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* The Greek letter m (mu) indicates multiplication by 10 6 .
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Inductance and Capacitance
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It is important to note that the amount of energy stored is proportional to the capacitance C and the square of the voltage v. Further notes, regarding resistance and capacitance, appear in the Appendix.*
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Capacitors in Series and in Parallel
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In practical work it is sometimes necessary to use both series and parallel connections of capacitors, Let us rst investigate the series connection with the aid of Figs. 123 and 124. In Fig. 123, the C s denote the capacitance, in farads, of each of the individual seriesconnected capacitors.
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Fig. 123
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Fig. 124
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In Fig. 124, CT denotes the capacitance of a single capacitor that would have the same capacitance as the series connection of the n individual capacitors in Fig. 123. This means that theoretically, for purposes of analysis, the n series-connected capacitors of Fig. 123 can be replaced with the single equivalent capacitor of CT farads of Fig. 124. A formula for nding the value of CT can be found by making use of the fact that the magnitude of charge is the same on both plates of a capacitor. This is because an amount of positive charge, owing into one plate, repels the same amount of positive charge out of the other plate (see discussion following Fig. 122). With this in mind, consider Fig. 123. When the switch is closed, a charge q ows into the left-hand plate of C1 , thus forcing the same amount of charge out of the right-hand plate of C1 into the left-hand plate of C2 . This forces the same amount of charge out of the right-hand plate of C2 into the left-hand plate of the next capacitor, and so on down the line, with the result that all the series-connected capacitors in Fig. 123 have the same magnitude of charge of q coulombs on their plates. Note that this satis es the basic requirement that, at all times, charge owing out of the positive terminal of the battery must be equal to the charge owing into the negative terminal. With the above in mind, now make use of the equation v q=C (eq. (184)). Using this equation, and remembering that all capacitors in Fig. 123 have the same charge q, we have, for Fig. 123, V1 q=C1 V2 q=C2 . . . Vn q=Cn where V1 voltage on capacitor C1 where V2 voltage on capacitor C2 . . . where Vn voltage on capacitor Cn
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* See note 14 in Appendix.
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CHAPTER 7 Inductance and Capacitance
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Since the sum of the left-hand sides of the above equations is equal to the sum of the right-hand sides, we have that V1 V2 Vn q 1=C1 1=C2 1=Cn Or, since V1 V2 Vn the battery voltage V, the last equation becomes   1 1 1 V q 186 C1 C2 Cn Now let us apply the same battery voltage V to the equivalent capacitance CT in Fig. 124. Since CT is to be equivalent to the circuit of Fig. 123, it must carry the same charge q, and hence, by eq. (184), it must be true that V q CT 187
Since the left-hand sides of the last two equations are equal, their right-hand sides are also equal, and upon making use of this fact we get the desired relationship 1 1 1 1 CT C1 C2 Cn Or, if we wish, we can invert both sides of the last equation and write that CT 1 1=C1 1=C2 1=Cn 189 188
Thus either equation, (188) or (189), allows us to calculate the equivalent capacitance, CT , of a series connection of n capacitors, where C1 ; C2 ; . . . ; Cn are the capacitances of the individual capacitors. When using series capacitors we must be able to calculate the voltage that will appear across each capacitor when the series string is connected to a battery of V volts, as in Fig. 123. This is important, because excessively high voltage across one of the capacitors could cause voltage breakdown of that capacitor, with subsequent failure of the whole series string. A formula that will allow us to calculate such a voltage can be found as follows. Let Cx be the capacitance of any one of the series capacitors in Fig. 123, and let Vx be the voltage on that capacitor. Then, from the general equation q Cv, we have that, for this capacitor, q Cx Vx but also, by eq. (187), q VCT where V is the battery voltage. From inspection of the above two equations we see that Cx Vx VCT giving us the important result Vx V CT Cx 190
Next, let us consider the problem of nding the equivalent capacitance of a parallel connection of n capacitors. Such a parallel connection is shown in Fig. 125, with the equivalent single capacitor of capacitance CT shown in Fig. 126.
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