Question

If on the concentric hollow spheres of radii r and $R(>r)$ the charge Q is distributed such that their surface densities are same then the potential at their common centre is

• A. $\frac{Q(R^2+r^2)}{4\pi {ϵ}_0(R+r)}$
  
• B. $\frac{QR}{R+r}$
• C. $Zero$
• D. $\frac{Q(R+r)}{4\pi {ϵ}_0(R^2+r^2)}$

Answer 1

The correct option is D $\frac{Q(R+r)}{4\pi {ϵ}_0(R^2+r^2)}$

$q_1+q_2=Q$ and $\frac{q_1}{4\pi r^2}=\frac{q_2}{4\pi R^2}$ (given)
$q_1=\frac{Q{r}^2}{R^2+r^2}$ and $q_2=\frac{Q{R}^2}{R^2+r^2}$
Potential at common centre
$\frac{1}{4\pi {ϵ}_0}[\frac{Q{r}^2}{(R^2+r^2)r}+\frac{Q{R}^2}{(R^2+r^2)R}]=\frac{Q(R+r)}{4\pi {ϵ}_0(R^2+r^2)}$