Question

A charge $Q$ is uniformly distributed over the surface of two concentric conducting spheres of radii $R$ and $r(R>r)$ Such that surface charge densities are same for both spheres.Then potential at common centre of these spheres-

• A. $\frac{KQ(R+r)}{Rr}$
• B. $\frac{KQ(R+r)}{R^2+r^2}$
• C. $\frac{KQ}{\sqrt{R^2+r^2}}$
• D. $KQ(\frac{1}{R}-\frac{1}{r})$

Answer 1

The correct option is B $\frac{KQ(R+r)}{R^2+r^2}$

Let the charge on first and second sphere be $Q_1$ and $Q_2$ respectively.
$\therefore$ $Q_1+Q_2=Q$
Let the surface charge density be $\sigma$.
$\therefore$ $\sigma \left(4\pi r^2\right)+\sigma \left(4\pi R^2\right)=Q$
$⟹\sigma =\frac{Q}{4\pi (r^2+R^2)}$
$⟹Q_1=\sigma \left(4\pi r^2\right)=\frac{Q{r}^2}{(r^2+R^2)}$
Similarly, we get $Q_2=\frac{Q{R}^2}{(r^2+R^2)}$
Potential at their common center $V_o=\frac{KQ}{r}+\frac{KQ}{R}$
$⟹V_o=\frac{KQ{r}^2}{r(r^2+R^2)}+\frac{KQ{R}^2}{R(r^2+R^2)}=\frac{KQ(r+R)}{(r^2+R^2)}$
Hence option B is correct.