δ(*mv*) the force,*F*, on the ball can be calculated. - Alternatively, film the flight of the ball using a camcorder, and use frame-by-frame playback mode to calculate its speed.
- Once students have learnt about the conservation of momentum in a collision then a different method can be used to calculate the force. The football is kicked into a cardboard box which is fixed to roller-skates or a skateboard.
- The box should be made massive so that it moves slowly enough for the time of motion to be measured with a stopwatch. The flaps on the box should trap the ball. All the momentum of the ball is shared with the box.
- The momentum of the box is calculated from its mass and velocity (= distance travelled in the measured time). This is equal to the initial momentum of the ball after it is kicked. Using
*Ft = Δmv*then the force can be calculated if the time of contact is measured on the scaler.

Forces and Motion

Lesson for 16-19

- Activity time 120 minutes
- Level Advanced

This episode introduces the concept of momentum and its conservation.

Lesson Summary

- Demonstration and discussion: An introduction making plausible the idea of conservation of momentum (20 minutes)
- Student experiment: For them to find the law of conservation of momentum for themselves (40 minutes)
- Worked examples: Showing how to apply conservation of momentum in simple cases (20 minutes)
- Student questions: Momentum conservation (30 minutes)
- Discussion: Relating conservation of momentum to Newton II and III (10 minutes)

Start by establishing experimentally the plausibility of the idea of conservation of momentum, by looking at some simple collisions first of all visually and then with some means of measuring velocities.

Demonstrate Newton’s cradle. Ask for an explanation in terms of forces, observing that if n balls are swung in, n balls swing out. This is a toy with limited possibilities, so move on to a better experimental system.

Demonstrate some collisions and explosions using trolleys on a flat bench or runway. (Alternatively use gliders on an air track.) Start with inelastic collisions, in which the trolleys stick together. Describe these as sticky

collisions. Point out that energy stored kinetically is not conserved. Try simple combinations such as trolleys of equal mass, or a single trolley colliding with a double one. How does velocity change? What quantity remains constant?

Remember that trolley runways and large air tracks require two people to manipulate and carry them safely. Some air track blowers also require two people to carry them.

Note that you are asking students to judge changes in velocity by eye. If the mass of the trolley doubles, its velocity halves, and so on.

It should become apparent that mass × velocity is constant is plausible. Name this quantity as momentum.

Emphasize that you are looking at events

and that you are comparing before

with after

. This will feed into the standard approach for solving numerical problems.

Now try explosions, in which the spring of one trolley is released to push the two apart. Try two single trolleys, then a single pushing on a double. What rule can you see? (The heavier (more massive) trolley comes away more slowly.) But what about momentum? Has it been created out of nothing? Emphasize the need to think of momentum as a vector (because velocity is a vector, mass is a scalar). Before the explosion, there is no momentum in the system; after, there are equal but opposite momenta, so the vector sum is zero.

Now you can state the principle of conservation of momentum in simple terms: in a sticky (inelastic) collision, the momentum of the moving object is shared between the colliding masses; in the Newton’s cradle case the momentum is clearly transferred, and in an explosion, there is no initial momentum, and the moving masses have equal but opposite momenta after the collision.

So total momentum before an event = total momentum after the event in all the cases so far.

Note that with air-tracks it is very difficult to change the mass of the gliders by much as they then tend to sink and drag on the track. Trolleys are more flexible in this respect but friction effects are larger.

The students can now investigate the idea of conservation of momentum experimentally. If you have sufficient apparatus it is very worthwhile getting the students to perform this experiment for themselves, especially if there is a quick and accurate way to measure speeds, such as light gates, or use of a motion sensor with a computer.

Episode 220-1: Observing collisions (Word, 56 KB)

Ask the students to look for a relationship in their mass and speed; if they find this difficult, suggest they look at the value of mass × speed. (Draw up a suitable table to help them with this.)

When students have made a few measurements, you may need to show how to calculate the total momentum for two masses,

i.e.
total momentum = *m*_{1}*v*_{1} + *m*_{2}*v*_{2} , not

(*m*_{1} + *m*_{2}) × (*v*_{1} + *v*_{2})
or any other combination of masses and velocities.

You could ask one group of students to demonstrate to the class the conservation of momentum in an inelastic collision, and another to demonstrate this for an explosion.

If time permits, ask students to extend the experiment to look at springy

(elastic) collisions. Is momentum still conserved?

The closer the light gates are to the trolleys or gliders at the time of collision or explosion, the less friction will distort the results. Alternatively a computer and motion sensor could be used.

Show how to calculate momentum from values of mass and velocity. Emphasize that units are kg m s^{-1} (no special name in SI system).

Establish a sign convention (e.g. velocities to the right are positive; to the left they are negative).

Work through examples to show:

- calculation of velocity of moving mass after inelastic collision
- calculation of velocity of one mass after explosion, given velocity of the other

Emphasise the need to draw diagrams showing the situation before and after an event (collision or explosion) when solving numerical problems.

Emphasise the predictive power of the principle of conservation of momentum. Mention that it works in 3 dimensions, as well as in these simple 1-D situations.

(You could look at other examples of conservation they will have met, such as energy and electric charge. They should be aware of the utility of such conservation laws in calculations and also that they are established experimentally.)

Episode 220-2: Worked Examples (Word, 42 KB)

Students should now be able to do more questions of the type described above. Elastic collisions are not dealt with until Episode 220.

Episode 220-3: Inelastic Collisions (Word, 21 KB)

(This is a rather abstract discussion, which you may wish to omit.)

The principle of conservation of momentum can be thought of as a consequence of Newton’s second and third laws. Try to prompt students to contribute to each stage in this argument.

Think about two trolleys of different masses exploding apart. From Newton’s third law it is clear that the trolleys are acted on by equal forces, in opposite directions.

Both forces must act for the same time – the time the trolleys are in contact.

These forces produce accelerations in inverse proportion to the masses, from Newton’s second law. So the bigger trolley has a smaller acceleration than the smaller one.

So the change in velocity of the bigger trolley is less than that of the smaller trolley, and in the opposite direction.

Because change in velocity ∝ 1mass, it follows that mass × velocity change for the two trolleys is equal and opposite. So total momentum is constant.

A moving glider on a linear air track collides with a stationary glider, thus giving it some momentum. This datalogging experiment explores the relationship between the momentum of the initially moving glider, and the momentum of both gliders after the collision.

**Demonstration**

A moving glider on a linear air track collides with a stationary glider, thus giving it some momentum. This datalogging experiment explores the relationship between the momentum of the initially moving glider, and the momentum of both gliders after the collision.

Apparatus and Materials

- Light gates, interface and computer, 2
- Linear air track with two gliders, each fitted with a black card
- Glider accessories: magnetic buffers, pin and Plasticine
- Clamps for light gates, 2
- Electronic balance

Health & Safety and Technical Notes

The most significant hazard is that of setting up the linear air track on the bench, especially if it is stored on a high shelf. Two people may be needed to achieve this safely.

Read our standard health & safety guidance

*Photograph courtesy of Mike Vetterlein*

Set up the linear air track in the usual manner, taking care to adjust it to be perfectly horizontal. A stationary glider should not drift in either direction when placed on the track.

Select two air track gliders of equal mass. Attach to each a magnetic buffer at one end, and a black card in the middle.

Prepare each card accurately to a width of 5.0 cm, and enter this value into the software.

The mass of the gliders must also be measured and entered into the software to prepare for the calculations (see below). If magnets are not available, crossed

rubber band catapults are an acceptable alternative.

Connect the light gates via an interface to a computer running data-logging software.

The program should be configured to obtain measurements of momentum, derived from the interruptions of the light beams by the cards.

The internal calculation within the program uses the interruption times from each light gate to obtain two velocities. These are multiplied by the appropriate glider masses to give two values of momentum, one before the collision, and one after. This assumes that the measurements for the width of the card and the masses of the gliders have been entered into the program correctly.

For the elastic collision (first part), the momentum measured at A depends upon the mass of the moving glider only. The momentum measured at B depends upon the mass of the initially stationary glider only.

For the inelastic collision (second part), the momentum measured at A depends upon the mass of the moving glider, whereas the momentum measured at B depends upon the combined mass of both gliders.

Students accumulate a series of results in a table with two columns, showing the momentum before and after each collision. It is informative to display successive measurements on a simple bar chart.

Procedure

**Data collection**

- Position the light gates A and B either side of the midpoint of the track as shown.
- Place one glider at the left hand end of the track, and the second between the light gates, with the magnetic buffers facing. The second glider should remain stationary.
- Give the first glider a short push so that it passes through light gate A. It then collides with the stationary glider. This then moves and passes through light gate B. If necessary, adjust the positions of the light gates to make sure that the sequence is correct. (As the magnetic buffers approach each other they repel so that there is no real contact between the two gliders. This creates the condition for 'elastic' collisions.)
- Return the gliders to their starting positions, set the software to record data, and repeat the sequence. Observe the measurements of momentum before and after the collision. Repeat this whole process several times to obtain measurements for a series of collisions.
- Replace the magnetic buffers with a pin on one glider and a lump of Plasticine on the other. (This will cause the gliders to stick together after the collision, making it an 'inelastic' collision.) The black card may be removed from the initially stationary glider.
- Reset the program so that the measurements at B use the combined mass of both gliders.
- Use the same procedure as for Part 1 to obtain measurements for a series of inelastic collisions.
- Depending upon the software, the results may be displayed on a bar chart as the experiment proceeds. Note the very similar values for momentum before and after each collision of either type.
- The results can be displayed as a graph of 'momentum before collision' against 'momentum after collision'. A straight line graph would demonstrate that the relationship does not depend upon the magnitude of the initial momentum. If the graph is at 45°, this confirms the conservation of momentum.

Teaching Notes

- This is a computer-assisted version of the classic experiment, using light gates and electronic timers. The great advantage of this version is the instant presentation of momentum values using the software. This avoids preoccupation with the calculation process and allows attention to focus on the results.
- It is unusual for the measured values of momentum before and after each collision to be identical. It is wise to limit the number of decimal places displayed, so that the discrepancy does not appear exaggerated. Note how small the discrepancy is, compared with the magnitude of each value. A bar chart display makes this comparison very plain to see. Thus it can be argued that momentum is conserved in each case.
- A discussion of the measurement errors must consider the residual friction affecting the motion of the gliders. Errors may be kept to a minimum by strategically placing the light gates so that they capture the motion as close as possible to before and after a collision.

*This experiment was safety-tested in June 2007*

The topic draws on what students already know about simple dc circuits. There may be a need to remind students about charge while it is possible that ideas about (uniform) electric fields can be reinforced.

Electricity and Magnetism

Teaching Guidance for 16-19

- Level Advanced

Many of the basic ideas can be studied with a range of capacitors (at least one with a large value, 10 000 mF or more) and cells, plus ammeters and voltmeters (some multimeters will have the ability to measure capacitance directly). A coulombmeter is most useful. Datalogger(s) will be an advantage. A reasonable oscilloscope can do a similar job but does not provide a permanent record

Look out for:

- Capacitors of different sizes; check that you can identify those which are electrolytic.
- Electrolytic capacitors may explode if they are connected the wrong way round. The material inside becomes a gas and the pressure is more than the case can contain. You may wish to demonstrate this effect; you should only do so in a fume cupboard. This can also occur when a capacitor carries a large ripple current. Those designed to cope with this are labelled
high ripple capacity

and are used as smoothing capacitors with lab power supplies. - A change-over reed switch mounted in a box, for measuring capacitance.
- Two large (30 cm square) metal plates, drilled to take 4 mm plugs.
- Some activities make use of spreadsheets.
- Many activities make use of electrolytic capacitors. If these have not been used for 12 months, it is worth reforming them before use. See
Electrolytic Capacitors

in the CLEAPSS Lab Handbook.

Students will:

- define capacitance in terms of charge stored per volt
- calculate values of charge and energy stored
- calculate values of capacitances connected in series and parallel
- interpret exponential and logarithmic graphs for capacitor discharge
- relate their understanding to analogous phenomena, including springs and radioactive decay

The topic draws on what students already know about simple dc circuits. There may be a need to remind students about charge while it is possible that ideas about (uniform) electric fields can be reinforced.

(If your specification introduces the formula for calculating capacitance, then students will have to use * ε *

The exponential equations for capacitor discharge are similar to those for radioactive decay (and for damped SHM). Depending on the order in which you are tackling these topics, you can either refer back to previous use of exponentials, or refer ahead to future work. It is helpful to students to feel that they are getting two for the price of one

, particularly if they find these mathematical ideas tricky.

This resource builds students’ understanding of exponential processes, through experiments, and through graphical and algebraic approaches, all related to the underlying physical processes involved. For the more mathematically able students, you may even be able to use calculus.

Electricity and Magnetism

Lesson for 16-19

- Activity time 220 minutes
- Level Advanced

Students will have already seen that the discharge is not a steady process in episode 125, but it is useful to have graphical evidence before discussing the theory.

You need to build up your students’ understanding of exponential processes, through experiments, and through graphical and algebraic approaches, all related to the underlying physical processes involved. For the more mathematically able students, you may even be able to use calculus.

This episode is a long one, and may spread over several teaching sessions.

A spreadsheet is included as part of the student questions for activity 129-7.

Lesson Summary

- Student experiment: Exponential discharge (30 minutes)
- Discussion: Characteristics of exponentials (20 minutes)
- Student activity: Spreadsheet model (20 minutes)
- Student experiment: Varying R and C (30 minutes)
- Discussion: Deriving exponential equations (30 minutes)
- Worked example: Using the equations (20 minutes)
- Discussion: Time constant (15 minutes)
- Student activity: Analysing graphs (15 minutes)
- Student questions: Practice with equations (30 minutes)
- Discussion: Back to reality (10 minutes)

The suggestion to look for a pattern by measuring halving times is worth pursuing. It forms a basis for further discussion, and shows that the patterns for current and voltage are similar. (They can be related to the idea of exponential decay in radioactivity, episode 513, if students have met this previously.)

Even though some specifications require the use of data logging for this, it is worth collecting data manually from a slow discharge and then getting the students to plot the graphs of current against time and voltage against time for the decay.

The specifications do not require details of the charging process but data for this is easily collected in the same experiment.

Episode 129-1: Slow charge and discharge (Word, 31 KB)

Draw out the essential features of the discharge graphs. Sketch three graphs, for *Q*, *I* and *V* against *t*. All start at a point on the y-axis, and are asymptotic on the t-axis. All have the same general shape. How are they related?

The *Q* graph is simply the *V* graph multiplied by *C* (since
*Q* = *C* × *V*
).

The *I* graph is the *V* graph divided by *R* (since
*I* = *V**R*
).

The *I* graph is also the gradient of the *Q* graph (since
*I* = d *Q* d *t*
).

Add small tangents along the *Q* graph to show this latter pattern. A large charge stored means that there is a large pd across the capacitor; this makes a large current flow, so the charge decreases rapidly. When the charge is smaller, the pd must be lower and so a smaller current flows. (Students should see that this will result in quantities which get gradually smaller and smaller, but which never reach zero.)

Students can use an iterative approach, with the help of a spreadsheet, to see the pattern of potential difference across the capacitor while it is discharging (top graph), and charging (bottom graph).

