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- 1. Capacitors
- 2. <ul><li>A capacitor is a device for storing electrical charge. </li></ul><ul><li>Capacitors consist of a pair of conducting plates separated by an insulating material (oil, paper, air) </li></ul><ul><li>The measure of the extent to which a capacitor can store charge is called Capacitance. </li></ul><ul><li>(measured in farads F, or more usually microfarads F or picofarads pF. </li></ul>
- 3. <ul><li>For a charge of Q coulombs and a potential difference V across the capacitor, the capacitance C is defined as: </li></ul><ul><li>C = Q/V </li></ul>
- 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. <ul><li>Total charge Q = Q 1 + Q 2 + Q 3 </li></ul><ul><li>= V(C 1 + C 2 + C 3 ) </li></ul><ul><li>Therefore equivalent capacitor </li></ul><ul><li>C = Q/V = Q 1 /V + Q 2 /V + Q 3 /V = C 1 + C 2 + C 3 </li></ul><ul><li>So for capacitors in parallel </li></ul><ul><li>C = C 1 + C 2 + C 3 </li></ul>
- 6. <ul><li>You can think about this another way. </li></ul><ul><li>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. </li></ul>
- 7. Capacitors in Series V C 1 C 2 C 3 V 1 V 2 V 3 Q Individual charges are equal. Why?
- 8. <ul><li>V 1 = Q/C 1 ; V 2 = Q/C 2 ; V 3 = Q/C 3 </li></ul><ul><li>But V = V 1 + V 2 + V 3 = Q(1/C 1 + 1/C 2 + 1/C 3 ) </li></ul><ul><li>AND V/Q = 1/C so </li></ul><ul><li>1/C = 1/C 1 + 1/C 2 + 1/C 3 </li></ul>
- 9. <ul><li>All capacitors in series carry the same charge which is equal to the charge carried by the system as a whole. </li></ul><ul><li>The potential difference is divided amongst the capacitors in inverse proportion to their capacitance. </li></ul>
- 10. Energy of a Charged Capacitor <ul><li>When a capacitor is charged, work is done in charging it. </li></ul><ul><li>So energy must be stored in the capacitor. </li></ul><ul><li>Now Q is proportional to V since </li></ul><ul><li>Q = CV, so a graph of Q against V is a straight line. </li></ul>
- 11. Graph of Q against V 0 V Q
- 12. Calculating work done <ul><li>Work done = area under curve = integral of QV dV = ½ QV. </li></ul><ul><li>Alternatively we can say that the total charge Q moves through an average p.d. of (0+V)/2 </li></ul><ul><li>So work done = energy stored = ½ V x Q </li></ul><ul><li>I.e. W = ½ QV = ½ CV 2 = ½ Q 2 /C </li></ul><ul><li>Notice that this is just the area under the graph. </li></ul>
- 13. <ul><li>Now the energy produced by the battery is QV </li></ul><ul><li>(energy = current x time x voltage) </li></ul><ul><li>But the energy stored by the capacitor I just ½ QV. </li></ul><ul><li>So where has half of the energy gone? </li></ul><ul><li>HEAT! </li></ul><ul><li>½ QV is lost as heat whether you have a high resistance circuit or a low one. </li></ul><ul><li>For low R, the charging time is short, for high R it is long but the energy loss is the same ½ QV. </li></ul>

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