Question

Two spherical conductors $A$ and $B$ of radii a and $b(b>a)$ are placed concentrically in air. $B$ is given a charge $+Q$ and $A$ is earthed. The equivalent capacitance of the system is -

• A. $4\pi {\epsilon }_0\left[\frac{ab}{b-a}\right]$
• B. $4\pi {\epsilon }_0[a+b]$
• C. $4\pi {\epsilon }_{0}b$
• D. $4\pi {\epsilon }_0\left(\frac{b^2}{b-a}\right)$

Answer 1

Answer: (D)

The charge $Q$ given to outer sphere distributes as $Q_1$ outside and $Q_2$ inside which induces charge $-Q_2$ on outside of inner sphere, $+Q_2$ on inside of inner sphere which is earthed.
The inside of outer and the inner sphere constituting a spherical condenser having capacitance $4\pi {ϵ}_0\frac{ab}{b-a}$ and the outside of the outer constitutes an isolated sphere of capacitance $4\pi {ϵ}_{0}b$.
$\therefore$ the effective capacitance is
$4\pi {ϵ}_0\frac{ab}{b-a}+4\pi {ϵ}_{0}b$
$=4\pi {ϵ}_{0}b\left[\frac{a}{b-a}+1\right]$
$=4\pi {ϵ}_{0}b\left[\frac{a+b-a}{b-a}\right]$
$C=4\pi {ϵ}_0\frac{b^2}{b-a}$