Question

A charge $Q$ is uniformly distributed over the surface of two conducting concentric spheres of radii $R$ and $r\left(R>r\right).$ Then potential at the 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. $kQ\left(\frac{1}{R}-\frac{1}{r}\right)$

Answer 1

Answer: (B)

Let the surface charge density is $\sigma .$
And, $q_A$ and $q_B$ are charges on shell $A$ and $B$

$\therefore Q=q_A+q_B$
$orQ=4\pi r^2\sigma +4\pi R^2\sigma$
$\therefore \sigma =\frac{Q}{4\pi \left(r^2+R^2\right)}$
$\therefore V_C=V_A+V_B$
$\therefore V_C=\frac{q_A}{4\pi {ϵ}_{0}r}+\frac{q_B}{4\pi {ϵ}_{0}R}$
$=\frac{1}{4\pi {ϵ}_0}[\frac{q_A}{r}+\frac{q_B}{R}]=k(\frac{4\pi r^2\sigma }{r}+\frac{4\pi R^2\sigma }{R})$
$=k\left(4\pi r\sigma +4\pi R\sigma \right)$
$=4\pi k\left[\frac{Qr}{4\pi \left(R^2+r^2\right)}+\frac{QR}{4\pi \left(r^2+R^2\right)}\right]$
$=\frac{kQ(R+r)}{(R^2+r^2)}\left(Here,k=\frac{1}{4\pi {ϵ}_0}\right)$