Question

Two concentric spherical shells of radii $R$ and $r$ have similar charges with equal surface charge densities $\left(\sigma \right)$.
What is the electric potential at their common centre ?

• A. $\sigma R/{ϵ}_0$
• B. None of these
• C. $\frac{\sigma }{{ϵ}_0}(R-r)$
• D. $\frac{\sigma }{{ϵ}_0}(R+r)$

Answer 1

The correct option is D $\frac{\sigma }{{ϵ}_0}(R+r)$


Charge on smaller shell $S_1$ is

$Q_{S_1}$ = $\sigma \times 4\pi r^2$

Similarly, charge on $S_2$ is

$Q_{S_2}$ = $\sigma \times 4\pi R^2$

We know that potential at the centre due to shells will be

$V_{centre}=V_{\text{due to}{S}_1}+V_{\text{due to}{S}_2}$

$\Rightarrow V_{centre}=\frac{k{Q}_{S_1}}{r}+\frac{k{Q}_{S_2}}{R}$

$\Rightarrow V_{centre}=k[\sigma .4\pi r+\sigma .4\pi R]$

$\Rightarrow V_{centre}=\frac{\sigma }{{ϵ}_0}(r+R)$

Hence, option $\left(c\right)$ is correct.