Episode 129-2: One step at a time (Word, 33 KB)

The previous experiment produced graphs of the discharge for a particular combination of resistor and capacitor. This can be extended by looking at the decay for a range of values of *C* and *R*. If a datalogger is available, this can be done quickly and can include some rapid decays. If a datalogger is not available, measurements can be taken with the apparatus used earlier in episode 129-1.

Before the experiment, ask your students how the graphs would be affected if the value of *R* was increased (for a particular value of pd the current will be less, and the decays will be slower); and if the value of *C* was increased (more charge stored for a given pd; the initial current will be the same, but the decay will be slower, because it will take longer for the greater quantity of charge to flow away.)

Episode 129-3: Experiment analyzing the discharge of a capacitor (Word, 41 KB)

At this point, you have a choice:

- you can jump directly to the exponential equations and show that they produce the correct graphs
- alternatively, you can work through the derivation of the equations, starting from the underlying physics

We will follow the second approach.

To explain the pattern seen in the previous experiment you will have to lead your pupils carefully through an argument which will call on ideas about capacitors and about electrical circuits.

Consider the circuit shown:

When the switch is in position A, the capacitor C gains a charge *Q*_{0} so that the pd across the capacitor *V*_{0} equals the battery emf.

When the switch is moved to position B, the discharge process begins. Suppose that at a time *t*, the charge has fallen to *Q*, the pd is *V* and there is a current *I* flowing as shown. At this moment:

*I* = *V**R*
(equation 1)

In a short time Δ *t*, a charge equal to Δ *Q* flows from one plate to the other so:

*I* = - Δ *Q* Δ *t*
(equation 2)

(with the minus sign showing that the charge on the capacitor has become smaller)

For the capacitor:

*V* = *Q**C*
(equation 3)

Eliminating *I* and *V* leads to:

Δ *Q* = -*Q**C**R* × Δ *t*
(equation 4)

Equation 4 is a recipe for describing how any capacitor will discharge based on the simple physics of equations 1 – 3. As in the activity above, it can be used in a spreadsheet to calculate how the charge, pd and current change during the capacitor discharge.

Equation 4 can be re-arranged as:

Δ *Q**Q* = 1*C**R*

(Showing the constant ratio property characteristic of an exponential change i.e. equal intervals of time give equal fractional changes in charge.)

We can write equation 4 as a differential equation:

d *Q* d *t* = - 1*C**R*

Solving this gives:

*Q* = *Q*_{0}e^{- t/_CR_}

where
*Q*_{0} = *C* × *V*_{0}

Current and voltage follow the same pattern. From equations 2 and 3 it follows that

*I* = *I*_{0}e^{- t/_CR_}

where
*I*_{0} = *V*_{0}*R*

and
*V* = *V*_{0}e^{- t/_CR_}

A 200 mF capacitor is charged to 10 V and then discharged through a 250 kW resistor. Calculate the pd across the capacitor at intervals of 10 s.

(The values here have been chosen to give a time constant of 50 s.)

First calculate *C**R*, which is 50 s

Draw up a table and help students to complete it. (Some students will need help with using the x^{y} function on their calculators.) They can then draw a *V*-*t* graph.

t / s | 0 | 10 | 20 | 30 | 40 | 50 | 60 | etc |

V / V | 10 | 8.2 | 6.7 | 5.5 | etc | |||

I / mA | 40 | |||||||

Q / mC |

Explain how to calculate
*I*_{0} = *V*_{0}*R*
and
*Q*_{0} = *C* × *V*_{0}
, so that they can complete the last two rows in the table.

It is useful to draw a *Q*-*t* graph and deduce the gradient at various points. These values can then be compared with the corresponding instantaneous current values.

Similarly, the area under the *I*-*t* graph can be found (by counting squares) and compared with the values of charge *Q*.

For radioactive decay, the half life is a useful concept. A quantity known as the time constant

is commonly used in a similar way when dealing with capacitor discharge.

Consider:
*Q* = *Q*_{0}e^{- t/_CR_}

When
*t* = *C**R*
, we have
*Q* = *Q*_{0}e^{-1}

(i.e. this is the time when the charge has fallen to 1e = 0.37 (about ⅓ ) of its initial value. *C**R* is known as the time constant – the larger it is, the longer the capacitor will take to discharge.)

The units of the time constant are seconds

. Why? (
F × W = C V^{-1} × V A^{-1},
which simplifies to C A^{-1}, and then again to C C^{-1} s, so just s)

(Your specifications may require the relationship between the time constant and the halving time

*T*_{ ½ }:

*T*_{ ½ } = ln(2) × *C**R*

or,

*T*_{ ½ } = 0.69 × *C**R*)

Students should look through their experimental results and determine the time constant from a discharge graph. They should check whether the experimental value is equal to the calculated value *C**R*. Why might it not be? (Because the manufacturer’s values of *R* and *C*, are only given to a specified range, or tolerance, and that range is rather large for *C*.)

Questions on capacitor discharge and the time constant, including a further opportunity to model the discharge process using a spreadsheet.

Episode 129-4: Discharge and time constants (Word, 31 KB)

Episode 129-5: Discharging a capacitor (Word, 56 KB)

Episode 129-6: Capacitors with the exponential equation (Word, 30 KB)

Episode 129-7: Discharge of high-value capacitors (Word, 69 KB)

Episode 129-8: Spreadsheet for 129-7 (Word, 54 KB)

After a lot of maths, there is a danger that students will lose sight of the fact that capacitors are common components with a wide range of uses.

Some of these can now be explained more thoroughly than in the initial introduction. Ask students to consider whether large or small values of C and R are appropriate in each case. Some examples are:

Back-up power supplies in computers, watches etc, where a relatively large capacitor (often > 1 F) charged to a low voltage may be used.

Some physics experiments need very high currents delivered for a very short time (e.g. inertial fusion). A bank of capacitors can be charged over a period of time but discharged in a fraction of a second when required.

Similarly, the rapid tranfer of energy needed for a flash bulb

in a camera often involves capacitor discharge. Try dismantling a disposable camera to see the capacitor.

This resource provides two potential methods for students to investigate the energy stored in a capacitor.

Electricity and Magnetism

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

So far, we have not considered the question of energy stored by a charged capacitor. Take care; students need to distinguish clearly between charge and energy stored.

Lesson Summary

- Demonstration: Energy changes (15 minutes)
- Discussion: Calculating energy stored (15 minutes)
- Worked example: Energy stored (10 minutes)
- Student experiment: Energy stored – two alternatives (20 minutes)
- Student questions: Calculations on the energy formula (30 minutes)

The idea that a capacitor stores energy may have already emerged in previous sections but it can be made clear by using the energy stored in a capacitor to lift a weight attached to a small motor. The energy transfer process is not very efficient but it should be possible to show that a larger pd (or a capacitance and pd.

Emphasise the link between work and energy. How do we know that the charged capacitor stores energy? (It can do work on the load.) How did the energy come to be stored in the capacitor? (The power supply did work on the charges in charging the capacitor.)

Episode 128-1: Using a capacitor to lift a weight (Word, 30 KB)

Having seen that the energy depends on the voltage, there are several approaches which lead to the relationship for the energy stored. Start with a reminder of the idea that joules = coulombs × volts

.

The simplest argument is that with a pd *V*, a capacitor *C* will store charge *Q*, but the energy stored is not *Q* × *V*. Why not? (As the capacitor charges, both *Q* and *V* increase so we have not moved all the charge with a pd of *V* across the capacitor.)

What does this graph tell us? At first, it is easy to push charge on to the capacitor, as there is no charge there to repel it. As the charge stored increases, there is more repulsion and it is harder (more work must be done) to push the next lot of charge on.

Can we make this quantitative? A first try says that the pd was on average *V*2, so the energy transferred was *Q* × *V*2.

A more general approach says that in moving the charge ΔQ, the pd does not change significantly, so the energy transferred is *V* × Δ *Q*. But this is just the area of the narrow strip, so the total energy will be the triangular area under the graph.

i.e.
Energy stored in the capacitor = 12*Q**V*

or

Energy stored in the capacitor = 12*C**V*^{ 2}

or

Energy stored in the capacitor = 12*Q*^{ 2}*C*

If your pupils are strong mathematically, this summation can be replaced by integration.

A 10 mF capacitor is charged to 20 V. How much energy is stored?

Emphasise how to choose the correct version of the equation, in this case:

Energy stored = 12*C**V*^{ 2}

energy = 2000 mJ

Ask you students to calculate the energy is stored at 10 V (i.e. at half the voltage). Answer:

500 mJ, one quarter of the previous value, since it depends on *V*^{ 2}.

The formula can be checked with either or both of the following experiments.

The first experiment is straightforward. It could be used as the basis of a demonstration in which you ask the pupils to suggest how many extra bulbs are required at each stage and how they should be connected.

Episode 128-2: How many bulbs will a capacitor light (Word, 53 KB)

The second experiment needs more apparatus and time, and needs patience to obtain accurate measurements; it has benefit in terms of thinking about experimental design and systematic errors.

Episode 128-3: Energy stored by a capacitor (Word, 39 KB)

These give practice with the energy formulae.

Episode 128-4: Energy stored in a capacitor (Word, 64 KB)

Episode 128-5: Energy to and from capacitors (Word, 34 KB)

This resource provides guidance on using a CRO but does not give any specific details on capacitors.

Electricity and Magnetism

Lesson for 16-19

- Activity time 50 minutes
- Level Advanced

Students learn to use an oscilloscope to measure voltages.

Lesson Summary

- Demonstration and/or student experiment (30 minutes)
- Student questions: Understanding the CRO (20 minutes)

Bear in mind that most students (and teachers?) find oscilloscopes daunting at first. If they are not already familiar with them then try to use the simplest ones available (single beam, non-storage). Whatever you use, be prepared to spend a lot of time helping them with settings and familiarising them with the adjustments.

It may help if you start by demonstrating the experiment first, and then allow your students time to repeat it for themselves. The most important ideas to get across are:

- Used normally, an oscilloscope plots a graph of voltage (y-axis) against time (x-axis).
- The scales on the x- and y- axes can be adjusted using the timebase (time per division) and voltage gain controls (voltage per division). (Check what these are called on the oscilloscopes you use.)

Episode 122-1: The cathode ray oscilloscope (Word, 32 KB)

There are two main aims here:

- Familiarity with the oscilloscope so that they can get a stable trace and adjust settings.
- Taking measurements of voltage and time from the screen.

Questions on use of the oscilloscope.

Episode 122-2: Questions on use of the oscilloscope (Word, 26 KB)

**Also: Investigation of charge and discharge of capacitors. Analysis techniques should include log-linear plotting leading to a determination of the time constant RC (AQA)**

Electricity and Magnetism

Lesson for 16-19

- Activity time 70 minutes
- Level Advanced

There are many examples of exponential changes, both in physics and elsewhere. Your specification may require that you make a detailed comparison of the energy stored by a capacitor and a spring and of exponential decay in radioactivity and capacitors.

Lesson Summary

- Discussion: Energy stored (20 minutes)
- Discussion: Exponential decrease (20 minutes)
- Student questions: Exponential decrease (30 minutes)

Comparing the energy stored by capacitors and springs: The key point in the discussion is that the graphs of charge against pd

for a capacitor and force against extension

for a spring are both straight lines through the origin. For capacitors, the energy stored is the area under the charge/pd graph. A similar argument can be used to show that the energy stored in a spring is the area under the force/extension graph.

It follows that there are similar equations:

Energy stored in the capacitor = 12*Q**V*

Energy stored in the spring = 12*F**x*

Energy stored in the spring = 12*k**x*^{ 2}

Although it is not specifically mentioned in the specifications, the energy can be released steadily but there are many occasions where oscillations occur. Students are likely to have seen this for a spring but may not have seen any electrical circuits involving oscillations. The section could be concluded with a demonstration of this.

Episode 130-1: Electrical oscillations (Word, 108 KB)

Comparing exponential decay for radioactivity and capacitors: You could build up the the connections using contributions from members of the class.

For capacitor decay:

*Q* = *Q*_{0}e^{- t/_CR_}

Current, Δ *Q**Q* = 1*C**R*

Time constant, *C**R*: time for charge to fall by 1e

*T*_{ ½ }*C**R* = ln(2)

For radioactive decay:

*N* = *N*_{0}e^{- λ_N_}

Activity, Δ *N**N* = λ*N*

Half life, *T*_{ ½ }: time for no. of atoms to fall by 12.

*T*_{ ½ }λ = ln(2)

Any such comparison needs to highlight the similarities in the patterns for two very different physical processes by comparing the graphs of the decays. (This is a good point to remind pupils that testing for exponentials, either by a constant ratio property

or from a log graph, is an important skill.)

This worksheet has a good survey of a number of processes involving exponential decay: radioactivity, capacitor discharge and more.

Episode 130-2: Exponential changes (Word, 75 KB)

Resistance and temperature: 110-2 gives specific details on calibrating a thermistor.

Electricity and Magnetism

Lesson for 16-19

- Activity time 115 minutes
- Level Advanced

This episode looks at the resistance of a metal and a semiconductor, giving a microscopic explanation of the variation with temperature. There is also a brief look at superconductivity and its applications.

Lesson Summary

- Demonstration and discussion: Resistance and temperature (10 minutes)
- Discussion: Free-electrons in metals (10 minutes)
- Student experiment: Thermistor behaviour (20 minutes)
- Discussion and demonstration: Conduction in semiconductors (5 minutes)
- Discussion: Superconductivity (20 minutes)
- Student activity: Researching superconductivity (30 minutes plus time for reporting back)
- Student questions: Using these ideas (20 minutes)

This episode picks up on the variation of resistance of the filament lamp. Students’ own results should show that the resistance increases with current. Link this to the change of temperature of the wire and remind them that metals obey Ohm’s Law if the temperature is constant. (When they measured the resistance of the constantan wire in Episode 109, the current was always small, so temperature was almost constant.) You can reinforce the idea of resistance change in metals by cooling a wire and showing that its resistance decreases. This can be done using a cooling spray or, more dramatically, using liquid nitrogen (if this is available).

Episode 110-1: Metal resistance decreases as temperature falls (Word, 43 KB)

It is worth pausing at this point to discuss the mechanism of metallic resistance. Remind students of the model whereby as temperature increases the thermal vibrations in the lattice increase causing more electron scattering. (Be aware that there is more here than meets the eye in terms of quantum, as opposed to classical, free electron theory). This increases the resistance of the metal.

Next consider semiconductors. Students are unlikely to know much about semiconductors so it may be worth giving a brief introduction by saying that, compared to metals, they have only a few free electrons, so resistance (resistivity is the more appropriate term here, but they have not yet met it) is much higher. However, semiconductors such as silicon are central to the electronics industry so it is well worth considering their electrical characteristics. For example, how does their resistance depend on temperature?

Students can investigate the temperature dependence of the resistance of a thermistor for themselves.

The results should show a clear decrease of resistance with increasing temperature. This is the opposite of what happened with the metal.

NB These thermistors are n.t.c. types (negative temperature coefficient). Other types exist which have a non-linear positive temperature coefficient.

