20.12

Capacitors in Series and in Parallel

Figure 20-37 shows two different capacitors connected in parallel to a battery. Since the capacitors are in parallel, they have the same voltage V across their plates. However, the capacitors contain different amounts of charge. The charge stored by a capacitor is q = CV (Equation 19.8), so q1 = C1V and q2 = C2V.

Figure 20-37 In a parallel combination of capacitances C1 and C2, the voltage V across each capacitor is the same, but the charges q1 and q2 on each capacitor are different.

As with resistors, it is always possible to replace a parallel combination of capacitors with an equivalent capacitor that stores the same charge and energy for a given voltage as the combination does. To determine the equivalent capacitance CP, note that the total charge q stored by the two capacitors is

This result indicates that two capacitors in parallel can be replaced by an equivalent capacitor whose capacitance is . For any number of capacitors in parallel, the equivalent capacitance is

| (20.18)

Capacitances in parallel simply add together to give an equivalent capacitance. This behavior contrasts with that of resistors in parallel, which combine as reciprocals, according to Equation 20.17. The reason for this difference is that the charge q on a capacitor is directly proportional to the capacitance , whereas the current I in a resistor is inversely proportional to the resistance .
The equivalent capacitor not only stores the same amount of charge as the parallel combination of capacitors, but also stores the same amount of energy. For instance, the energy stored in a single capacitor is (Equation 19.11b), so the total energy stored by two capacitors in parallel is

which is equal to the energy stored in the equivalent capacitor CP.
When capacitors are connected in series, the equivalent capacitance is different from when they are in parallel. As an example, Figure 20-38 shows two capacitors in series and reveals the following important fact: All capacitors in series, regardless of their capacitances, contain charges of the same magnitude, +q and −q, on their plates. The battery places a charge of +q on plate a of capacitor C1, and this charge induces a charge of +q to depart from the opposite plate a′, leaving behind a charge −q. The +q charge that leaves plate a′ is deposited on plate b of capacitor C2 (since these two plates are connected by a wire), where it induces a + q charge to move away from the opposite plate b′, leaving behind a charge of −q. Thus, all capacitors in series contain charges of the same magnitude on their plates. Note the difference between charging capacitors in parallel and in series. When charging parallel capacitors, the battery moves a charge q that is the sum of the charges moved for each of the capacitors: . In contrast, when charging a series combination of n capacitors, the battery only moves a charge q, not nq, because the charge q passes by induction from one capacitor directly to the next one in line.

Figure 20-38 In a series combination of capacitances C1 and C2, the same amount of charge q is on the plates of each capacitor, but the voltages V1 and V2 across each capacitor are different.

Problem solving insight

The equivalent capacitance CS for the series connection in Figure 20-38 can be determined by observing that the battery voltage V is shared by the two capacitors. The drawing indicates that the voltages across C1 and C2 are V1 and V2, so that V = V1 + V2. Since the voltages across the capacitors are V1 = q/C1 and V2 = q/C2, we find that

Thus, two capacitors in series can be replaced by an equivalent capacitor whose capacitance CS can be obtained from 1/CS = 1/C1 + 1/C2. For any number of capacitors connected in series the equivalent capacitance is given by

| (20.19)

Equation 20.19 indicates that capacitances in series combine as reciprocals and do not simply add together as resistors in series do. It is left as an exercise (problem 91) to show that the equivalent series capacitance stores the same electrostatic energy as the sum of the energies of the individual capacitors.
It is possible to simplify circuits containing a number of capacitors in the same general fashion as that outlined for resistors in Example 12 and Figure 20-24. The capacitors in a parallel grouping can be combined according to Equation 20.18, and those in a series grouping can be combined according to Equation 20.19.

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