## Episode 126: Capacitance and the equation Q=C/V

Lesson for 16-19

- Activity time 150 minutes
- Level Advanced

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

^{-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 ε

_{0}ε

_{r}

*R*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* = ε_{0}ε_{r} × *A**d*
, 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 *ε*_{0}} , the permittivity of free space. Deduce its units of F m^{-1} or C^{2} 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)