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 ($\pm Q$) 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

$$V=V_1+V_2+\dots +V_{\mathrm{n}}=\frac{Q}{C_1}+\frac{Q}{C_2}+\dots +\frac{Q}{C_{\mathrm{n}}}$$ (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:

$$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots +\frac{1}{C_n}$$ (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 ($\pm Q_1$) on capacitor 1 and the plate charges ($\pm Q_2$) on the capacitor 2 are not necessarily the same:

$$Q_1=C_{1}V,Q_2=C_{2}V$$ (2.61)

The equivalent capacitor is one with charge

$$Q=Q_1+Q_2$$(2.62)

and potential difference $V$.

$$Q=CV=C_{1}V+C_{2}V$$(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., $$CV=C_{1}V+C_{2}V+...+C_{n}V$$(2.66)

which gives

$$C=C_1+C_2+...+C_n$$(2.67)