Question

A spherical conducting shell of inner radius r₁ and outer radius r₂ has a charge Q.

(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?

(b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.

Answer 1

(a) Charge placed at the centre of a shell is +q. Hence, a charge of magnitude −q will be induced to the inner surface of the shell. Therefore, the total charge on the inner surface of the shell is −q.

Surface charge density at the inner surface of the shell is given by the relation,

`sigma_1 = "Total Charge"/"Inner surface area" = (-q)/(4pi"r"_2^2)` ..........(1)

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

`sigma_2 = "Total Charge"/"Outersurface area" = ("Q" + "q")/(4pi"r"_2^2)` ..........(2)

(b) Yes

The electric field intensity inside a cavity is zero, even if the shell is not spherical and has any irregular shape. Take a closed loop such that a part of it is inside the cavity along a field line while the rest is inside the conductor. Net work done by the field in carrying a test charge over a closed loop is zero because the field inside the conductor is zero. Hence, the electric field is zero, whatever is the shape.