Capacitors
5.0 Capacitance of a earthed sphere by a concentric spherical shell
5.0 Capacitance of a earthed sphere by a concentric spherical shell
A charge $q$ is given to the shell. Let $-q_1$ be the charge appears on the outer surface of the earthed sphere, then $+q_1$, charge will come on the inner surface of the shell and the remaining ($q-q_1$) charge will appear on the outer surface of shell. Potential on the inner sphere is zero. Therefore,$$\frac{1}{{4\pi {\varepsilon _ \circ }}}\left[ {\frac{{ - {q_1}}}{a} + \frac{{{q_1}}}{b} + \frac{{q - {q_1}}}{c}} \right] = 0$$ $${q_1} = q\left( {\frac{{ab}}{{bc + ab - ac}}} \right)$$
From the principle of generator, potential difference between the two is,$$V = \frac{{{q_1}}}{{4\pi {\varepsilon _ \circ }}}\left( {\frac{{b - a}}{{bc + ab - ac}}} \right)$$ $$ = \frac{{{q_1}}}{{4\pi {\varepsilon _ \circ }}}\left( {\frac{{b - a}}{{ab}}} \right)$$
Substituting the value of $q_1$,we have$$V = \frac{{{q_1}}}{{4\pi {\varepsilon _ \circ }}}\left( {\frac{{b - a}}{{bc + ab - ac}}} \right)$$
and hence the capacitance of the system is,$$C = \frac{q}{V} = 4\pi {\varepsilon _ \circ }\left( {\frac{{bc + ab - ac}}{{b - a}}} \right)$$
or $$C = 4\pi {\varepsilon _ \circ }\left( {\frac{{ab}}{{b - a}} + c} \right)$$
- The above expression can also be obtained by assuming the system as the combination of two capacitor $(1)$ that between the inner sphere and the inner surface of the shell and $(2)$ that between the outer surface of the shell and infinity. Capacitance of the former is $\frac{{4\pi {\varepsilon _ \circ }ab}}{{b - a}}$ and that of later is $4\pi {\varepsilon _ \circ }C$.
- Capacitance of the system $ = 4\pi {\varepsilon _ \circ }\left( {\frac{{ab}}{{b - a}} + c} \right)$. Further, if the thickness of the shell is negligible or, $c \simeq b$, then capacitance of the system is $$C \simeq 4\pi {\varepsilon _ \circ }\left( {\frac{{ab}}{{b - a}} + b} \right)$$ or $$C = 4\pi {\varepsilon _ \circ }\left( {\frac{{{b^2}}}{{b - a}}} \right)$$ when $c \simeq b$.
