2.14 COMBINATION OF CAPACITORS

We can combine several capacitors of capacitance \( C_1\),\( C_2\),...,\( C_n\) to obtain a system with some effective capacitance \( C \). The effective capacitance depends on the way the individual capacitors are combined. Two simple possibilities are discussed below.

2.14.1 Capacitors in series

Figure 2.26 shows capacitors \(C_{1}\) and \(C_{2}\) combined in series. The left plate of \(C_{1}\) and the right plate of \(C_{2}\) are connected to two terminals of a battery and have charges \(Q\) and \(–Q\) , respectively. It then follows that the right plate of \(C_{1}\) has charge \(–Q\) and the left plate of \(C_{2}\) has charge \(Q\). If this was not so, the net charge on each capacitor would not be zero. This would result in an electric field in the conductor connecting \(C_{1}\) and \(C_{2}\). Charge would flow until the net charge on both \(C_{1}\) and \(C_{2}\) is zero and there is no electric field in the conductor connecting \(C_{1}\) and \(C_{2}\). Thus, in the series combination, charges on the two plates (\(±Q\)) are the same on each capacitor. The total potential drop \(V\) across the combination is the sum of the potential drops \(V_{1}\) and \(V_{2}\) across \(C_{1}\) and \(C_{2}\), respectively.

\[ V=V_{1}+V_{2}=\frac{Q}{C_{1}}+\frac{Q}{C_{2}} \] (2.55)

i.e., \[ \frac{V}{Q}=\frac{1}{C_{1}}+\frac{1}{C_{2}} \] (2.56)

Now we can regard the combination as an effective capacitor with charge \( Q \) and potential difference \( V \). The effective capacitance of the combination is

\[ C= \frac{Q}{V} \] (2.57)

We compare Eq. (2.57) with Eq. (2.56), and obtain

\[ \frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}} \] (2.58)

The proof clearly goes through for any number of capacitors arranged in a similar way. Equation (2.55), for \( n \) capacitors arranged in series, generalises to

\[ V=V_{1}+V_{2}+\ldots+V_{\mathrm{n}}=\frac{Q}{C_{1}}+\frac{Q}{C_{2}}+\ldots+\frac{Q}{C_{\mathrm{n}}} \] (2.59)

Following the same steps as for the case of two capacitors, we get the general formula for effective capacitance of a series combination of \( n \) capacitors:

\[ \frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\ldots+\frac{1}{C_{n}} \] (2.60)

2.14.2 Capacitors in parallel

Figure 2.28 (a) shows two capacitors arranged in parallel. In this case, the same potential difference is applied across both the capacitors. But the plate charges (\( ±Q_1 \)) on capacitor 1 and the plate charges (\( ±Q_2 \)) on the capacitor 2 are not necessarily the same:

\[ Q_1=C_1 V, \text{ } Q_2=C_2 V \] (2.61)

The equivalent capacitor is one with charge

\[ Q=Q_1+Q_2 \](2.62)

and potential difference \( V \).

\[ Q=CV=C_1V+C_2V \](2.63)

The effective capacitance \( C \) is, from Eq. (2.63),

\[ C=C_1+C_2 \] (2.64)

The general formula for effective capacitance \( C \) for parallel combination of \( n \) capacitors [Fig. 2.28 (b)] follows similarly,

\[ Q=Q_1+Q_2+...+Q_n \](2.65)

i.e., \[ CV=C_1V+C_2V+...+C_nV \](2.66)

which gives

\[ C=C_1+C_2+...+C_n \](2.67)