It is helpful to start this topic by discussing capacitors, rather than the more abstract notion of capacitance.

Lesson Summary

• Demonstration: A super-capacitor (10 minutes)
• Demonstration: Some capacitors in use (10 minutes)
• Student experiment and discussion (40 minutes): Charging and discharging capacitors
• Student questions: Charge storage (20 minutes)

Demonstration: A super capacitor

You should be able to capture the attention of your students with a short demonstration of a 'super-capacitor'. This will allow the term capacitor to be introduced and shows that these devices store energy (electrically).

Episode 125-1: Super capacitor (Word, 27 KB)

Demonstration: Some capacitors in use

To emphasize the wide range of situations in which capacitors are used, show a few examples.

Episode 125-2 Where to find capacitors in everyday use (Word, 24 KB)

Student experiment and discussion (40 minutes): Charging and discharging capacitors

The transient nature of the charge/discharge process can be looked at in a qualitative way using a range of capacitors and resistors and monitoring the current with an (analogue) ammeter. This is a good student experiment but you may have to give some initial guidance in how to discharge the capacitor between observations by connecting a lead across its terminals.

Episode 125-3 Charging and discharging capacitors (Word, 42 KB)

The experimental work can be followed by a discussion which should bring out the following observations:

• that current flows for a short time (meters deflect briefly)
• that the current is initially large and then decreases
• that there is the same current in the wires connecting the capacitor to both the positive and negative terminals of the supply (meters deflect identically)
• that the value of the capacitor and resistor alter the magnitude of the current and the time for which it flows

Further discussion should emphasise the underlying physics explanations:

• that electrons are being removed from one plate, while others are being added to the other, during the charging process
• that as this charge increases the potential difference across the capacitor increases (the experiment uses a number of cells and assumes that each has the same EMF)
• that this causes a reduction in the flow of charge until the pd across the capacitor equals the EMF of the supply
• thus a charged capacitor has equal but opposite charges on the two plates so that the total charge is zero.

(There is some advantage in looking at the time variation of the current and/or voltage using either an oscilloscope or datalogger but this can be left until Episode 129.)

Student questions: Charge storage

Questions will reinforce the discussion or get the students to think through the ideas about charge and its storage for themselves.

Episode 125-4: Questions on charging capacitors (Word, 49 KB)

Having established that there is charge on each capacitor plate, the next stage is to establish the relationship between charge and potential difference across the capacitor.

Lesson Summary

• Demonstration: Charging a capacitor (10 minutes)
• Discussion: Defining capacitance and the farad (20 minutes)
• Student experiment: Charge proportional to voltage – two alternatives (30 minutes)
• Discussion: Factors affecting C (10 minutes)
• Student experiment: Factors affecting C (30 minutes)
• Discussion: Permittivity (20 minutes)
• Discussion: Working with real capacitors (10 minutes)
• Student questions and discussion: Calculations with real capacitors (20 minutes)

Demonstration: Charging a capacitor

The experimental demonstration charging a capacitor at a constant rate shows that the potential difference across the capacitor is proportional to the charge.

Episode 126-1: Charging a capacitor at constant current (Word, 34 KB)

Discussion: Defining capacitance and the farad

The experiment shows that Q  ∝  V, or Q = constant × V. This constant is called the capacitance, C, of the capacitor and this is measured in farads (F). So capacitance is charge stored per volt, and

farads = coulombsvolts.

It is a good idea to point out that 1 farad is a very large capacitance and that most capacitors will be micro, μ, – (10^-6), nano

(10^-9), or pico- (10^-12) farads. The capacitance of the planet Earth, considered as an isolated sphere of radius R, calculated using

C = 4 π ε₀ ε_rR is 710 mF.

Student experiment: Charge proportional to voltage – first alternative

The relationship between charge and potential difference can be investigated further by the students themselves. Two experiments are possible; this one makes use of a coulomb meter.

By charging a suitable capacitor to different voltages and measuring the charge stored each time, you have a rapid confirmation of the relationship Q  ∝  V. The experiment can be repeated with different capacitors. Plot a graph of Q against V.

Episode 126-2: Measuring the charge on a capacitor (Word, 47 KB)

Charge proportional to voltage – second alternative

The second investigation of the relationship between charge and pd makes use of a change-over reed switch. Students may have met simple on/off reed switches in technology or even in primary school.

Although this is a more difficult experiment to perform, it has value because it can be extended to investigate the factors determining capacitance of a parallel plate capacitor if this is needed for your specification.

From either experiment, a graph of Q against V can be plotted. This is helpful later when discussing the energy stored in a capacitor. (N.B. The graph from a reed switch experiment will not pass through the origin so the effect of stray capacitance in the experiment will have to be explained)

Episode 126-3: Using a reed switch to measure capacitance (Word, 46 KB)

Discussion: Factors affecting C

If your specification requires the study of the equation C = ε₀ε_r  ×  Ad , then this is a convenient point to cover that work.

It is a good time to introduce the idea that many tubular shaped capacitors are, in fact, a parallel plate capacitor which has been rolled up and filled with a dielectric. Why? (A large area with a small gap gives reasonable values of capacitance; dielectric increases capacitance; rolling reduces the overall size.)

Student experiment: Factors affecting C

Using a reed switch, or a digital capacitance meter, investigate the factors determining capacitance for a parallel plate capacitor.

