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,

`sigma_"outer" = (Q - q)/(4 pi R^2)`

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

`V_1 = (kQ)/r`

= `Q/(4 pi epsilon_0 r)`

= `(2 Q)/(4 pi epsilon_0 R)`    ...(i)

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

`V_2 = (kQ)/r`

= `q/(4 pi epsilon_0 R)`    ...(ii)

Total potential (V) = V₁ + V₂ 

From equation (i) & (ii),

`V = (2 Q)/(4 pi epsilon_0 R) + q/(4 pi epsilon_0 R)`

`V = 1/(4 pi epsilon_0 R)(2Q + q)`