Episode 128: Energy stored by a capacitor

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.

Discharge of capacitor through a motor

Summary

  • Demonstration: Energy transformations (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 is a store of electrical 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 higher capacitance) lifts the weight further. Hence energy stored depends on both C and V.

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 capacitor gain its energy store? (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.)

PD across capacitor against charge

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 transformed was Q × V/2.

A more general approach says that in moving the charge ΔQ, the pd does not change significantly, so the energy transformed is V × ΔQ. 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 =  1/2 QV = 1/2 CV2 = 1/2 Q2/C

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

Worked example: 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 = 1/2 CV2 = 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.

Charge in electric field of capacitor

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

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

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Episode 128: Energy stored by a capacitor (Word, 192 KB)