Question

A spherical conducting shell of inner radius `r_(1)` and outer radius `r_(2)` has a charge Q.
(a) A charge q is placed at the center of the shell. What is the surface charge density on the inner and outer surfaces of the shell ?
(b) Is the electric field intensity inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape ? Explain.

Answer 1

(a) The charge `+Q` resides on the outer surface of the shell. The charge q placed of the center of the shell induces charge `-q` on the inner surface and charge `+q` on the outer surface of the shell. Fig.
Therefore, total charge on inner surface of the shell is `-q` and total charge on the outer surface of the shell is (`Q +q)`.
`:. sigma_(1) = (q)/(4pi r_(1)^(2))`
and `sigma_(2) = (Q +q)/(4pi r_(2)^(2))`
(b) Electric field intensity inside a cavity with no charge is zero, even when the shell has any irregular shape. If we were to take a closed loop, part of which is inside the cavity along a field line, and the rest outside it, then net work done by the field in carrying a test charge over the closed loop will not be zero. This is impossible for an electrostaic field. Hence electric field intensity inside a cavity with no charge is always zero.