Episode 110-2: Calibration of a thermistor (Word, 39 KB)

Ask whether the atoms in the semiconductor vibrate more at higher temperature. Of course they do – so this contribution to resistance must increase in the same way as for a metal. So what else could make the semiconductor conduct better? The answer is: more charge carriers. Whereas the number of free electrons in a metal is constant the effect of heating a semiconductor frees additional electrons (and holes, but it’s probably not worth mentioning them yet!). For silicon in this temperature range the effect of additional charge carriers outweighs the effect of additional vibrations.

An interesting additional demonstration can be done using a different semiconductor (carbon). This shows that the two effects compete with each other. At lower temperatures the increase in resistance due to vibration dominates, as temperature rises, more and more electrons are freed and the resistance begins to fall.

Having introduced the idea that metallic resistance is caused by electron scattering from ions as they vibrate you should return to what happens as a metal is cooled down.

You are looking for an argument that runs along the lines of: lower temperature, smaller amplitude of vibration so reduced scattering and therefore reduced resistance. Refer back to the initial demonstration.

The students ought to predict that thermal vibrations will eventually stop (at absolute zero on a simple mechanical model). This implies a very low resistance at low temperatures (but not necessarily zero).

Lead into Kammerlingh Onnes’s work and his surprise that mercury’s resistance disappears at a very low temperature (a few degrees above absolute zero: 4.15 K).

This sudden transition was unexpected and is a quantum effect. It occurs for some but not all metals. It has also been observed at much higher temperatures (around 150 K) in certain ceramics. These are called high temperature

superconductors (even though we are still talking about temperatures more than 100 degrees below zero Celsius! The mechanism for high temperature superconductivity is not fully understood and it is hoped that in future we may be able to manufacture room temperature superconductors.

Rather than lecture them about superconductors this would be a good opportunity to set them some research tasks which can be reported back to the class. Here is a work sheet that could be used:

Episode 110-3: Researching superconductivity (Word, 27 KB)

Episode 110-4: Filament lamp and thermistor in series (Word, 31 KB)

This experiment, for advanced level students, shows that the current through a thermistor increases with temperature, as more charge carriers become available. There are similar investigations for other components and an LDR could be explored when illuminated by a lamp at different heights.

Quantum and Nuclear
| Electricity and Magnetism

Practical Activity for 14-16

**Class practical**

This experiment, for advanced level students, shows that the current through a thermistor increases with temperature, as more charge carriers become available.

Apparatus and Materials

- timer or clock
- Leads, 4 mm
- Crocodile clip holder
- Thermometer -10°C to 110°C
- Thermistor - negative temperature, coefficient, e.g. 100 ohm at 25°C (available from Rapid Electronics).
- Power supply, 5 V, DC or four 1.5 V cells
- Beaker, 250 ml
- Kettle to provide hot water
- Digital multimeter, used as a milliammeter
- Heat-resistant mat
- Power supply, low voltage, DC, continuously variable or stepped supply with rheostat (>1 A)

Health & Safety and Technical Notes

Read our standard health & safety guidance

A thermistor may be described as:

**ntc**negative temperature coefficient

: its resistance decreases as the temperature increases**ptc**positive temperature coefficient

: its resistance increases as the temperature increases

If you have both types available, students may be interested in comparing them.

Procedure

- Set up the circuit as shown below.
- Pour boiling water into the beaker and take readings of the current through the thermistor as the temperature falls. Record the results.
- Plot a graph of current/ mA (y-axis) against temperature/ °C (x-axis).
- Assuming that the voltage is constant, describe how the conductance or resistance varies with temperature.

Teaching Notes

- The thermistor is made from a mixture of metal oxides such as copper, manganese and nickel; it is a semiconductor. As the temperature of the thermistor rises, so does the conductance.
- The increase in conductance is governed by the Boltzmann factor. Whether or not your students need to understand Boltzmann, they should be able to grasp that
- as the temperature goes up, the resistance goes down
- in this case, it happens because more charge carriers are released to engage in conduction.

*This experiment comes from AS/A2 Advancing Physics. It has been re-written for this website by Lawrence Herklots, King Edward VI School, Southampton.*

Uses a method of melting ice to determine the specific latent heat of fusion of ice.

Energy and Thermal Physics
| Properties of Matter

Lesson for 16-19

- Activity time 130 minutes
- Level Advanced

Energy is involved in changes of phase, even though there is no change of temperature.

Lesson Summary

- Discussion: Defining specific latent heat (10 minutes)
- Demonstration: Boiling water (15 minutes)
- Student experiment: Measuring
*l*(30 minutes) - Student experiment: Cooling curves (30 minutes)
- Worked example: Latent and specific heat (5 minutes)
- Student questions: Involving
*c*and*l*(40 minutes)

The final point in this topic is to return to the original definition of (internal energy

) as being a combination of both the energy stored kinetically (kinetic energy) and energy stored chemically i.e. in the motion of the particles and the bonds between them. In talking about ideal gases all the energy was assumed to be stored kinetically because there were assumed to be no bonds between the atoms. However, in a solid or liquid there are bonds and clearly some energy is needed to break those bonds. That means that, in melting a solid or boiling a liquid, a substantial amount more energy needs to be transferred *which does not raise the temperature*. This is the hidden heat

or latent heat

.

The energy you need to transfer to a mass *m* of a substance to melt it is given by

Δ *E* = *m* × *L*

Or the 'specific latent heat' is the energy you need to transfer to change the unit mass from one phase to another.

Ask your class to watch some water boiling and think about what is going on. Energy is being transferred, but the temperature is not rising. Intermolecular bonds are breaking, and, as a physicist would say, work is being done to separate the particles against intermolecular attractive forces.

The key point from these is that, for certain materials, there is a *phase transition* where the energy transferred no longer raises the temperature (adds to each molecule's kinetic energy) but instead breaks bonds and separates the particles. This should be made quantitative. Likewise, the reverse processes involve energy being transferred from the substance. So evaporating liquids are good coolants and freezing water to make ice is considerably more of an effort than cooling water to 0 ° C.

Episode 608-1: Examination of boiling (Word, 34 KB)

It is useful to have measured a specific latent heat – for example, that of melting ice.

Episode 608-2: The specific latent heat of fusion of ice (Word, 67 KB)

If you have a class set of data-loggers for recording temperature, determination of the cooling curve of stearic acid, naphthalene or lauric acid is worthwhile. Even as a demonstration this is good and can be left running in the background while the students work on calculations.

Episode 608-3: Heating and cooling curves (Word, 52 KB)

Scalds from water and steam

We assume that our hand is at 37 ° C, and that we put 10 g of water at 100 ° C accidentally on our hand. The water will cool to 37 ° C. Assuming that all the energy lost

by the water will be gained

by our hand:

Energy shifted from water = *m**c* Δ *T*

Energy shifted = 1.5 kg × 4.2 kJ kg^{-1} ° C^{-1} × 63 ° C

Energy shifted = 2 646 J.

But if the 10 g had been steam then the steam would first have to condense.

Energy shifted in condensing = *m**L*

Energy shifted in condensing = 1.5 kg × 2 260 kJ kg^{-1}

Energy = 22 600 J

So the energy lost in 10 g of steam turning to water at 37 ° C is 25,246 J.

This is nearly ten times as much as the water alone!

The worked example is based on one from Resourceful Physics.

Practice in situations involving specific heat capacity and specific latent heat.

Episode 608-4: Questions on specific heat capacity and specific latent heat (Word, 27 KB)

Episode 608-5: Further specific and latent heat questions (Word, 30 KB)

Episode 607 has a straightforward approach to specific heat capacity, an interesting extension would be to do this by a method of mixtures.

Energy and Thermal Physics
| Properties of Matter

Lesson for 16-19

- Activity time 95 minutes
- Level Advanced

Energy must be transferred to (or from) a material to increase (or decrease) its temperature. Here is how to calculate how much.

Lesson Summary

- Discussion: Energy and change of phase (15 minutes)
- Student experiment: Measuring specific heat capacities (40 minutes)
- Worked example: Calculation involving
*c*(10 minutes) - Student questions: Calculations (30 minutes)

Up until this point the link between internal energy

and temperature has been qualitative, except for gases. In order to extend the discussion to solids and liquids we need to get more quantitative in two ways. One is to discuss how much the temperature of a body changes when its internal energy

in increased by a certain amount. The other is to ask what happens when a substance *changes phase* from a solid to a liquid or liquid to a gas.

Start by introducing the equation for specific heat capacity *c* (SHC) and defining the terms. The word specific

is an old fashioned way of saying per unit mass

. Work through a simple calculation.

Understanding this equation will help solidify ideas about temperature and energy and how they differ. A possible analogy was supplied by Richard Feynman. He suggested thinking of energy as being like water, and temperature as wetness. A towel can have different amounts of fluffiness, so take more or less water to make it wet. When we dry ourselves, we dry until the towel is as wet as we are (same temperature

).

The anomalously large SHC of water should also be discussed as it is particularly important for the development and maintenance of life on Earth.

NB nomenclature: there isn’t any agreed way to name *c*. Some use *specific thermal capacity* , others favour *specific heating capacity* to emphasise the fact that heat

is not an entity but a short hand name for a process (heating as oppose to working). Perhaps the most common is specific heat capacity.

Another source of confusion is treating *state* and *phase* as synonyms (as in changes of state / phase). Solids, liquids and gases are three of the different phases of matter (superfluids and plasmas are two others. NB Here, by a plasma, we mean an ionised gas, not a biological fluid). Thus melting, boiling etc are changes of phase. Each phase can exist in a variety of states depending upon e.g. the temperature and pressure. Thus the Ideal Gas Equation of State *P**V* = n*R**T* summarises the physically possible combinations of *P*, *V* and *T* for n moles of the ideal gas.

Students should carry out an experiment to measure the specific heat capacity of a solid and/or a liquid very soon after meeting the expression. There are a number of points to note here:

- If specific heat capacity is constant, the temperature will rise at a uniform rate so long as the power input is constant and no energy is dissipated (stored thermally in the surroundings)
- If the substance is not well insulated a lot of energy will be dissipated. This can be accounted for but in most cases students will not do so quantitatively
- They should calculate their value and make a comparison with data book values. They should be able to think of a number of reasons why their value does not match that in the data book.

Several different methods of determining the SHC of liquids and solids are given in the links below. Choose those best suited to your pupils and available equipment.

Episode 607-1: Measuring the specific heat capacity of a metal (Word, 33 KB)

Episode 607-2: The specific heat capacity of water and aluminium (Word, 37 KB)

It is useful to compare electrical methods of measuring the specific heat capacity of a solid and liquid including the continuous flow calorimeter for a liquid.

Episode 607-3: Measurement of specific heat capacities (Word, 56 KB)

This example deals with the mixing of liquids at different temperatures.

1.5 l of water from a kettle at 90 ° C is mixed with a bucket of cold water (10 l at 10 ° C) to warm it up for washing a car. Find the temperature of the mixed water, assuming no significant dissipation during the mixing. *c* of water is 4.2 kJ^{kg-1} °^{C-1}.

Answer: If there is no energy dissipated and no work done then the total energy of the system at the end is the same as at the start. Working with temperature differences from 10 ° C we have:

initial energy = *m**c* Δ *T*

as 1.5 l has mass of 1.5 kg:

Energy stored initially due to rise in temperature of the water in the kettle = 1.5 kg × 4.2 kJ kg^{-1} ° C^{-1} × (90-10) ° C

Energy stored initiallly = 504 kJ

This is the amount of energy available to raise the temperature of all 11.5 l of water. Hence:

504 kJ = 11.5 l × 4.2 kJ kg^{-1} ° C^{-1} × (*T*_{final}-10) ° C}

(where *T*_{final} is the final temperature)

*T*_{final} = 20.4 ° C.

You can invent similar problems using a mixture of substances – a hot brick in water for example. However, be careful to consider realistic situations where not too much is dissipated (due to production of steam).

A range of questions involving specific heat capacity.

Episode 607-4: Specific heat capacity: some questions (Word, 30 KB)

Episode 607-5: Thermal changes (Word, 46 KB)

Using an aluminium block and immersion heater to estimate the specific thermal capacity (also called the 'specific heat capacity') of aluminium.

**Class practical**

Using an aluminium block and immersion heater to estimate the specific thermal capacity (also called the specific heat capacity

) of aluminium.

Apparatus and Materials

*For each group of students*

- Aluminium calorimeter with holes for heater and thermometer (see discussion below)
- Thermometer -10°C to 110°C
- Stopwatch or stopclock
- Immersion heater, 12 V 100 W (older, 60 W types will do)
- Low voltage power supply or transformer (to supply 8A)
- Lever-arm or domestic balance (+/- 2g)
- Insulation/cladding for the metal block OPTIONAL

Health & Safety and Technical Notes

The immersion heaters should have been allowed to cool in air after heating water, to eliminate the (small) risk that water has been drawn inside through a cracked seal.

Read our standard health & safety guidance

If bespoke insulation is not available, then scraps of material and or newspaper can be held on with string/elastic bands to provide a thick insulating jacket

for the block.

If you drop some paraffin-oil into the thermometer hole it will ensure good thermal contact between the block and the thermometer. It is not necessary to use oil with the immersion heater. In fact, as there is a danger of cracking

any oil which is left on the heater when it is removed from the block, it is wiser not to use it.

Procedure

- Find the mass of the aluminium block on the balance. Place a small drop of oil in the thermometer hole. (This will provide good thermal contact between the block and the thermometer bulb.) Insert the thermometer and immersion heater in the appropriate holes. Read the thermometer. Connect the heater to the 12 volt supply and switch it on for 5 minutes. Note the maximum temperature rise obtained after the supply has been switched off.
- Many suppliers can provide similar 1 kg blocks made of steel, copper, brass etc. If these are all set up at the same time they will show that different materials of the same mass will achieve different temperature rises when the same amount of energy is transferred to them.

Teaching Notes

- Change in energy stored thermally (due to the temperature rise) =
*mass*x*specific thermal capacity*x*temperature rise* - The temperature of I kilogram of aluminium rises about four times that of a kilogram of water. If the heater does not behave differently in aluminium compared to water there must be another factor which is peculiar to the aluminium. This is the specific thermal capacity (also called
specific heat capacity

) of the aluminium. - The specific thermal capacity of aluminium is 900 J/kg °C
- The specific thermal capacity of water is 4200 J/kg °C
- It takes more energy to raise the same temperature of water by each °C than it does to raise the temperature of the same mass of aluminium.
**How Science Works extension:**After collecting data, students calculate the specific thermal capacity of the aluminium (or other material) used. To assess the accuracy of their measured data, they can compare their value of specific thermal capacity with its accepted (true

) value from data tables. You can also ask them to calculate the percentage difference between the two values, to show how the accuracy of measurements can be expressed quantitatively. Differences between the two values can also be used to prompt a discussion about errors and uncertainties in their measurements, identifying the main sources.- Energy dissipated so that it is stored thermally in the surroundings is something that students can investigate further, to obtain a more accurate value of the specific thermal capacity. The Guidance note:

suggests a procedure for controlling such transfers. A less sophisticated, but equally valid, approach is to repeat the experiment (the block needs time to cool}, using insulation around the block. Their second set of data will enable them to assess whether this gives a more accurate result for specific thermal capacity. - Before you make this comparison remember that power supplies may only give unidirectional potential differences and not fully smoothed values. The power measured is on DC meters as VI is only 0.8 of what it should be. See the guidance note...
Explaining rms voltage and current

*This experiment was safety-tested in December 2006*

Measuring the specific thermal capacity (also called the 'specific heat capacity') of aluminium, including the use of a cooling correction.

