Capacitors
4.0 Capacitance of spherical conductor
4.0 Capacitance of spherical conductor
Capacitance of a spherical conductor enclosed by an earthed concentric spherical shell
If a charge $q$ is given to the inner spherical conductor it spreads over the outer surface of it and a charge $-q$ appears on the inner surface of the shell. The electric field is produced only between the two.
From the principle of generator the potential difference between the two will depends on the inner charge $q$ only and given by,$$V = \frac{q}{{4\pi {\varepsilon _ \circ }}}(\frac{1}{a} - \frac{1}{b})$$
Hence the capacitance of the system $$C = \frac{q}{V}$$
or $$C = 4\pi {\varepsilon _ \circ }(\frac{{ab}}{{b - a}})\ \ ...(1)$$
Capacitance of a spherical conductor
When a charge $q$ is given to a spherical conductor of radius $R$, the potential on it is,$$V = \frac{1}{{4\pi {\varepsilon _ \circ }}}\frac{q}{R}$$
From this expression we can find that,$$\frac{q}{V} = 4\pi {\varepsilon _ \circ }R = C$$
Thus the capacitance of the spherical conductor is $$C = 4\pi {\varepsilon _ \circ }R$$
This expression can be obtain by putting $b \to \infty $ in equation $(1)$ of above heading.$$C = 4\pi {\varepsilon _ \circ }(\frac{a}{{1 - \frac{a}{b}}})$$
let's $b \to \infty $ and $a=R$
$$C = 4\pi {\varepsilon _ \circ }R$$
i.e. the charged sphere may be regarded as a capacitor in which the outer surface has been removed to infinity.
From the expression $$C = 4\pi {\varepsilon _ \circ }R$$ we can draw the following conclusions.
- $C$ depends on $R$ only. Which we have already stated that $C$ depends on the dimensions of the conductor. Moreover if two conductor have radii ${R_1}$ and ${R_2}$ then,$$\frac{{{C_1}}}{{{C_2}}} = \frac{{{R_1}}}{{{R_2}}}$$
- Earth is also a spherical conductor of radius $R = 6.4 \times {10^6}m$. The capacitance of earth is therefore,$$C = (\frac{1}{{9 \times {{10}^9}}})(6.4 \times {10^6}) \simeq 117 \times {10^{ - 6}}F$$$$C = 711\mu F$$ From here we can see that faraday is a large unit. As capacity of such a huge conductor is only $C = 711\mu F$.
