Question

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

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

Answer 1

The correct option is D $4\pi {ϵ}_0\left(\frac{b^2}{b-a}\right)$

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
$4\pi {ϵ}_0\frac{ab}{b-a}+4\pi {ϵ}_{0}b$
$=4\pi {ϵ}_{0}b[\frac{a}{b-a}+1]=4\pi {ϵ}_{0}b\left[\frac{a+b-a}{b-a}\right]$
$C=4\pi {ϵ}_0\frac{b^2}{b-a}$