Energy and Thermal Physics

Practical Activity for 14-16

**Class practical**

Measuring the specific thermal capacity (also called the specific heat capacity

) of aluminium, including the use of a cooling correction.

Apparatus and Materials

*For each group of students*

- Immersion heater, 12 V 100 W (older, 60 W types will do)
- Aluminium calorimeter with holes for heater and thermometer (see discussion below)
- Ammeter (0-5 amp)
- Stopwatch or stopclock
- Thermometer
- Voltmeter, 0-15 volt (see discussion below)
- Rheostat (10-15 ohms, rated at 5A or more)
- Top pan balance
- 12 volt supply, i.e. LV power supply with high current smoothing unit

Health & Safety and Technical Notes

Read our standard health & safety guidance

If you drop some paraffin-oil into the thermometer hole it will ensure good thermal contact between the block and the thermometer. It is not necessary to use oil with the immersion heater. In fact, as there is a danger of cracking

any oil which is left on the heater when it is removed from the block, it is wiser not

to use it.

Procedure

- Find the mass of the aluminium block using the top pan balance. Connect the immersion heater to the 12-volt supply in series with an ammeter and a rheostat. The immersion heaters are 12 volt, 60 or 100 watt, so adjust the rheostat to give a current of about 4 amps. Switch the heater off.
- Insert the immersion heater in the aluminium block and place the thermometer into its hole. Before switching on for the experimental run, wait for five minutes before taking the temperature of the block. Switch on the heater and start the clock.
- The easiest way to measure the temperature rise is to leave the heater switched on until a rise of about 10°C is achieved. Switch off the heater and continue to monitor the temperature until it begins to fall. Note the maximum temperature reached by the block.
- A more accurate method is to take temperature readings every half minute and to plot a graph of the results both whilst the heater is on and for approximately the same time after the heater is switched off. A cooling correction can be applied to the temperature rise measured, using a standard technique.

Teaching Notes

- The specific thermal capacity can be determined from the relationship:
*mass x specific thermal capacity x rise in temperature/ time = current x p.d.* - The ratio
*temperature rise / time*can be obtained from the slope of a graph of*temperature*plotted against*time*. - The aluminium blocks can be lagged by enclosing them in foamed polystyrene.
- You can attach the immersion heater to a joulemeter and so measure directly the energy transferred from the electrical supply to the material.
- The choice of power supply makes a lot of difference in this experiment. You will get an accurate result with AC supplies and AC meters because the AC meters have been constructed to read correct values.
- However, most school power supplies providing currents high enough to warm up immersion heaters are not smoothed at all and the DC terminals give voltages which are extremely bumpy. External smoothing units will help.
- If you use moving coil meters, students will need to make a correction for the electrical power calculation.
- A moving coil meter reads time-averaged values, which are (2/π) x peak value.
- So actual power = 1.2 x power calculated from meter readings.
- For further information, see the guidance note:
Explaining rms voltage and current

*This experiment was safety-tested in August 2007*

Transferring energy mechanically to lead shot and measuring its temperature rise.

**Class practical**

Transferring energy mechanically to lead shot and measuring its temperature rise.

Apparatus and Materials

*For each group of students*

- Lever-arm or top pan balance (+/- 10g is sufficiently accurate)
- Cardboard tube (approx 50 - 100 cm long)
- Lead shot, 500 g
- Plastic or cardboard cups
- Thermometer
- Metre rule
- Corks or bungs to fit tube, 2

Health & Safety and Technical Notes

There is no need to handle the lead shot. However, if it is spilt and is collected by hand (for example), the hands must be washed thoroughly before eating.

Read our standard health & safety guidance

Procedure

- Measure out about 500 g of lead shot in one of the plastic or cardboard cups.
- Put a thermometer into the lead shot to find the temperature.
- Seal one end of the cardboard tube with a cork or bung. Put the lead shot into the tube and seal the open end.
- Measure the length of the tube, Δh.
- Turn the tube over 20, 40 or 50 times, counting the number. Quickly pour the shot into the cup and measure the maximum temperature again.

Teaching Notes

- When the lead is falling take care that it falls vertically rather than sliding along the tube, when friction will come into play. Also a hand should not cushion the bottom of the tube when the shot hits the bottom, otherwise
energy

will betransferred

into the hand. - Turning the tube over more times in order to achieve higher temperature rises is not advisable. The longer time and higher temperature differences will allow more thermal transfer to the surroundings.
- The energy transferred is equal to the number of falls
*(n) x mgΔh*and therefore*n x mg x Δh = m x specific thermal capacity x temperature rise*. - Note that there is no need to record the mass of the lead,
*m*. - Lead is used because its specific thermal capacity is about 1/30 that of water. It is the thermal capacity per unit mass which is important. (Most metals have approximately the same thermal capacity per unit volume.) It is also an inelastic metal so that the energy of the shot stored gravitationally is then stored thermally. The temperature rises.
- When Joule was investigating energy conservation he is said to have measured the temperature at the top and bottom of a waterfall on his honeymoon in Switzerland. This is a useful model of his experiment.

*This experiment was safety-tested in January 2006*

The kinetic theory section of the energy episodes includes both Boyles law & the other ideal gas laws.

Specifically episode 601-4 describes relationship between pressure and volume. It looks at it both macroscopically and microscopically with reference to density of the gas (number of particles) which is not traditional

Properties of Matter

Lesson for 16-19

- Activity time 140 minutes
- Level Advanced

This episode looks at Brownian motion as evidence for the particulate nature of matter, and the macroscopic gas laws.

Lesson Summary

- Demonstration and discussion: Brownian motion and what this tells us about air (and other gases) (30 minutes)
- Demonstration and student experiments: The gas laws (60 minutes)
- Discussion: Boyle’s law – a particle explanation (20 minutes)
- Discussion: Extrapolating to absolute zero (10 minutes)
- Student activity: A computer model of Boyle’s law (20 minutes)

A reasonable place to start is by reviewing the evidence that matter is made up of particles (molecules and atoms). (It is probably best to refer to particles

in general, and to think of them as spherical; point out that, by this, we mean either atoms or molecules.) Students will have met these ideas at a lower level in chemistry as well as physics. Evidence includes the combination laws of gases, and Brownian motion, which can be demonstrated in the classroom.

According to your students’ previous experience, you may wish to demonstrate Brownian motion, the expansion of bromine into a vacuum, and a measurement of the density of air.

Episode 601-1: Brownian motion in a smoke cell (Word, 58 KB)

Episode 601-2: Diffusion of bromine (Word, 72 KB)

Episode 601-3: The density of air (Word, 72 KB)

Brownian motion is evidence not just for the existence of atoms or molecules but also for their movement, which is random. This random motion can be modelled mathematically and leads to a test for the size of the atoms from measured diffusion rates – this was one of Einstein’s great papers of 1905.

Question: What is the mass of air in the room? Answering this will require the estimation of the volume of the room, and the use of the density of air. Students are often surprised by the result: perhaps 100 kg, more than the mass of a typical student. If all of the air in the room condensed to form a liquid, it would make a layer perhaps 5 mm deep on the floor.

Now you can introduce a gas as a simple system. In crystalline solids, all the atoms are nicely ordered in an array, making calculations possible. In a gas, the motion is random and again this simplifies calculations as the laws of statistics can be applied. (Disordered solids and liquids are more difficult to treat mathematically, because they are neither well-ordered nor completely disordered.)

Now move on to the gas laws. Strictly speaking it is only necessary to look at Boyle’s law (*P**V* = constant at constant temperature).

You may well have an apparatus specifically designed to show this such as an oil-filled column attached to a pump and pressure gauge. You pump air in to pressurise the oil, which compresses an air space at the top. A series of about 10 *P* and *V* readings usually gives a good fit to a straight line when *P* is plotted against 1*V*.

Episode 601-4: Changes in volume, changes in pressure (Word, 58 KB)

The other laws follow from this and the definition of *thermodynamic* temperature. However, at this stage it is likely that the only idea of temperature students are familiar with is that which is measured by a thermometer

. (This is not a totally silly definition as it relies on the fact that when two bodies of different material, temperature and size are in contact, their temperatures equalise. This is usually referred to as the Zeroth Law of Thermodynamics.)

So it is worth pressing ahead with demonstrations or class experiments of Charles’ law (*V* proportional to *T*) and/or the pressure law (*P* proportional to *T*).

Episode 601-5: Changes in temperature, changes in pressure (Word, 96 KB)

Episode 601-6: Changes in temperature, changes in volume (Word, 112 KB)

You may need to discuss how Boyle’s law can be explained in terms of particles.

Episode 601-7: Boyle’s law, density and number of molecules (Word, 65 KB)

Both Charles’ law and the pressure law lead to the concept of an absolute zero of temperature. Note that Absolute Zero (0 K) is the temperature at which the energy of the particles of a material has its minimum value; this is not zero, as the particles have so-called zero point energy due to quantum effects, which cannot be removed. They still vibrate.

Remember real gases turn into liquids and solids before absolute zero is reached. However extrapolating back often gives reasonable values.

Episode 601-8: Changing pressure and volume by changing temperature (Word, 53 KB)

Episode 602: Ideal gases and absolute zero (Word, 56 KB)

If suitable apparatus for demonstrating Boyle’s law is not available it can be simulated using computer packages or applets but this is a poor relation to the real experiment and should only be used to illustrate the effect, not to demonstrate it.

See a model on the

Uses standard Boyles Law apparatus to show the relationship between pressure and volume of a gas at fixed temperature.

**Demonstration**

Apparatus and Materials

- Boyle's law apparatus
- Foot pump and adaptor
- Kinetic theory model kit (transparent cylinder with small steel balls)

Health & Safety and Technical Notes

It has been known for the glass tube to fly upwards when the gas is at maximum pressure. To prevent this, check the compression joint holding the tube and any tube supports before use. (The apparatus is filled and emptied by removing the pressure gauge.)

Read our standard health & safety guidance

The apparatus has been specially designed to give quick, clear readings which the class can see.

A sample of dry air is confined in a tall, wide glass tube by a piston of oil. The volume is found from the length of the air column, which should be clearly visible at the back of the class.

The pressure is read from a Bourdon gauge connected to the air over the oil reservoir. This is calibrated to read absolute pressure and is also visible from the back of the class.

The foot pump is attached to the oil reservoir and is used to change the pressure. The gauge reads up to 3 x 10 ^{ 5 } N m ^{ -2 } and the pressure can safely be taken up to this value but must not be taken beyond.

To fill the apparatus with oil, unscrew the Bourdon gauge with a spanner and fill the chamber with a low vapour pressure oil. Tilt the apparatus in the final stage of filling in order to get enough oil into the main tube. When refixing the gauge, tighten the nut to get a good seal, but not so much that the thread is damaged.

Procedure

- Give a quick demonstration to show that doubling the pressure halves the length of the air column, and so its volume.
- Increase the pressure to its maximum value, and then record it and the (minimum) length of the air column.
- Next, disconnect the pump and release a little air using the valve on the oil reservoir, so that the oil level in the tube falls a few centimetres.
- Before taking the next pair of readings, wait a while so that the air temperature recovers and the oil left behind has fallen down the wall of the tube.
- Keep repeating step 3 until the gauge returns to atmospheric pressure.

Teaching Notes

- It is important to ensure that students have grasped that the volume of the air column is directly proportional to its length, so that the way the length changes tells us how the volume alters. (It is not hard to get a good estimate of the internal diameter of the tube, if finding the cross-sectional area of the air column would help.)
- It is helpful if students plot a graph of pressure (
*P*) against lengths of air columns (*V*). This can lead them to see that trying a graph of*P*against*I/length (I/V)*might be a good idea. - Students might then also use a spreadsheet to find how the product of pairs of values
*P x length (P x V)*compare.

*This experiment was safety-tested in January 2006*

Measuring the change in pressure when air is heated at constant volume.

**Demonstration**

Measuring the change in pressure when air is heated at constant volume.

Apparatus and Materials

- Crushed ice
- Bourdon gauge
- Round-bottomed flask, 250 ml
- Rubber bung and tubing
- Aluminium container or water bath
- Tripod
- Electric kettle to provide hot water OR Bunsen burner and tripod
- Stand and clamp to hold the flask down
- Heatproof gloves
- Rotary vacuum pump (optional)
- Thermometer -10°C to 110°C

Health & Safety and Technical Notes

Heatproof gloves will be needed to handle the flask and water bath after the experiment unless they can be left to cool down.

If a rotary vacuum pump is used, remember that it is too heavy for one person to lift or carry.

Read our standard health & safety guidance

You can improve the accuracy of measurements in this demonstration by ensuring that the neck of the flask is in the water. This probably means using a clamp to hold the flask in the water.

Procedure

- Connect the Bourdon gauge to the flask.
- Note the gauge reading when the flask is first immersed in cold water (preferably at or near the freezing point). Then note it with the flask in hot water (preferably at or near the boiling point).

Teaching Notes

- As the air inside the flask is cooled, its molecules move more slowly. Collisions with the walls become less frequent and less violent, meaning pressure falls. When the air is warmed, molecules move faster. Collisions with the walls become more frequent and more violent, meaning pressure rises.
- If you measure the water temperature, you can take a set of pressure readings against temperature and plot a graph. The Bourdon gauge scale will probably have to be interpolated because it may not be sensitive enough. About 40 kN m
^{ -2 }pressure change should be obtained between 0 and 100°C. If you start counting temperature at minus 273°C the line will pass through the origin, showing that pressure is proportional to temperature. - If you attach a T-piece and tap or clip to the flask, you can pump out about two-thirds of the air. Then you can try the experiment at another density. The change in pressure should be in the same proportion.
- Having noticed that the pressure falls as the temperature decreases, ask whether we could predict a temperature at which the pressure would be zero. (To get students to see this is not a daft idea, get them to consider what happens to the motion of the molecules as the temperature falls.)
- Either by calculation or - better - by drawing onto an extra sheet of graph paper, get them to extrapolate to find values of the temperature at which the pressure would be zero. Discuss its significance.
- The Bourdon gauge may be calibrated in a variety of units, lb in
^{ -2 }, kg m^{ -2 }, N m^{ -2 }, etc. Check that you can translate these into units your students are used to. See CLEAPSS Laboratory Handbook, section 20.24.

*This experiment was safety-tested in August 2007*

This resource provides a method for investigating the expansion of gas at constant pressure using hot and cold water from the tap.

**Class practical**

Apparatus and Materials

*For each group of students*

- Washing-up bowls with hot and cold water from taps
- Round-bottomed flask
- Bung with narrow-bore tubing to fit
- Access to lubricating oil

Health & Safety and Technical Notes

Warn the class to handle the long glass tubes carefully as they are (relatively) easily broken.

Read our standard health & safety guidance

Glass test-tubes and corks with capillary tubing can be used in place of the flasks. The drop of oil is added to the bung-end of the tube before insertion into the flask.

Procedure

- Trap the air in the flask with a small bead of oil in the glass tubing. Gently heat the flask with your hand. This will produce a sufficient temperature rise for the oil index to move up the tubing.
- Plunge the flask first into cold and then into warm (not hot) water.

