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Energy of a charged capacitor

Since capacitors have the ability to store charge they are also a source of electrical energy. This energy may be released slowly in some electrical circuits, in a precise time as in timing circuits or rapidly in a camera flash unit. Care must be taken when touching capacitors because you can't tell if they are charged simply by looking at them. Although they may be disconnected from a supply they may still retain a charge, and this stored energy can give you a serious shock!

Consider a parallel-plate capacitor as shown in Figure 1. Imagine that one plate carries a charge +Q and that the other plate is earthed. If we take a small charge dq from one plate to the other, the work done will be v dq where v is the potential across the plates. If the initial charge on the positive plate is Q then the total energy lost in completely discharging the capacitor is:

Energy stored = - integral [v dq] = ½Q²/C
(taking the integral from 0 to Q)

Using Q = CV gives some alternative formulae:

Energy stored in a capacitor = ½Q²/C = ½CV² = ½QV

If the plates had been moved so as to increase the area but keep their separation the same the new energy would be: Energy = ½ Q²/C
Q would be constant but C would increase ( C = εA/d with a bigger area) and so the overall energy would decrease. C = εA/d = Q/V.
The charge density would have decreased. Moving apart charges of the same sign (i.e those on each plate) would require no work.

Example problems
Example problem A parallel-plate air capacitor of area 25 cm² and with plates 1 mm apart is charged to a potential of 100 V.
(a) Calculate the energy stored in it.
(b) The plates of the capacitor are now moved a further 1 mm apart with the power supply connected. Calculate the energy change.
(c) If the power supply had been disconnected before the plates had been moved apart, what would have been the energy change in this case?

(a) Energy = ½ CV² = ½ε_oAV²/d = [8.85 x 10^-12 x 25 x 10^-4 x 10⁴]/2x0.001 = 1.1x10^-7J

(b) New plate separation = 0.002 m; potential across plates is still 100 V.
New energy = 0.5 x original energy = 0.55x10^-7 J
The difference in energy is explained by the movement of charge in the wires as the capacitor partly discharges to maintain the potential.
(c) New plate separation = 0.002 m; the charge on plates is unchanged but the potential increases. New energy = ½ Q²/C = ½ Q²t/ε_oA and since t is doubled the energy will be doubled to 2.2 x 10^-7 J.

This increase in energy is explained by the addition of energy in the movement of the plates apart against their mutual attraction.

The energy stored in a capacitor used for camera flash unit is about 1 J. This is released in around 1 ms giving an output power of 1 kW!

Measurement of the energy in a stored capacitor

The energy stored by a large capacitor may be studied using the following three experiments using a capacitor of large capacitance - 10 000 mF is suitable. In the first experiment the energy is converted to potential energy as a small motor lifts a small load, while in the second heat energy is used to light one or more light bulbs. In the final experiment heat energy is produced in a heating coil.

Experiment 1

Set up the apparatus shown in Figure 2. First charge the capacitor to 10 V and then by throwing the switch allow it to discharge through the motor. Measure the height through which the weight is raised and hence calculate the mechanical energy gained.

The initial electrostatic energy stored in the capacitor was ½ CV = 0.5 J
Compare this with the mechanical energy gained. Where else has the initial energy gone?

Experiment 2

Set up the circuit shown in Figure 3. Charge the capacitor to 3 V and then discharge it through one lamp. Then charge it to 6 V and discharge it first through two lamps in series and then two lamps in parallel. Compare and contrast the brightness and the time for which the lamps light in each case.

Experiment 3

Set up the apparatus shown in Figure 4.

Charge the capacitor to 30 V and then discharge it through the heating coil. This coil should consist of 2 m of 32 s.w.g constantan wire. The temperature rise produced in the coil should be measured with a copper-constantan thermocouple. The effect on the temperature of a number of charges and discharges should be investigated.

Do these experiments confirm the preceding equations for the energy stored in a charged capacitor?

A VERSION IN WORD IS AVAILABLE ON THE SCHOOLPHYSICS CD

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