If you do not have a reed switch many cheap digital multimeters now have a capacitance meter that covers the pF and nF range, which will work effectively here.

By using parallel plates as the capacitor in this experiment, the relationship between capacitance and area can be found by altering the area of overlap while using spacers leads to the relationship between capacitance and separation. Placing plastic sheets between the plates shows the effect of a dielectric and shows why the relative permittivity appears in the formula. If time is short, these three experiments could be done as group activities, with groups reporting back on their findings.

Discussion: Permittivity

Discuss the outcomes of the experiments and the significance of ε₀} , the permittivity of free space. Deduce its units of F m^-1 or C² N^-1 m^-2.

Discussion: Working with real capacitors

Take a selection of capacitors and look at the information written on each. This will include the capacitance and the maximum working voltage. On an electrolytic capacitor there will also be an indication of the polarity for each terminal (and there may be a maximum ripple current).

Discuss what the markings mean and compare the charge stored by each capacitor at maximum voltage (practice in using

Q = C × V.

How does this relate to the physical size of the capacitor? (This is unlikely to be simply that the larger the capacitance the bigger the capacitor. The working voltage is important, as is the material between the plates.)

Student questions: Calculations with real capacitors

Follow-up questions will round off this episode.

Episode 126-4: Charging capacitors questions (Word, 62 KB)

Episode 126-5: Problems on capacitors (Word, 37 KB)

The derivation of formulae for capacitors in series and parallel will help to reinforce your students’ understanding of circuits involving capacitors.

Lesson Summary

• Discussion: Deriving the formulae (20 minutes)
• Worked examples: Using the formulae (10 minutes)
• Student questions: Using the formulae (30 minutes)
• Student experiment: Checking the formulae (20 minutes)

Discussion: Deriving the formulae

Your students will have encountered the idea of replacing resistors in series and parallel by a single resistor which has the same effect in the circuit. Remind them of this as an introduction but be ready to dispel any confusion that may arise because the formulae are reversed for capacitors.

For capacitors in parallel the pd across each is the same. For capacitors in series, it is the charge stored that is the same.

Episode 127-1: Capacitors in series and parallel formula derivations (Word, 36 KB)

Worked examples: Using the formulae

Choose a couple of simple examples; say, 20 mF and 30 mF in parallel (50 mF), and then in series (12 mF). Point out that capacitors in parallel combine to give a greater capacitance; in series, the resultant is less than either. What about equal capacitors? (In series, half the capacitance of either.)

Student questions: Using the formulae

Questions 1 and 2 can reinforce the above discussion. Questions 3 and 4 give practice in using the formulae.

Episode 127-2: Capacitors in series and parallel questions (Word, 60 KB)

Episode 127-3: More capacitors in series and parallel questions (Word, 28 KB)

Student experiment: Checking the formulae

If a capacitance meter is available, the results of some of the calculations above can be checked experimentally and/or further combinations can be tried. Provide students with two (or more) capacitors whose values they can measure. They can then connect them together, first in parallel and then in series. Does the meter reading agree with calculated values?

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)

Demonstration: Energy transformations

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)

Discussion: Calculating energy stored

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 V2, so the energy transferred was Q  ×  V2.

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 = 12QV

or

Energy stored in the capacitor = 12CV²

or

Energy stored in the capacitor = 12Q²C

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

Worked examples: Energy stored

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 = 12CV²

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 V².

Student experiment: Energy stored – first alternative

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)

Student experiment: Energy stored second alternative

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)

Student questions: Calculations on the energy formula

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)

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)

Student experiment: Exponential discharge

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)

Discussion: Characteristics of exponentials

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 = VR ).

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.)

Student activity: Spreadsheet model

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)

Student experiment: Varying R and C

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)

Discussion: Deriving exponential equations

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 Q₀ so that the pd across the capacitor V₀ 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 = VR (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 = QC (equation 3)

Eliminating I and V leads to:

Δ Q = -QCR  ×  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 QQ = 1CR

(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 = -  1CR

Solving this gives:

Q = Q₀e^- t/_CR_

where Q₀ = C × V₀

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

I = I₀e^- t/_CR_

where I₀ = V₀R

and V = V₀e^- t/_CR_

Worked examples: Using the equations

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 CR, 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 I₀ = V₀R and Q₀ = C × V₀ , 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.

Discussion: Time constant

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 = Q₀e^- t/_CR_

When t = CR , we have Q = Q₀e^-1

(i.e. this is the time when the charge has fallen to 1e = 0.37 (about ⅓ ) of its initial value. CR 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) × CR

or,

T _½  = 0.69 × CR)

Student activity: Analysing graphs

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 CR. 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.)

Student questions: Practice with equations

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)

Discussion: Back to reality

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.

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)

Discussion: Energy stored

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 = 12QV

Energy stored in the spring = 12Fx

Energy stored in the spring = 12kx²

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)

Discussion: Exponential decrease

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 = Q₀e^- t/_CR_

Current, Δ QQ = 1CR

Time constant, CR: time for charge to fall by 1e

T _½ CR = ln(2)

For radioactive decay:

N = N₀e^{-  λ_N_}

Activity, Δ NN = λN

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.)

Student questions: Exponential decrease

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)