A capacitor is a device that stores electrical charge between two conducting plates separated by an insulating material. The capacitance of a capacitor determines how much charge it can store and is measured in Farads. For capacitors in parallel, the total capacitance is the sum of the individual capacitances. For capacitors in series, the reciprocal of the total capacitance is the sum of the reciprocals of the individual capacitances. When a capacitor is charged, it stores electrical potential energy equal to 1/2 CV^2, with half the energy used to charge it dissipated as heat.

1. Capacitors

2. A capacitor is a device for storing electrical charge. Capacitors consist of a pair of conducting plates separated by an insulating material (oil, paper, air) The measure of the extent to which a capacitor can store charge is called Capacitance. (measured in farads F, or more usually microfarads  F or picofarads pF.

3. For a charge of Q coulombs and a potential difference V across the capacitor, the capacitance C is defined as: C = Q/V

4. Capacitors in Parallel We put 3 capacitors with capacitances C 1 , C 2 and C 3 in parallel V Q 1 Q 2 Q 3 C 1 C 2 C 3 Charges on individual capacitors: Q 1 = C 1 V Q 2 = C 2 V Q 3 = C 3 V

5. Total charge Q = Q 1 + Q 2 + Q 3 = V(C 1 + C 2 + C 3 ) Therefore equivalent capacitor C = Q/V = Q 1 /V + Q 2 /V + Q 3 /V = C 1 + C 2 + C 3 So for capacitors in parallel C = C 1 + C 2 + C 3

6. You can think about this another way. All capacitors in parallel have the same potential difference across them but the stored charge is divided amongst them in direct proportion to the capacitance.

7. Capacitors in Series V C 1 C 2 C 3 V 1 V 2 V 3 Q Individual charges are equal. Why?

8. V 1 = Q/C 1 ; V 2 = Q/C 2 ; V 3 = Q/C 3 But V = V 1 + V 2 + V 3 = Q(1/C 1 + 1/C 2 + 1/C 3 ) AND V/Q = 1/C so 1/C = 1/C 1 + 1/C 2 + 1/C 3

9. All capacitors in series carry the same charge which is equal to the charge carried by the system as a whole. The potential difference is divided amongst the capacitors in inverse proportion to their capacitance.

10. Energy of a Charged Capacitor When a capacitor is charged, work is done in charging it. So energy must be stored in the capacitor. Now Q is proportional to V since Q = CV, so a graph of Q against V is a straight line.

11. Graph of Q against V 0 V Q

12. Calculating work done Work done = area under curve = integral of QV dV = ½ QV. Alternatively we can say that the total charge Q moves through an average p.d. of (0+V)/2 So work done = energy stored = ½ V x Q I.e. W = ½ QV = ½ CV 2 = ½ Q 2 /C Notice that this is just the area under the graph.

13. Now the energy produced by the battery is QV (energy = current x time x voltage) But the energy stored by the capacitor I just ½ QV. So where has half of the energy gone? HEAT! ½ QV is lost as heat whether you have a high resistance circuit or a low one. For low R, the charging time is short, for high R it is long but the energy loss is the same ½ QV.