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# Charging C

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### Charging C

1. 1. Lecture 21 - Capacitors in circuits <ul><li>Charging a capacitor (semi-qualitative). </li></ul><ul><li>Charging a capacitor (quantitative). </li></ul><ul><li>The time constant. </li></ul><ul><li>Discharging a capacitor. </li></ul><ul><li>Energy Considerations. </li></ul><ul><li>The End </li></ul>
2. 2. Charging a capacitor - diagram.
3. 3. Charging a capacitor (semi-qualitative). <ul><li>At time t=0 the switch is closed, with the capacitor initially uncharged. </li></ul><ul><li>A current will flow  =V c +V R =I 0 R , as initially V c =0. Thus the initial current is I 0 =  /R . </li></ul><ul><li>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  . </li></ul><ul><li>Final charge is given by &quot; Q=CV &quot; => Q 0 =C  . </li></ul>
4. 4. Charging a capacitor (quantitative). <ul><li>Apply Kirchoff's loop rule. </li></ul>
5. 5. Charging a capacitor (cont) <ul><li>Where Q 0 = C  = the final charge on the capacitor. </li></ul>
6. 6. Charging a capacitor (cont). <ul><li>To find the current, differentiate since I=dQ/dt . </li></ul><ul><li>By considering time zero, when the current is I 0 , </li></ul>
7. 9. The time constant. <ul><li>The time constant  =RC . </li></ul><ul><li>The units are seconds ( t/RC is dimensionless). </li></ul><ul><li>The time taken for the charge to rise to 1-(1/e) of the final value in the circuit. </li></ul><ul><li>The current to fall by 1/e of its initial value. </li></ul>
8. 10. Discharging capacitor - diagram.
9. 11. Discharging a capacitor. <ul><li>Apply Kirchoff's loop rule. </li></ul>
10. 12. Discharging a capacitor (cont) <ul><li>To find the current... </li></ul>
11. 13. Discharging a capacitor (cont) <ul><li>To find the current... </li></ul><ul><li>Note the sign, the current flow has reversed! </li></ul><ul><li>But, when t=0, I=I 0 , so </li></ul>
12. 14. Energy Considerations. <ul><li>During charging, a total charge Q=C  flows through the battery. </li></ul><ul><li>The battery does work W=Q 0  =C  2 . </li></ul><ul><li>The energy stored in the capacitor is ½ QV= ½ Q 0  = ½ C  2 . </li></ul><ul><li>Where's the other half? </li></ul>
13. 15. Energy considerations (cont). <ul><li>Solve by setting x=2t/RC . </li></ul><ul><li>Which, when added to the energy stored on the capacitor, equals the work done by the battery. </li></ul>
14. 16. Finally… <ul><li>E-M depends a lot on integrals, vectors etc. shows how useful they are. </li></ul><ul><li>It is one of the foundations of physics but: </li></ul><ul><ul><li>it can be rather formal, encouraging the precise thinking that we expect of any academic training; </li></ul></ul><ul><ul><li>it is rather far removed from the everyday, but that develops the imagination we expect from a physicist. </li></ul></ul>