Question

A thin spherical conducting shell of radius R has a charge q. A point charge Q is placed at the centre of the shell. Find

1. The charge density on the outer surface of the shell and
2. the potential at a distance of (R/2) from the centre of the shell.

Answer 1

i. A charge of q is induced on the outer surface of the sphere. A charge of magnitude Q is placed on the outer surface of the sphere. Therefore, the total charge on the outer surface of the sphere is Q − q. Surface charge density at the outer surface,

${\displaystyle {\sigma }_{\text{outer}}=\frac{Q-q}{4\pi R^2}}$

ii. Potential (V₁) at point P due to charge Q

${\displaystyle V_1=\frac{kQ}{r}}$

= ${\displaystyle \frac{Q}{4\pi {\epsilon }_{0}r}}$

= ${\displaystyle \frac{2Q}{4\pi {\epsilon }_{0}R}}$    ...(i)

Potential (V₂) at point P due to sphere (i.e., same at all points inside the shell)

${\displaystyle V_2=\frac{kQ}{r}}$

= ${\displaystyle \frac{q}{4\pi {\epsilon }_{0}R}}$    ...(ii)

Total potential (V) = V₁ + V₂ 

From equation (i) & (ii),

${\displaystyle V=\frac{2Q}{4\pi {\epsilon }_{0}R}+\frac{q}{4\pi {\epsilon }_{0}R}}$

${\displaystyle V=\frac{1}{4\pi {\epsilon }_{0}R}(2Q+q)}$