Teaching Notes

- The oil plug will rise up the glass tube as the air in the flask expands (in warm water) and fall as the air contracts (in cold water). The volume expansion of a gas is approximately 500 times that of glass, so it is unlikely that the expansion of the flask will have any noticeable effect.
- Students may already know that a gas is made up of rapidly moving molecules which hit the surfaces of the container and exert a force on the container so creating a pressure. If the temperature of the container is raised, molecules move faster. The pressure exerted by the gas will therefore increase.
- Plunging the flask into hot water may increase the volume of the gas so much that either the oil plug flies out of the tube, or it breaks up and runs down the inside of the tube.

*This experiment was safety-tested in August 2006*

**Also:**

**Absorption of α or β or γ radiation (OCR)**

**and**

**Investigation of the inverse square law for gamma radiation (AQA)**

Quantum and Nuclear

Lesson for 16-19

- Activity time 65 minutes
- Level Advanced

The focus of this episode is the properties of ionizing radiation. It is a good idea to introduce these through a consideration of safety.

Lesson Summary

- Discussion: Ionising radiation and health (10 minutes)
- Demonstration: Deflection of beta radiation (10 minutes)
- Student activity: Completing a summary table (10 minutes)
- Student experiment: Inverse square law for gamma radiation (30 minutes)
- Discussion: Safety revisited (5 minutes)

Why are radioactive substances hazardous? It is the ionising property of the radiation that makes it dangerous to living things. Creating ions can stimulate unwanted chemical reactions. If the radiation transfers sufficient energy it can split molecules. Disrupting the function of cells may give rise to cancer. Absorption of radiation exposes us to the *risk* of developing cancer.

Thus it is prudent to avoid all unnecessary exposure to ionising radiation. All deliberate exposure must have a benefit that outweighs the risk.

Radioactive *contamination* is when you get a radioactive substance on, or inside, your body (by swallowing it or breathing it in or via a flesh wound). The contaminating material then irradiates you.

How can you handle sources safely in the lab? Point out that you will be safe if you follow your local rules which will incorporate the following:

- Always handle sources with tongs
- Point the sources away from your body (and not at any anybody else)
- Fix the source in a holder which is not adjacent to where your body will be when you take measurements
- Replace sources in lead-lined containers as soon as possible
- Wash hands when finished

Follow the local rules for using radioactive sources, in particular do not handle radioactive sources without a tool or place them in close proximity to your body.

Show the deflection of β by a magnetic field. (Make sure you have a small compass to determine which are the N and S poles of the magnet.) Is the deflection consistent with the LH rule? (Yes; need to recall that electron flow is the opposite of conventional current.) This demonstration is no good with the α particles, as they are absorbed too quickly by the air.)

NB The diagram above is common in textbooks, but is *only* illustrative. For the curvature shown of beta particles, the curvature of the alpha tracks would be immeasurably small.

Display the table, with only the headings and first column completed. Ask for contributions, or set as a task; compile results.

Episode 510-1: α, β and γ radiation (Word, 39 KB)

name | symbol | nature | electric charge | stopped by | ionising power | What is it? |
---|---|---|---|---|---|---|

Alpha | α | particle | +2 | mm air; paper | very good | He nucleus |

Beta | β | particle | -1 | mm Al | medium | very fast electron |

Gamma | γ | wave | 0 | cm Pb* | relatively poor | electromagnetic radiation |

Can you see any patterns in the table? (Most ionising – the largest electrical charge – is the least penetrating.) Can you explain this? (The most ionising transfer energy the quickest.) How can the electrical charge determined? Deflection in a magnetic field.)

*NB Gamma radiation is never completely absorbed (unlike alpha and beta) it just gets weaker and weaker until it cannot be distinguished form the background.

Note: since you are unlikely to have sufficient gamma sources for several groups to work simultaneously, this experiment can be part of a circus with others in the next episode. Alternatively, it could be a demonstration.

Gamma radiation obeys an inverse square law in air since absorption is negligible. (Radiation spreads out over an increasing sphere. Area of a sphere = 4 π *r*^{ 2} , so as *r* gets larger, intensity will decrease as 1/ *r*^{ 2} . The effect of absorption by the air will be relatively small.

Episode 510-2: Range of gamma radiation (Word, 38 KB)

(Some students could do an analogue experiment with light, with an LDR or solar cell as a detector.)

When detecting g radiation with a Geiger tube you may like to aim the source into the *side* of the tube rather than the window at the end. The metal wall gives rise to greater secondary electron emission

than the window and this increase the detection efficiency.

Correct readings for background.

How can we get a straight line graph? We expect

*I* ∝ *r*^{ -2},

so a plot of *I* versus *r*^{ –2} should be direct proportion (i.e. a straight line through the origin). It is much easier to see if a graph is a straight line, rather than a particular curve. Lift the graph and look along the line – it’s easy to spot a trend away from linear. However, two points are worth noting:

- Sealed γ sources do not radiate in all directions, so do not expect perfect 1
*r*^{ 2}behaviour - You do not know exactly where in the Geiger tube the detection is taking place, so plotting
*I*^{ - ½ }against*r*gives an intercept, the systematic error in the measurement.

Return briefly to the subject of safe working. Background radiation is, say, 30 counts per minute. How far from a gamma source do you have to be for the radiation level to be twice this? Would this be a safe working distance? (Probably.) How much has your lifetime dose of radiation been increased by an experiment like the above? (Perhaps one hour at double the background radiation level – a tiny increase. It will be safe enough to carry out a few more experiments.)

**Also:**

**Absorption of α or β or γ radiation (OCR)**

**and**

**Investigation of the inverse square law for gamma radiation (AQA)**

Quantum and Nuclear

Lesson for 16-19

- Activity time 40 minutes
- Level Advanced

This gives students the opportunity to work with radioactive sources.

Lesson Summary

- Student experiments: Absorption of radiation and report back (40 minutes)
- Demonstration: Absorption of radiation by living matter
- Student experiment (optional)

Groups could work in parallel and report back to a plenary session.

Remind them to correct for the background count (taken at least twice – at the start and end of the main experiments and the two results averaged).

Range of alpha radiation

Episode 511-1: Use a spark counter (Word, 62 KB)

Range of beta radiation

Episode 511-2: The range of beta particles in aluminium and lead (Word, 38 KB)

Range of gamma radiation

An optical analogue for the absorption of γ particles by lead is the absorption of light by successive microscope slides.

Episode 511-3: Absorption in a liquid (Word, 54 KB)

Absorption of γ particles is an example of exponential decrease – check the data for a constant half thickness

, thus suggesting the type of physics involved. (Each mm of absorber is reducing the intensity by the same fraction.)

Episode 511-4: Absorbing radiations (Word, 38 KB)

To simulate the absorption of radiation by living matter use slices of different vegetables as absorbers, or a slice of bacon to represent human flesh.

Episode 511-5: Absorption in biological materials (Word, 53 KB)

The first requires a sealed radium-226 source. Because Ra-226 is the parent to a chain of radioactive daughters, granddaughters and so on, you get a mixture of αs, βs and γs emitted. Challenge students to use absorbers to establish that all three radiation types are being emitted. (The maximum energies are:
*E*_{ α } = 7.7 MeV
*E*_{ β } = 3.3 MeV
*E*_{ γ } = 2.4 MeV)

The second is an extension of the β absorption experiment. You could speculate that some β particles might be back-scattered

(like Rutherford’s α particle scattering that first demonstrated the existence of the nucleus). A quick try shows that some β particles are indeed back-scattered.

Follow the local rules for using radioactive sources, in particular do not handle radioactive sources without a tool or place them in close proximity to your body. Deliberately placing a radioactive source in contact with the skin would increase your dose of ionising radiation unnecessarily and increase the risks to your health. This is a criminal offence.

**Also:**

**Absorption of α or β or γ radiation (OCR)**

**and**

**Investigation of the inverse square law for gamma radiation (AQA)**

**Demonstration**

Gamma radiation is part of the electromagnetic spectrum. It is not absorbed by the air, but its intensity decreases because it spreads out. Therefore, the intensity varies with the inverse square of distance: it follows an inverse square law. You can show this in the laboratory and use it as evidence to support the fact that gamma radiation is a part of the electromagnetic spectrum.

Apparatus and Materials

- Holder for radioactive sources
- Geiger-Müller tube
- Holder for Geiger-Müller tube
- Scaler
- Metre rule
- Sealed "pure" gamma source, cobalt-60 (
^{60}Co), 5 μCi or sealed radium source - Set of absorbers (e.g. paper, aluminium and lead of varying thickness)

Health & Safety and Technical Notes

See guidance notes on...

Managing radioactive materials in schools

Read our standard health & safety guidance

Note that 5 μCi is equivalent to 185 kBq.

Cobalt-60 is the best pure gamma source. However, you may have a sealed radium source in your school. This gives out alpha, beta and gamma radiation. You can use it for this experiment by putting a thick aluminium shield in front of it. This will cut out the alpha and beta radiations.

An alternative is to try using a Geiger-Muller tube sideways. The gamma radiation will pass through the sides of the tube but alpha and beta will not. You can do a quick check by doubling and tripling the distance between the source and the axis of Geiger-Muller tube and seeing if the count follows an inverse square law (by dropping to a quarter and a ninth).

Using the Geiger-Muller tube sideways has an added advantage that you have an accurate measure of where the distance is zero. It is along the axis of the tube.

Education suppliers stock a set of absorbers ranging from tissue paper to thick lead. This is a useful piece of equipment to have in your prep room. You can make up your own set. This should include: tissue paper, plain paper, some thin metal foil (e.g. cigarette paper, wrapping from a chocolate from an assortment box and a small piece of gold leaf}

Procedure

**Setting up...**

- Set up the Geiger-Muller tube and attach it to the scaler.
- Clamp a metre rule to the bench and line it up with your zero point (in the Geiger-Muller tube).
- With some Geiger-Muller tubes, the gamma radiation will pass through the side. So set the Geiger-Muller tube up at right angles to the metre rule. The zero point is then the axis of the tube.
- You can check your zero point by doing some quick readings before the lesson. When you double the distance, the count should be a quarter. If it is more than a quarter, then move the tube towards the source to re-zero it. If it is less than a quarter, then your zero point is closer than you reckoned: move the tube away from the source to re-zero it.
- Measure the background count with the source far away.
- Start with the gamma source 10 cm from the zero point.
- Increase the distance and take measurements of count rate at 20 cm, 30 cm, 40 cm, 60 cm and 80 cm.
- Correct the count rates for the background count.
- Plot a graph of corrected count-rate against distance. You could use a spreadsheet program to do this.

Teaching Notes

- The shape of the graph shows that count rate decreases with distance. You can show that it is an inverse square by checking that the count rate quarters when the distance doubles (10 cm to 20 cm; 20 cm to 40 cm; 30 cm to 60 cm), falls to a ninth when it trebles (10 cm to 30 cm; 20 cm to 60 cm) and drops to a sixteenth when the distance is quadrupled (10 cm to 40 cm; 20 cm to 80 cm). (This is only true assuming the source is a small area compared with the cross-section of the detector. Keep minimum distance large!)
- A graph of count rate against 1distance
^{2}is a straight line. - This is the same law that governs all electromagnetic radiation (see, for example

This is some evidence that gamma radiation is part of the electromagnetic spectrum. - The moral of this story is that in order to protect yourself from gamma radiation the best thing to do is to move farther away. At 10 times the distance you will be 100 times as safe.

*This experiment was safety-tested in May 2006*

**Demonstration**

This demonstration focuses on the properties of gamma radiation. You can show that it is much more penetrating than alpha or beta radiation and has a much longer range.

Apparatus and Materials

- Holder for radioactive source
- Geiger-Müller tube
- Holder for Geiger-Müller tube
- Scaler (if needed by Geiger-Müller tube)
- Sealed "pure" gamma source, cobalt-60 (
^{60}Co), 5 μCi or sealed radium source - Set of absorbers (e.g. paper, aluminium and lead of varying thickness)

Health & Safety and Technical Notes

See guidance notes on...

Managing radioactive materials in schools

Read our standard health & safety guidance

Note that 5 μ Ci is equivalent to 185 kBq.

Cobalt-60 is the best gamma source. However, you may have a sealed radium source in your school. This gives out alpha, beta and gamma radiation. You can use it for this experiment by putting a thick aluminium shield in front of it. This will cut out the alpha and beta radiations.

An alternative is to try using a Geiger-Muller tube sideways. The gamma radiation will pass through the sides of the tube but alpha and beta radiation will not. Some gamma particles interact with the tube wall and knock electrons into the tube gas, where they are detected. This effect enhances the detection efficiency of the gamma particles. You can do a quick check by doubling and tripling the distance between the source and the axis of the Geiger-Muller tube and seeing if the count follows an inverse square law (by dropping to a quarter and a ninth).

Some education suppliers now stock all-in-one Geiger-Muller tubes with a counter. See e.g.

Education suppliers stock a set of absorbers that range from tissue paper to thick lead. This is a useful piece of equipment to have in your prep room. You can make up your own set. This should include: tissue paper, plain paper, some thin metal foil (e.g. cigarette paper, wrapping from a chocolate from an assortment box and a small piece of gold leaf}

Procedure

- Set up the Geiger-Muller Tube and attach it to the scaler if needed.
- Put the source in its holder and clamp it a few centimetres from the Geiger-Muller tube.
- Show that the gamma radiation has a long range in air - at least 80 cm. You could show that the count is falling off with distance, and gets smaller and smaller rather than stopping altogether.
- Show that the gamma radiation will penetrate paper, cardboard, aluminium and thin lead, but is greatly reduced by thick lead.

Teaching Notes

- The moral of this story is that in order to protect yourself from gamma radiation the best thing to do is to move a long way away.
- Discuss the uses of gamma radiation in industry and for medical imaging and treatment. The applications are based on its penetrating power.
- Remind students that gamma radiation is much less ionising than alpha.

*This experiment was safety-tested in August 2007*

This demonstration focuses on the properties of alpha particles. It follows on closely from the experiment Identifying the three types of ionising radiation

**Demonstration**

This demonstration focuses on the properties of alpha particles. It follows on closely from the experiment Identifying the three types of ionising radiation.

Apparatus and Materials

- Power supply, EHT, 0–5 kV (with option to bypass safety resistor)
- Spark counter (or Geiger-Müller tube and counter
- Sealed pure alpha source, plutonium-239 (
^{ 239}Pu), 5 μCi (if available) or sealed (semi-pure) alpha source, americium-241 (^{241}Am), 5μCi - Holder for radioactive source (e.g. forceps)
- Connecting leads
- Set of absorbers (e.g. paper, aluminium and lead of varying thickness)
Alpha particle tracks showing their short range

Health & Safety and Technical Notes

See guidance note on...

Managing radioactive materials in schools

A school EHT supply is limited to a maximum current of 5 mA., which is regarded as safe. For use with a spark counter, the 50 MΩ. safety resistor can be left in the circuit. This reduces the maximum shock current to less than 0.1 mA..

Although the school EHT supply is safe, shocks can make the demonstrator jump. It is therefore wise to see that there are no bare high voltage conductors. Use female 4 mm connectors where required. Read our standard health & safety guidance...

