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# Charging C

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### Transcript

• 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 &quot; Q=CV &quot; => 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 ,
• 7.
• 8.
• 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.