This document discusses charging and discharging capacitors in circuits. When a capacitor is charging, current flows through the circuit and decreases over time as charge builds up on the capacitor. The time it takes for the charge to reach 63% of its final value is called the time constant, represented by τ, which equals the resistance R multiplied by the capacitance C. When discharging, the current direction reverses but also decreases over time as the stored charge depletes from the capacitor. The energy stored in the capacitor is 1/2 CV^2 and equals the work done by the battery during charging.

1. Lecture 21 - Capacitors in circuits Charging a capacitor (semi-qualitative). Charging a capacitor (quantitative). The time constant. Discharging a capacitor. Energy Considerations. The End

2. Charging a capacitor - diagram.

3. Charging a capacitor (semi-qualitative). At time t=0 the switch is closed, with the capacitor initially uncharged. A current will flow  =V c +V R =I 0 R , as initially V c =0. Thus the initial current is I 0 =  /R . Now a charge begins to build on the capacitor, introducing a reverse voltage. The current falls, and stops when the P.D. across C is  . Final charge is given by " Q=CV " => Q 0 =C  .

4. Charging a capacitor (quantitative). Apply Kirchoff's loop rule.

5. Charging a capacitor (cont) Where Q 0 = C  = the final charge on the capacitor.

6. Charging a capacitor (cont). To find the current, differentiate since I=dQ/dt . By considering time zero, when the current is I 0 ,

9. The time constant. The time constant  =RC . The units are seconds ( t/RC is dimensionless). The time taken for the charge to rise to 1-(1/e) of the final value in the circuit. The current to fall by 1/e of its initial value.

10. Discharging capacitor - diagram.

11. Discharging a capacitor. Apply Kirchoff's loop rule.

12. Discharging a capacitor (cont) To find the current...

13. Discharging a capacitor (cont) To find the current... Note the sign, the current flow has reversed! But, when t=0, I=I 0 , so

14. Energy Considerations. During charging, a total charge Q=C  flows through the battery. The battery does work W=Q 0  =C  2 . The energy stored in the capacitor is ½ QV= ½ Q 0  = ½ C  2 . Where's the other half?

15. Energy considerations (cont). Solve by setting x=2t/RC . Which, when added to the energy stored on the capacitor, equals the work done by the battery.

16. Finally… E-M depends a lot on integrals, vectors etc. shows how useful they are. It is one of the foundations of physics but: it can be rather formal, encouraging the precise thinking that we expect of any academic training; it is rather far removed from the everyday, but that develops the imagination we expect from a physicist.