Read our standard health & safety guidance

Note that 5μCi is equivalent to 185 kBq.

Sealed sources for radium and plutonium are no longer available. However, if you have them in your school, you can use them as long as you follow your school safety policy and local rules.

If you do not have a pure alpha source (
^{
239
}
Pu), you need to be careful about trying to show the properties of alpha using a Geiger-Müller tube. The radiation from a mixed source like
^{
241
}
Am can penetrate aluminium and has a long range. This is because it gives out gamma as well as alpha radiation:

Radioactive sources: isotopes, radiation and availability

The most effective way of demonstrating the properties of alpha radiation is to use the spark counter. If you do not have a pure alpha source (i.e. you are using radium or americium-241), this is the recommended method because the spark counter does not respond to beta or gamma radiation:

The Geiger-Müller tubes are very delicate, especially if they are designed to measure alpha particles. The thin, mica window needs a protective cover so that it is not accidentally damaged by being touched.

Education suppliers stock a set of absorbers that range from tissue paper to thick lead. This is a useful piece of equipment to have in your prep room. You can make up your own set. This should include: tissue paper, plain paper, some thin metal foil (e.g. cigarette paper, wrapping from a chocolate from an assortment box, and a small piece of gold leaf}.

Teaching Notes

- This experiment can be done in conjunction with
Beta radiation: range, and stopping

andGamma radiation: range and stopping

You might decide to merge these three experiments with {Identifying three types of ionizing radiation}{/identifying-three-types-ionizing-radiation} so that you do the range of all three types of radiation. You can then show the effects of a magnet on beta radiation separately. - You should find that the range of the alpha particles is between 3 and 10 cm. The alphas from americium have a range of about 3 cm, from plutonium 5 cm, and the most energetic ones from radium, 7 cm. Refer to the Diffusion cloud chamber experiment to reinforce this evidence.
- You should find that the alpha particles are stopped by anything except the very thinnest of paper or foil leaf. The gold leaf reduces the range of the alpha particles, because they lose energy getting through the gold leaf.
- Remind students that this is alpha radiation, which is the most ionizing of the three main ionizing radiations. Link this with the observations that you have made. Alpha radiation collides with and ionizes a lot of particles in the material through which it passes. Because of this, it loses its energy quickly and is slowed down and absorbed.
- Refer to cloud chamber photographs of alpha particle tracks, showing them being deflected in a magnetic field
- An alpha particle is a helium nucleus with two positively charged protons and two neutral neutrons. The atomic mass of the radiating atom falls by four units when an alpha particle is emitted. The speed of an alpha particle can be up to 15 x 10
- You can discuss the dangers of radioactivity in general. Radiation harms people by making ions in our flesh and thereby upsetting or killing cells. The more ionizing the radiation, the more harmful it is. This makes sources of alpha radiation very hazardous – especially if they are ingested.
- Relate the hazard to the safety precautions that you are taking during the demonstration.

Alpha particle tracks bent in a very strong magnetic field

The deflection is too small to measure in the school laboratory, but shows that they have a positive charge. The small deflection shows that they have a relatively large mass. Collisions with helium produce 90° forks showing that they have the same mass as a helium (nucleus). You can say that

Alpha particle tracks including a collision with a helium nucleus

If the students have already met the idea of nuclei, then you can call alpha particles a helium nucleus.

*This experiment was safety-tested in August 2006*

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

Forces and Motion
| Light, Sound and Waves

Lesson for 16-19

- Activity time 115 minutes
- Level Advanced

This episode and the next focus on practical simple harmonic motion (SHM) systems.

Lesson Summary

- Discussion: Hooke’s law leads to SHM (15 minutes)
- Student experiment: Testing the relationship
*T*= 2 π*m**k*} (30 minutes) - Student activity: Using a computer model (20 minutes)
- Discussion: Modelling in physics (10 minutes)
- Worked example: Applying the relationship
*T*= 2 π*m**k*} (10 minutes) - Student questions: Calculations involving
*T*= 2 π*m**k*} (30 minutes)

If your specification requires it, here is where you can derive the expression for the period of a mass-spring system.

We have
*ω* = 2 π *f*, so
*ω* = *F* *mx* as the requirement for SHM, and *F* = *k* × *x* from Hooke’s law.

It is easiest to deal with a horizontal mass-spring system first (because you can ignore gravity).

Students can test the relationship

*T* = 2 π *m* *k* for a mass-spring system. (Note that this expression is independent of *g*.)

Students may find that there is a systematic error, caused by the finite mass of the spring. Try modifying the simple theory to take into account the mass of the spring *m*_{S} :

*T* = 2 π *m + m_{S}*

Episode 303-1: Loaded spring oscillator (Word, 59 KB)

They can also use a computer model.

Episode 303-2: Oscillating freely (Word, 26 KB)

and the model

Episode 303-3: Modelling springs and masses (Word, 24 KB)

The study of SHM may well be the first occasion that students meet detailed mathematical modelling. It may be worth spelling out for them what is happening. There is a physical behaviour we want to understand. First we simplify the actual situation to an idealised physical model by making assumptions, e.g. no friction

, pendulum strings or springs that have no mass, etc. Then we make a mathematical model to represent the physical model. The mathematical model is then analysed (solved

) and has to be interpreted in terms of the physical model. Experiments try to mirror the physical model but they cannot do this exactly (e.g. make a pendulum string as light as possible while still being strong enough to support the bob

). So care is needed when comparing the theory with experimental measurements.

A vibrating atom in a solid can be modelled as a mass m between two tensioned springs, the springs representing the interatomic forces.

For typical interatomic forces *k* = 60 N m^{-1}

Mass of an atom (Na in NaCl) ~ 3.8 × 10^{-26} kg

Estimate the natural vibration frequency of atoms.

*f* = 1*T*

*f* = 12 π × *2k* *m*

*f* ~9 × 10^{12} Hz, which is in the IR region of the electromagnetic spectrum. We will return to atomic vibrations when discussing resonance.

These questions reinforce basic ideas about SHM.

Episode 303-4: Harmonic oscillators (Word, 35 KB)

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

Forces and Motion
| Light, Sound and Waves

Lesson for 16-19

- Activity time 120 minutes
- Level Advanced

This episode reinforces many of the fundamental ideas about simple harmonic motion (SHM).

Lesson Summary

- Student experiment: Measuring the restoring force (20 minutes)
- Student experiment: Testing the relationship
- Student activity: Using an applet of a pendulum (30 minutes)
- Discussion: Gravitational and inertial mass (10 minutes)
- Student questions: Calculations involving pendulums (30 minutes)

Note a complication: a simple pendulum shows SHM only for small amplitude oscillations.

Measure the restoring force for a simple pendulum.

Episode 304-1: The simple pendulum (Word, 57 KB)

Test the relationship *T* = 2 π *l* *g* for a simple pendulum. Students could decide for themselves which measurements to make, which quantities to vary, and how to process and interpret the results. Encourage them to look for deviations from linear behaviour, arising from large-amplitude oscillations.

Investigate a virtual pendulum; this allows you to vary *g* . You can also force the pendulum, which is useful later when studying resonance.

NB: the analysis of the data uses log-log plots, so this may not be suitable for all students.

Episode 304-2: Virtual pendulum (Word, 31 KB)

The fact that the period of a simple pendulum is independent of the mass of the bob is an example of the Principle of Equivalence – something still not understood today and being tested by very sophisticated experiments involving astronomical measurements on the one hand and how single atoms fall due to gravity on the other.

The basic puzzle is why the *m* in *F**m* = *a* (where *m* is the inertial mass which determines how an object responds to any unbalanced force) has exactly the same magnitude as the *m* in *m**g* (where the *m* is the gravitational mass, the source of the gravitational force).

In deriving the equation for the period of a simple pendulum, we have used both, and used the fact that numerically they cancel out.

These questions reinforce basic ideas about SHM.

Episode 304-3: Pendulum (Word, 25 KB)

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

**Demonstration**

A mass suspended on a spring will oscillate after being displaced. The period of oscillation is affected by the amount of mass and the stiffness of the spring. This experiment allows the period, displacement, velocity and acceleration to be investigated by datalogging the output from a motion sensor. It is an example of simple harmonic motion.

Apparatus and Materials

- Motion sensor, interface and computer
- Slotted masses on holder, 100 g-400 g
- Clamp and stand
- String
- Springs, 3
- Card

Health & Safety and Technical Notes

Unless the stand is very heavy, use a G-clamp to anchor it to the bench.

Read our standard health & safety guidance

Suspend the spring from a clamp and attach a mass to the free end. Adjust the height of the clamp so that the mass is about 30 cm above the motion sensor, which faces upwards.

The clarity of measurements depends upon the choice of spring stiffness and mass. Good results can be obtained with three springs linked in series, and masses in the range 100 to 400 g. With this choice, it is necessary to place the sensor on the floor and allow the mass and spring to overhang the edge of the bench.

When the mass is displaced and released, its vertical motion is monitored by a motion sensor connected via an interface to a computer. In general, the magnitude of the initial displacement should not exceed the extension of the spring. It is best to lift the mass to displace it, rather than pull it down.

The mass may acquire a pendulum type of motion from side to side. Eliminate this by suspending the spring from a piece of string up to 30 cm.

For the collection and analysis of the data, data-logging software is required to run on the computer. Configure the program to measure the distance of the mass from the sensor, and to present the results as a graph of distance against time. Scale the vertical axis of the graph to match the amplitude of oscillation.

Procedure

- Lift and release a 400 g mass to start the oscillation. Start the data-logging software and observe the graph for about 10 seconds.
- Before the oscillation dies away, restart the data-logging software and collect another set of data, which can be overlaid on the first set.
- Repeat the experiment with 300 g, 200 g and 100 g masses.
- The period of the sinusoidal graph may be measured using a time-interval analysis tool in the software. Measure the period from peak to peak.
- Take measurements at several different places on the time axis, and observe that the period does not vary with elapsed time.
- Take similar measurements on the set of results with a smaller amplitude, and observe that the period appears to be independent of amplitude.
- Measure the period for each of the other graphs resulting from using different masses. Plot a new graph of period against mass. (Y axis: period; X axis: mass.)
- Use a curve-matching tool to identify the algebraic form of the relationship. This is usually of the form 'period is proportional to the square root of mass'.
- Use the program to calculate a new column of data representing the square of the period. Plot this against mass on a new graph. A straight line is the usual result, showing that the period squared is proportional to the mass.
- On the 'distance vs. time' graph, the gradient at any point represents the velocity of the oscillating mass. Choose the clearest set of data and use the program to calculate the gradient at every point on the graph.
- A plot of the resulting data shows a 'velocity vs. time' graph. Note that the new graph is also sinusoidal. However, compared with the 'distance vs. time' graph, there is a phase difference - the velocity is a maximum when the displacement is zero, and vice versa.
- A similar gradient calculation based on the 'velocity vs. time' graph yields an 'acceleration vs. time' graph. Comparing this with the original 'distance vs. time' graph shows a phase difference of 180°. This indicates that the acceleration is always opposite in direction to the displacement.

Teaching Notes

- This experiment illustrates the value of rapid collection and display of data in assisting thinking about the phenomenon under investigation. Data is collected within a few seconds and the graph is presented simultaneously. Students can observe connections between features on the graph and the actual motion of the mass. For example, the crests and troughs on the graph represent the mass at the extremes of its displacement.
- The parameters suggested here usually produce displacements of a few centimetres. The motion sensors can detect these with suitable precision. Small amplitude oscillations produce rather noisy data. Starting with the largest mass, shows the clearest results first.
- Software tools for taking readings from the graph are employed: measuring gradients and time intervals. The detail available in the data allows the idea that the periodic time remains constant for a given mass to be tested.
- A particularly useful software function is that which calculates the velocity for all points on the graph and plots these as a new graph. A notable feature of the velocity graph is the phase difference from the distance data. This can provoke useful discussion about the change in magnitude and direction of velocity during each cycle of oscillation. The
noisiness

of the measurements begins to show more markedly on the velocity graph. The process by which the program calculates the velocity (usually by taking differences between distance readings) should be questioned. - The further derivation of acceleration from the gradients of the velocity graph usually shows even more measurement noise. Nevertheless, the form of the graph convincingly shows the antiphase relation with the distance graph. This is useful for prompting discussion about the conditions for simple harmonic motion (SHM). This can be reinforced by plotting a further graph of acceleration against displacement. The negative gradient straight line supports the basic condition for SHM: acceleration is proportional to displacement, but in the opposite direction.
- You could add a card to the bottom of the masses to increase the damping. Students can see if the presence or amount of damping affect the natural frequency. Secondly, the amplitude can be extracted from each peak and a damping curve plotted. This can be tested to see if it is exponential.

*This experiment was safety-checked in May 2006*

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

**Demonstration and Class practical**

To demonstrate S.H.M. of a mass on a spring and gather accurate data using a datalogger.

Apparatus and Materials

- Datalogger position sensor
- Slotted 50 g masses and hanger
- Expendable spring
- Card

Health & Safety and Technical Notes

Masses hanging from ceiling.

Read our standard health & safety guidance

Be careful to set the mass moving only vertically, not swinging side to side. Also many position sensors do not work if the object gets too close: be careful to maintain at least the minimum working distance at all times.

Procedure

- Attach an expendable spring to the ceiling or a very high retort stand and hang a 50 g slotted mass hanger from it. Place a motion sensor underneath, pointing upwards. Displace the mass a small distance downwards. Position-time data can be recorded swiftly and easily.
- The experiment can be repeated using different numbers of masses, springs in series, and adding card to the bottom of the masses to increase the damping.

Teaching Notes

- If you have motion sensors this is much easier to set up than a pendulum attached to a rotary potentiometer.
- This can work at every level. Initially data can be recorded over a few cycles and used to find the period and hence the frequency. The effect of changing the mass, springs and damping is quickly measured and compared directly to theory.
- Position-time data can be analyzed to see if it is sinusoidal and if the period is constant as the amplitude damps down. Also, period being independent of initial amplitude can be checked.
- Velocity and acceleration-time graphs can be plotted by the computer and shown to have the same shape and period but different phase to the position-time graph.
- The damping can be investigated in various ways. Initially students can look and see if the presence or amount of damping affects the natural frequency. Secondly the amplitude can be extracted from each peak and a damping curve plotted. This can be tested to see if it is exponential.
- As the data is on a computer it can be exported to a spreadsheet or mathematics package and fitted to a sinusoidal form (with exponential damping if appropriate).

As well as illustrating the fundamental relationships of SHM (displacement, velocity, acceleration, time etc.), this experiment can be used as the basis of several open-ended investigations. Questions to tackle include:

- How does the period of oscillation T depend on the mass m, spring constant k and amplitudeA? (Note that, to double the spring constant, connect two springs side-by-side; to halve it, connect two end-to-end.)
- How does the amplitude of damped oscillations ‘decay’?
- Does damping affect the period of oscillation?

The mathematical relationship involved is T = 2 π√ m/k and this gives the opportunity to discuss how to choose appropriate axes to obtain a straight line graph. T 2 is proportional to m and to 1/k, so you can introduce the idea of adding columns to results tables to calculate relevant quantities. You could also extend this to consider the quantities which can be deduced from the gradients of graphs.

