Question

Consider three concentric metallic spheres A, B and C of radii a, b, c respectively where a < b < c. A and B are connected whereas C is grounded. The potential of the middle sphere B is raised to V then the charge on the sphere C is

• A. $-4\pi {\epsilon }_{0}V\frac{bc}{c-b}$
• B. $-4\pi {\epsilon }_{0}V\frac{ac}{c-a}$
• C. $+4\pi {\epsilon }_{0}V\frac{bc}{c-b}$
• D. $zero$

Answer 1

Answer: (A)

$-4\pi {\epsilon }_{0}V\frac{bc}{c-b}$

Since the spheres A and B are connected, the charge on A is $\text{zero}$

Let the charge on B be $+Q$. As the sphere C is grounded, by charge induction, charge on sphere C is $-Q$.

For the given charge distribution, Potential on the surface of B is given by sum of potentials duet to charges on A, B and C.

Charge on A = 0 $\Rightarrow$ Potential at B due to A = 0.

Charge on B = $+Q\Rightarrow$ Potential at B due to B = $\frac{Q}{4\pi {ϵ}_{0}b}$.

Charge on C = $-Q\Rightarrow$ Potential at B due to C = $\frac{-Q}{4\pi {ϵ}_{0}c}$.

The potential on B is given by

$V=\frac{Q}{4\pi {ϵ}_0}(\frac{1}{b}-\frac{1}{c})=\frac{Q}{4\pi {ϵ}_0}\times \frac{c-b}{bc}$

Therefore,

$Q=4\pi {ϵ}_{0}V\frac{bc}{c-b}$

Thus, Charge on Sphere C = $-Q=-4\pi {ϵ}_{0}V\frac{bc}{c-b}$