First slide

Capacitance

Question

Two spherical conductors A and B of radii a and b (b > o) are placed con-centrically in air as shown in fig. B is given a charge + Q and A is earthed. The equivalent capacitance of the system is

Difficult

A

$4 {πε}_{0} \frac{ab}{b - a}$
B

$4 {tπε}_{0} (a + b)$
C

$4 {πε}_{0} b$
D

$4 {πε}_{0} (\frac{b^{2}}{b - a})$

Solution

The charge Q given to outer sphere is distributed as e outside and Q₂ inside. The inside charge Q₂ induces a charge - Q₂ on outside of inner sphere and + Q₂ on inside
of inner sphere which is earthed. The inside of outer sphere and outside of inner sphere constitute a spherical condenser having capacitance $[4 {πε}_{0} {ab / (b - a)}]$ and
the outside of the outer constitutes an isolated sphere of capacitance $4 {πε}_{0} b$ .
Now the effective capacitance C is given by
$\begin{array}{l} C = 4 {πε}_{0} \{\frac{ab}{(b - a)}\} + 4 {πε}_{0} b \\ = 4 {πε}_{0} b [\frac{a}{(b - a)} + 1] \\ = 4 {πε}_{0} b [\frac{a + b - a}{(b - a)}] \\ = 4 {πε}_{0} \{\frac{b^{2}}{(b - a)}] \end{array}$

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