A graph of T 2 against 1/k has a gradient of 4π2 m; students can deduce m and compare it with the value of the mass they have used in the experiment. (Note that they may have to add the mass of the spring.)

You could point out that this is a method of determining mass which does not require gravity; fix a mass between two horizontal springs so that it oscillates from side to side, and deduce its mass from the period of oscillation. That would be useful in an orbiting spacecraft; it can also introduce the idea of inertial mass as opposed to gravitational mass.

The ‘decay’ of damped oscillations can be tested in two ways. Firstly, look for a ‘half-life’; does the decay of amplitude always halve over the same time interval? Next, plot a log-linear graph and show that it is a straight line. Points to make: a straight line graph makes use of all the data, rather than just selected points; and it will show up more clearly any errant data points.

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

**Class practical**

A circus with many different examples of simple harmonic motion.

Apparatus and Materials

**Station A: Simple Pendulum**

- stands, 3
- clamps, 3
- bosses, 3
- G-clamps, 3
- pairs of 5 cm wood or metal blocks as jaws, 3
- pendulum bobs, 3, on strings with lengths in ratio 1:2:4

**Station B: Torsional Pendulum**

- pair of 5 cm wood or metal blocks as jaws
- stand, clamp and boss
- G-clamp
- short wooden rod
- wire, Eureka, effective length 50 cm, 26 SWG

**Station C: Vibrating Lath**

- G-clamp
- metre rule
- boss

**Station D: Oscillating water column**

- U-tube of water 2.5 m, filled halfway up
- disposable mouthpieces, to protect hygiene or use a simple puffer bottle to start the oscillations

**Station E: Rolling ball**

- steel ball or marble
- bowl, shallow and spherical

**Station F: Wig-wag**

- wig-wag, with 3 removable masses
- G-clamps, 2

**Station G**

- curtain rail, 60 cm length, 3 (circular shape, parabolic shape & V-shape) mounted on an appropriate board
- safety screen holders, 2
- steel ball or marble

**Station H: Undamped light beam galvanometer**

- light beam galvanometer
- cell holder with one cell
- switch
- resistance substitution box
- leads

Health & Safety and Technical Notes

Read our standard health & safety guidance

At station B, the rod used for the torsional pendulum must be balanced. Two rods fastened together with elastic bands or a shorter length of wire may also be tried.

At station C, provide a second boss-head so that students can investigate the effect of increasing the load. The position of the boss head along the length of the metre rule could also be varied.

At station D, have students alter the water levels by blowing into the tube or use a simple puffer bottle. The water will then perform damped harmonic motion. Obtaining a time trace is not easy, since the period is short and damping is high. One possibility would be using a light beam and a scalar timer, with repeated timings.

At station G , set up the board leaning backwards a little, at about 10° to 15°.

At station H, set up the circuit with the galvanometer on its least sensitive scale; then increase the sensitivity until, with a resistance of over 500 kΩ, the spot reaches almost a full-scale deflection with the switch closed. Then, with the galvanometer on its direct

setting, open the switch: the spot will oscillate about its central zero position.

Procedure

At each station, displace the system from its equilibrium position and carefully observe what happens. Listen to the differences if a sound is made.

Teaching Notes

These experiments can give students a qualitative appreciation of a range of oscillators. Encourage them to use their own initiative to develop a description (graphical or otherwise) of the motion of an oscillator in its cycle. Careful work will provide the basis for discussions about the displacement, velocity and acceleration of the oscillator. You could introduce the terms displacement, amplitude, period, frequency.

**Features common to all harmonic oscillations are:**

- each complete oscillation of a system takes the same time
- a force returns the system to its equilibrium position when displaced
- an inertia factor makes the system overshoot its equilibrium position when in motion.

If the acceleration of a body is directly proportional to its distance from a fixed point, and is always directed towards that point, the motion is simple harmonic.

Some systems have a period of oscillation which depends on the mass. In many systems, the amplitude of oscillation decreases with time.

The link from acceleration of an oscillator to the force on the oscillator is obvious but should nonetheless be stressed as later modelling depends upon consideration of the changes in the force on an oscillator during its cycle.

**Expected results for some of the stations:**

- A: The periodic time,
*T*, depends on the length,*l*. (The motion is isochronous.)*T*∝*l*^{½}. - C: This behaves like a very large ticker-timer blade.
- D: The motion is damped by fluid friction but is clearly isochronous. Ask students whether period be the same if a denser liquid is used. The force tending to return the liquid to its equilibrium position will be
*rg*D*hA*, where*r*is the fluid density,*g*is gravitational field strength, D*h*is the difference in liquid column heights, and*A*is the column cross-section. - E: Listen to the sound: what does this tell you about the motion? The amplitude decreases but the frequency remains unaltered.
- F: Load the end of the wig-wag with a variable number of masses so that it oscillates sideways. Note the affect of mass on the time for one oscillation.
- G: Listen to the sound the ball makes as it rolls or slides along the tracks. The circular track will give what sounds like an isochronous motion; the parabolic track gives a frequency that increases as the amplitude decreases; the V-shaped track is not isochronous at all.
- H: A time trace for one oscillation can be obtained by photography, using a multi-slit stroboscope. Students could also record how the amplitude dies away, and isochronous property of the oscillations.

**Also **

**Investigating the factors affecting the period of a simple harmonic oscillator (OCR)**

**and **

**Investigation into simple harmonic motion using a mass-spring system and a simple pendulum (AQA)**

**Class practical**

This experiment could extend (or replace) the traditional pendulum or mass-on-a-spring experiments illustrating S.H.M.

Apparatus and Materials

- Metre rules
- G-clamps
- Slotted masses (100 g each)
- Sellotape
- Stopwatches
- Small, rough wooden blocks

Health & Safety and Technical Notes

It might be best advised to wear goggles in case something snaps.

Don't stand with toes underneath the slotted masses.

Read our standard health & safety guidance

Procedure

- G-clamp the metre rule securely to the bench using the wooden blocks to protect the rule and bench.
- Sellotape one or more slotted masses near the end of the rule.
- Twang and time several oscillations.
- Adjust the vibrating length (or mass attached) and repeat.

Teaching Notes

- You could use this experiment as a follow-up to the standard "g from a pendulum". There are more variables to play with so you can easily set up differentiated tasks for your students.
- In this case we can find
*E*for wood because for this cantilever we have ...
ω *x*= width of ruler*y*= thickness of ruler (scale to undersurface)*M*= mass Sellotaped on*L*= vibrating length*E*= Young modulus of wood*ω*= 2π/T- So T
^{2}v*M*(or*L*^{ 3}) gives you*E*from the gradient. - With a wide range of abilities you can have one group simply verifying it's S.H.M. (by proving -
*T*is independent of amplitude), another determining*E*, and another using log graphs to discover that*T*is proportional to -*L*^{3/2}. It's also a good one for error analysis; which term contributes the largest error in*E*(answers on a postcard)? - The vibrations are quite fast (especially at short lengths). To obtain an accurate result for
*T*, time many oscillations and find the average time for a single oscillation. - If you have the materials you can try things other than metre rules.

Exy

*Thank you to Wayne Morton for pointing out that there was an error in the formula that we previously printed.*

*This experiment was submitted by Jason Welch who is Head of Physics at County High School, Leftwich, Cheshire.*

Section 501-4 gives a method for students to use LEDs of different colours to investigate the relationship between frequency and photon energy for light.

Quantum and Nuclear

Lesson for 16-19

- Activity time 185 minutes
- Level Advanced

This episode discusses: quantisation; energy levels in a hydrogen atom; and, distinguishing quantisation and continuity. And is accompanied by worked examples, student question sets, a student experiment and a demonstration.

Lesson Summary

- Demonstration: Looking at emission spectra (20 minutes)
- Discussion: The meaning of quantisation (20 minutes)
- Demonstration: Illustrating quantisation (10 minutes)
- Discussion: Energy levels in a hydrogen atom (10 minutes)
- Worked example and Student Questions: Calculating frequencies (20 minutes)
- Discussion: Distinguishing quantisation and continuity (5 minutes)
- Worked example: Photon flux (10 minutes)
- Student calculations: Photon flux (20 minutes)
- Student experiment: Relating photon energy to frequency (30 minutes)

Show a white light and a set of standard discharge lamps: sodium, neon, hydrogen and helium. Allow students to look at the spectrum of each gas. They can do this using a direct vision spectroscope or a bench spectroscope, or simply by holding a diffraction grating up to their eye.

What is the difference? (The white light shows a continuous spectrum; the gas discharge lamps show line spectra.)

The spectrum of a gas gives a kind of finger print

of an atom. You could relate this to the simple flame tests that students will have used at pre-16 level. Astronomers examine the light of distant stars and galaxies to discover their composition (and a lot else).

Relate the appearance of the spectra to the energy levels within the atoms of the gas. Students will already have a picture of the atom with negatively charged electrons in orbit round a central positively charged nucleus. Explain that, in the classical model, an orbiting electron would emit radiation and spiral in towards the nucleus, resulting in the catastrophic collapse of the atom.

This must be replaced by the Bohr atomic structure – orbits are quantised. The electron’s energy levels are discrete. An electron can only move directly between such levels, emitting or absorbing individual photons as it does so. The ground state is the condition of lowest energy – most electrons are in this state.

Think about a bookcase with adjustable shelves. The bookshelves are quantised – only certain positions are allowed. Different arrangement of the shelves represents different energy level structures for different atoms. The books represent the electrons, added to the lowest shelf first etc

Throw a handful of polystyrene balls round the lab and see where they settle. The different levels on which they end up – the floor, on a desk, on a shelf – gives a very simple idea of energy levels.

Some useful clipart can be found below:

Episode 501-1: The emission of light from an atom (Word, 35 KB)

Resourceful Physics >Teachers>OHT>Emission of Light

The absorption of a photon can raise an electron to a higher energy level. Alternatively the electrons can change energy level as a result of heating (which, in turn, can either be by absorbing radiation or particle collision).

When the electrons fall back to a lower level they emit a quantum of radiation.

Show a scale diagram of energy levels. It is most important that this diagram is to scale to emphasise the large energy drops between certain levels.

The students may well ask the question, Why do the states have negative energy?

This is because the zero of energy is considered to be that of a free electron just outside

the atom. All energy states below

this – i.e. within the atom are therefore negative. The atom must absorb energy to raise the electron to the surface

of the atom and allow it to escape.

Calculate the frequency and wavelength of the quantum of radiation (photon) emitted due to a transition between two energy levels. (Use two levels from the diagram for the hydrogen atom.)

Δ *E* = *E*_{2} − *E*_{1}

*E*_{2} – *E*_{1} = *h* × *f*

Point out that this equation links a particle property (energy) with a wave property (frequency).

Ask your students to calculate the energy and frequency or a photon for one or two other transitions. Can they identify the colour or region of the spectrum of this light?

Emphasise the need to work in SI units. The wavelength is expressed in metres, the frequency in hertz, and the energy difference in joules. You may wish to show how to convert between joules and electronvolts.

The difference between the quantum theory and the classical theory is similar to the difference between using bottles of water (quantum) or water from a Episode (classical). The bottles represent the quantum idea and the continuous flow from the Episode represents the classical theory.

The quantisation of energy is also rather like the kangaroo motion of a car when you first learn to drive – it jumps from one energy state to another, there is no smooth acceleration.

It is all a question of scale. We do not see

quantum effects generally in everyday life because of the very small value of Planck's constant. Think about a person and an ant walking across a gravelled path. The size of the individual pieces of gravel may seem small to us but they are giant boulders to the ant.

We know that the photons emitted by a light bulb, for example, travel at the speed of light

(3 × 10^{8} m s^{-1} ) so why don’t we feel them as they hit us? (Although all energy is quantised we are not aware of this in everyday life because of the very small value of Planck’s constant.)

Students may worry about the exact nature of photons. It may help if you give them this quotation from Einstein:

All the fifty years of conscious brooding have brought me no closer to the answer to the question,

What are light quanta?

. Of course, today every rascal thinks he knows the answer, but he is deluding himself.

Calculate the number of quanta of radiation being emitted by a light source.

Consider a green 100 W light. For green light the wavelength is about 6 × 10^{-7} m and so Energy of a photon,*E* :

*E* = *h* × *f*

*E* = *h* × *c* *λ*

Energy of a photon = 3.3 × 10^{-19} J

The number of quanta emitted per second by the light

*N* = 100 × *λ* *h* × *c*

*N* = 3 × 10^{20} s^{-1}.

Episode 501-2: Photons streaming from a lamp (Word, 22 KB)

Episode 501-3: Quanta (Word, 27 KB)

Episode 501-4: Relating photon energy to frequency (Word, 70 KB)

Students can use LEDs of different colours to investigate the relationship between frequency and photon energy for light.

This resource a method of using the photoelectric effect to determine Planck's constant (502-3). The apparatus can be expensive but the experiment is great if you can get hold of it.

Quantum and Nuclear

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

This episode introduces an important phenomenon. Light releases electrons from metal surfaces.

Lesson Summary

- Demonstration: The basic phenomenon (15 minutes)
- Discussion: Summarising the phenomenon (10 minutes)
- Discussion: An analogy (5 minutes)
- Student questions: Using the photoelectric equation. The Millikan experiment: to verify Einstein’s photo-electric relationship (30 minutes)
- Student experiment: Measuring Planck’s constant (30 minutes)

Introduce the topic by demonstrating the electroscope and zinc plate experiment.

Episode 502-1: Simple photoelectric effect demonstration (Word, 36 KB)

Point out to the students that the photoelectric effect is apparently instantaneous. However, the light must be energetic enough, which for zinc is in the ultraviolet region of the spectrum. If light were waves, we would expect the free electrons to steadily absorb energy from the waves until they escape from the surface. This would be the case in the classical theory, in which light is considered as waves. However, we could wait all day and still the red light would not liberate electrons from the zinc plate.So what is going on? We picture the light as quanta of radiation (photons). A single electron captures a single photon. The emission of an electron is instantaneous as long as the energy of each incoming quantum is big enough. If an individual photon has insufficient energy, the electron will not be able to escape from the metal.

Summarise the important points about the photoelectric effect.

- There is a threshold frequency (and, hence, energy), below which no electrons are released.
- The electrons are released at a rate proportional to the intensity of the light (i.e. more photons per second means more electrons released per second).
- The kinetic energy of the emitted electrons is independent of the intensity of the incident radiation. They are emitted with a maximum velocity.

Try this analogy, which involves ping-pong balls, a bullet and a coconut shy. A small boy tries to dislodge a coconut by throwing a ping-pong ball at it – no luck, the ping-pong ball has too little energy! He then tries a whole bowl of ping-pong balls but the coconut still stays put! Along comes a physicist with a pistol (and an understanding of the photoelectric effect), who fires one bullet at the coconut – it is instantaneously knocked off its support.

Ask how this is an analogy for the zinc plate experiment. (The analogy simulates the effect of infrared and ultra violet radiation on a metal surface. The ping-pong balls represent low energy infrared

, while the bullet takes the place of high-energy ultra violet

.)

Now you can define the work function. Use the potential well model to show an electron at the bottom of the well. It has to absorb the sufficient energy in one go to escape from the well and be liberated from the surface of the material.

