Question

A charge Q is distributed over two concentric hollow spheres of radii r and $(R > r)$ such that the surface densities are equal. The potential at the common centre is

• A. $\frac{Q(R^{2} + r^{2})}{4\pi\varepsilon_{0}(R + r)}$
• B. $\frac{Q}{R + r}$
• C. Zero
• D. $\frac{Q(R + r)}{4\pi\varepsilon_{0}(R^{2} + r^{2})}$

Answer 1

Answer: (D)

$\frac{Q(R + r)}{4\pi\varepsilon_{0}(R^{2} + r^{2})}$

Answer 2

If $q_{1}$ and $q_{2}$ are the charges on spheres of radius r and R respectively, in accordance with conservation of charge

$Q = q_{1} + q_{2}$ ….(i)

and according to the given problem $\sigma_{1} = \sigma_{2}$

i.e., $\frac{q_{1}}{4\pi r^{2}} = \frac{q_{2}}{4\pi R^{2}}$ Þ $\frac{q_{1}}{q_{2}} = \frac{r^{2}}{R^{2}}$ …. (ii)

So equation (i) and (ii) gives $q_{1} = \frac{Qr^{2}}{(R^{2} + r^{2})}$and

$q_{2} = \frac{QR^{2}}{(R^{2} + r^{2})}$

Potential at common centre

$V = \frac{1}{4\pi\varepsilon_{0}}\left\lbrack \frac{q_{1}}{r} + \frac{q_{2}}{R} \right\rbrack = \frac{1}{4\pi\varepsilon_{0}}\left\lbrack \frac{Qr}{(R^{2} + r^{2})} + \frac{QR}{(R^{2} + r^{2})} \right\rbrack = \frac{1}{4\pi\varepsilon_{0}}.\frac{Q(R + r)}{(R^{2} + r^{2})}$