Capacitors

• 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.

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

C = Q/V

Capacitors in ParallelWe put 3 capacitors with capacitances C1, C2 and C3 in parallel

V

Q1

Q2

Q3

C1

C2

C3

Charges on individual capacitors:

Q1 = C1V

Q2 = C2V

Q3 = C3V

• Total charge Q = Q1 + Q2 + Q3

= V(C1 + C2 + C3)

Therefore equivalent capacitor

C = Q/V = Q1/V + Q2/V + Q3/V = C1 + C2 + C3

So for capacitors in parallel

C = C1 + C2 + C3

• 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.

Capacitors in Series

V

C1 C2 C3

V1 V2 V3

Q

Individual charges are equal. Why?

• V1 = Q/C1; V2 = Q/C2; V3 = Q/C3

• But V = V1 + V2 + V3 = Q(1/C1 + 1/C2 + 1/C3)

• AND V/Q = 1/C so

• 1/C = 1/C1 + 1/C2 + 1/C3

• 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.

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.

Graph of Q against V

0 V

Q

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 CV2 = ½ Q2/C• Notice that this is just the area under the graph.

• 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.