The electronvolt is introduced because it is a convenient small unit. You might need to point out that it can be used for any (small) amount of energy, and is not confined to situations involving electrically accelerated electrons.

It is useful to compare the electron with a person in the bottom of a well with totally smooth sides. The person can only get out of the well by one jump, they can't jump half way up and then jump again. In the same way an electron at the bottom of a potential well must be given enough energy to escape in one jump

. It is this energy that is the work function for the material.

Now you can present the equation for photoelectric emission:

Energy of photon *E* = *h* × *f*

Picture a photon transferring energy to one of the electrons which is least tightly bound in the metal. The energy of the photon does two things.

Some of it is needed to overcome the work function Φ.

The difference is the kinetic energy of the electron when it leaves the metal.

*h* × *f* = Φ + 12 *m* *v*^{ 2}

A voltage can be applied to bind the electrons more tightly to the metal. The stopping potential *V*_{s} is just enough to prevent any from escaping:

*h* × *f* = Φ + *e* × *V*_{s}

Set the students some problems using these equations.

Episode 502-2: Photoelectric effect questions (Word, 42 KB)

Episode 502-4: Student question. The Millikan experiment (Word, 51 KB)

The Millikan experiment question may best come after

Episode 502-3: Student experiment: Measuring threshold frequency (Word, 35 KB)

Episode 502-3: Measuring threshold frequency (Word, 35 KB)

Students can measure Planck’s constant using a photocell.

(Some useful clipart is given here below).

Research skills (two examples quoted);

- The principles behind the operation of the Global Positioning System
- The use of radioactive materials as tracers in medical imaging
**(OCR**)

*This is relevant only to those who use OCR ‘B’ Advancing physics and will be well understood by them, though the task would need to be shorter than the traditional two-week research report*

Specifically episode 414-5: Magnets falling through a coil

Electricity and Magnetism

Lesson for 16-19

- Activity time 200 minutes
- Level Advanced

Students will already have ideas about electromagnetic induction. In this episode, your task is to develop a picture of induction in which it is the cutting of lines of flux by a conductor that leads to an induced EMF or current.

Lesson Summary

- Student experiment: Wire, magnet, meter (10 minutes)
- Discussion and demonstration: Induction effects (20 minutes)
- Discussion: More about flux and flux linkage (40 minutes)
- Student questions: On flux linkage (20 minutes)
- Student experiments: Investigating induction (20 minutes)
- Demonstrations: Related effects (20 minutes)
- Student questions: Induced EMFs (30 minutes)
- Discussion and demonstration: Eddy currents (20 minutes)
- Student questions: Eddy currents and Lenz's law (20 minutes)

Start with a simple experiment involving a coil of wire and a voltmeter. This will give you a chance to assess the knowledge that students bring to this section.

Episode 414-1: Faraday’s law (Word, 26 KB)

A good starting point is to revise the pre-16 level ideas that your students should have about electromagnetic induction.

The first two demonstrations involve moving a wire in a magnetic field and then a permanent magnet into and out of a small coil. In both it is important to emphasise that:

electricity

is only produced while something is moving- the faster the movement, the more
electricity

we get

Introduce the idea of flux cutting

. Use your fingers to represent the flux lines; show how the conductor moves so as to cut the lines of flux. If you move the conductor *along* the lines of flux, no current is induced.

The third demonstration shows that movement is not essential and that changing the field near a coil has similar effects to a moving magnet.

(The demonstration with a dynamo adds little at this stage and could be delayed until generators are discussed further.)

Episode 414-2: Electromagnetic induction (Word, 62 KB)

electricitymade?

The demonstrations have shown that making

electricity involves magnetic fields, but what is really going on? Your students already know that charges moving across a magnetic field experience a force (the *B**I**L* force). Now, the metal of a conductor contains mobile charges, the conduction electrons. What happens to these if the conductor is moved across a magnetic field?

Consider a conducting rod PQ moving at a steady speed v perpendicular to a field with a flux density *B*. An electron (negative charge*e*) in the rod will experience a force (*B**e**v*) (Fleming's left hand rule) that will push it towards the end Q. The same is true for other electrons in the rod, so the end Q will become negatively charged, leaving P with a positive charge. As a result, an electric field *E* builds up until the force on electrons in the rod due to this electric field (*E**e*) balances the force due to the magnetic field.

*E**e* = *B**e**v*

so

*E* = *B**v*

For a rod of length *L*

*E* = *V**L*

Hence the induced EMF

*E* = *B**L**v*

Clearly what we have here is an induced EMF (no complete circuit so no current flows) and already we can see that more rapid movement gives a greater induced EMF.

Now consider what happens when the EMF drives a current in an external circuit. To do this, imagine that the rod moves along a pair of parallel conductors that are connected to an external circuit.

The EMF will now cause a current to flow in the external resistor R. This means that a similar current flows through the rod itself giving a magnetic force, BIL to the left.

(L is now the separation of the two conductors along which the rod PQ moves.) An equal and opposite force (to the right) is needed to keep PQ moving at a steady speed.

The work done in moving the rod will equal the energy dissipated in the resistor.

In a time *t*, the rod moves a distance
*d* = *v* × *t*

Work done on the rod = *B**I**L**v**t*

Energy dissipated in R = power × time

Energy dissipated in R = EMF*v**t*

giving

*B**I**L**v**t* = EMF*v**t*

or, as before,

*E* = *B**L**v*

But in this case it can be seen that the electrical circuit encloses more magnetic field as the rod is moved along and that in one second, the extra area enclosed will be

*v* × *l*.

i.e.
induced EMF, *E* = *B* × area swept out per second

E = *B**A**t*

We have already called *B* the flux density, so it is perhaps not surprising that the quantity *B* × *A* can be called the magnetic flux, *F*.

Thus
induced EMF = *F**t*

induced EMF = rate of change of flux

And more generally

*E* = d *Φ* d *t*.

How can the induced EMF be increased? Discussion should lead to:

- moving the wire faster – d
*A*d*t*increased – rate of change of flux increased - increasing the field (and hence the flux) – rate of change of flux increased

But there is a further possibility and this is to increase the number of turns of wire N in our circuit. By doing this, the flux has not been altered but the flux linkage (N × *F*) will have increased. Hence it is more correct to say that

induced EMF = rate of change of flux linkage

*E* = N × d *F* d *t*

This relationship is known as Faraday's law: – when the flux linked with a circuit changes, the induced EMF is proportional to the rate of change of flux linkage.

Finally, remind your students that the magnetic force on our simple generator (a) (b) was in a direction which would make the bar slow down unless an external force acted. This is an example of Lenz's law: – the direction of the induced EMF is such that it tends to oppose the motion or change causing it.

To include this idea in our formula, a minus sign has to be introduced, giving;

*E* = − N × d *F* d *t*

We have two formulae:

Flux,
*F* = *B* × *A*

Flux linkage
N*F* = N*B**A*

When using these formulae, it is important to realize that *B* should be at right angles to the area *A*. If this is not the case, then it is the component of the field perpendicular to *A* that should be used.

Recall that the tesla (T) is defined from *F* = *B**I**L* , so
1 T = 1 N A^{-1} m^{-1}.

The units for flux are thus N A^{-1} m. This unit is known as the Weber (Wb).

Flux linkage is measured in Weber-turns (Wb-turns).

Although it is better to delay questions about Faraday's law until after more experimental work has been done, the relationship between flux, flux density and flux linkage should be reinforced with a question or two.

Episode 414-3: Sketching flux patterns (Word, 269 KB)

To support the theory, it is important that students look at electromagnetic induction experimentally in more detail than was met in the initial demonstration. What you choose to do will depend on what apparatus is available.

The experiment suggested here is based on coils (120/240 turns) linked by iron cores. Again the basics of Faraday's law are shown and there is a very strong lead into transformers.

Episode 414-4: Investigating electromagnetic induction (Word, 219 KB)

A simple experiment (or demonstration) can be done by passing a permanent magnet through a coil of wire that is connected to a data logger.

This shows clearly that as the magnet moves into the coil an EMF is generated for a short time.

Episode 414-5: Magnet falling through a coil (Word, 27 KB)

Some ideas for quick demonstrations of effects related to electromagnetic induction.

Episode 414-6: Quick demonstrations of electromagnetic induction (Word, 39 KB)

The first link involves some qualitative work, sketching graphs and includes a falling magnet experiment

Episode 414-7: Rates of change (Word, 198 KB)

Some simple calculations.

Episode 414-8: EMF in an airliner (Word, 34 KB)

So far, the induced effects have been seen in wires with an associated change in flux. But does the conductor involved have to be a wire

? The answer is that there will be induced currents whenever the flux linked with a conductor of any shape or size changes. If the conductor is not a wire, then these induced currents are referred to as eddy currents

.

Several demonstrations show the effect. From these experiments it should become clear that Lenz's law applies, i.e. the induced effects oppose the motion that is producing them. One of the main uses for eddy currents is in electromagnetic braking.

When eddy currents flow in a resistive metal, eddy current heating results. It is put to good practical use, e.g. in the production of pure alloys where eddy current heating of a metal crucible replaces a dirty

furnace. More frequently the heating is a nuisance as it wastes energy in electromagnetic machines.

Episode 414-9: Introducing eddy currents (Word, 49 KB)

Episode 414-10: Further eddy current demonstrations (Word, 48 KB)

Some descriptive work can reinforce these ideas.

Episode 414-11: Eddy currents and Lenz’s law (Word, 69 KB)

Episode 412-2 gives a method of determining the force on a current carrying wire placed in a uniform magnetic field.

Electricity and Magnetism

Lesson for 16-19

- Activity time 70 minutes
- Level Advanced

Having reminded your students that magnetic fields can be found near permanent magnets and in the presence of an electric current, the next step is to show how the field

can be quantified. Again, students should know that a conductor carrying a current in a magnetic field will experience a force and will probably remember that Fleming's Left Hand Rule can be used to find the direction of that force.

Lesson Summary

- Demonstrations: Leading to
*F*=*B**I**L*(15 minutes) - Discussion: Factors affecting the force (15 minutes)
- Discussion: Formal definitions (20 minutes)
- Student questions:
*B**I**L*force calculations (20 minutes)

Several quick experimental reminders are possible.

Episode 412-1: Forces on currents (Word, 79 KB)

Episode 412-2: An electromagnetic force (Word, 53 KB)

These lead on to a further experiment in which the relationship *F* = *B**I**L* can be established.

Episode 412-3: Force on a current-carrying wire (Word, 43 KB)

The experiments above lead to the conclusion that the force *F* on the conductor is proportional to the length of wire in the field, *L* , the current *I* and the strength

of the field, represented by the flux density *B* . (There is also an angle factor

to consider, but we will leave this aside for now.)

Combining these we get *F* = *B**I**L*

(It can help students to refer to this force as the

.)*B**I**L* force

Students will probably know that electric and gravitational fields are defined as the force on unit charge or mass. So by comparison, *B* = *F**I**L*
, and this gives a way of defining the magnetic field strength

. Physicists refer to this as the *B*-field or magnetic flux density which has units of N A^{-1} m^{-1} or tesla (T).

A field of 1 T is a very strong field. The field between the poles of the Magnadur magnets that are used in the above experiment is about 3 × 10^{-2} T while the Earth's magnetic field is about 1 × 10^{-5} T.

If your specification requires, you will need to develop the angle factor seen in the experiment into the mathematical formula:

*F* = *B**I**L*sin( θ ).

For the mathematically inclined, it can be shown that the effective length of the wire in the field (i.e. that which is at right angles) is *L*sin( θ ). If students find this difficult, then it can be argued that the maximum force occurs when field and current are at right angles,

θ = 90 °

(sin( θ ) = 1),

and that this falls to zero when field and current are parallel,

θ = 0 °

( sin( θ ) = 0 )

Some specifications require a formal definition of magnetic flux density and/or the tesla.

The strength of a magnetic field or magnetic flux density *B* can be measured by the force per unit current per unit length acting on a current-carrying conductor placed perpendicular to the lines of a uniform magnetic field.

The SI unit of magnetic flux density *B*
is the tesla (T), equal to 1 N A^{-1} m^{-1}. This is the magnetic flux density if a wire of length 1 m carrying a current of 1 A as a force of 1 N exerted on it in a direction perpendicular to both the flux and the current.

Study of the force between parallel conductors leading to the definition of the ampere may be required. Students may already have seen the effect in your initial experiments but this may need to be repeated here. The effect can be explained by considering the effect of the field produced by one conductor on the other and then reversing the argument.

(The most common alternative approach relies upon field lines only and describes a catapult effect

from regions where the field lines are tightly packed into regions where the lines are more widely spaced.)

The force between parallel conductors forms the basis of the definition of the unit of current, the ampere. A formal definition is not usually required but students should realize that in a current balance (such as was used above) measurement of force and length can be traced back to fundamental SI units (kg, m, s) leaving the current as the only unknown

.

Some students are likely to be interested in the formal definition which is:

that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross-section, and placed 1m apart in a vacuum, would produce a force of 2 × 10

.^{-7} newton per metre of length

Episode 412-4: Forces on currents in magnetic fields (Word, 35 KB)

Although the electric motor could be discussed here, it is probably better to leave this until after electromagnetic induction has been covered so that the back emf can be included.

Electricity and Magnetism

Practical Activity for 14-16

**Class practical**

This effect is the basis of all electric motors.

Apparatus and Materials

*For each student group*

- Iron yoke
- Magnadur (ceramic) slab magnets, 2
- Copper wire, stiff, bare, SWG 32 and SWG 26
- Clamp, or wooden support blocks
- Crocodile clips, 2
- Leads, 4 mm, 2

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

- Make a long rectangular loop of thin copper wire.
- Clamp it in a wooden support block with wing nut, or between two pieces of wood in the jaws of a clamp. The closed end of the loop will project out horizontally, sagging a little.
- Connect the ends of the copper wire to the low-voltage DC supply, using cleaned crocodile clips and 4 mm leads.
- Place the slab magnets on the yoke, ensuring that opposite poles are facing each other. Bring it near the free end of the loop when a current is flowing.
- Find the position in which the magnets have the greatest effect on the current-carrying wire.
- Now, using two 5-cm lengths of the thicker copper wire, make a pair of parallel horizontal rails. Clamp them as shown, and connect up to the power supply, or clamp them directly to the DC terminals of a Westminster pattern power supply.
- Place a third piece of copper wire across the rails.
- Bring up the magnets; how should they be held to produce a force on the third wire?
- Investigate what happens if you reverse the current, or if you reverse the magnets.

Teaching Notes

- In this experiment, students may use the knowledge that a current-carrying wire has an associated magnetic field. When the wire is placed in a magnetic field it is likely that these two fields will interact.
- In practice, students will see the motion and know that there must be forces at play, but the three-dimensional geometry will remain obscure.
- Students will find that there is a force on the wire at right angles to both the current and the magnetic field. (If the current-carrying wire is not at right angles to the field, then only a component of the current will create a force.) If the wire lies along the magnetic field, there will be no force. If the wire is perpendicular to the magnetic field then the force will be maximum. A reversal of the current or of the field will reverse the direction of the force.
- You could introduce the left hand rule here in order to summarize what students have discovered.

*This experiment was safety-tested in July 2007*

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