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.

Explanation

(a) Charge palce at the centre of a shell is +q. hence, a charge of magnitude -q will be induce to the inner surface of the inner surface of the shell. Therefore, 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 realation,
`sigma_(1)=("Total charge")/("inner surface area")=(-q)/(4pir_(1)^(2)).....(i)`
A charge of +q is induced on the outer surface of the shell. a charge of magnitude Q is place on the outer surface of the shell . therefore , 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")/("Outer surface area")=(Q+q)/(4pir_(2)^(2)).....(ii)`
(b) yes
The electric field intensitinside a cavity is zero, even if the shell is not spherical and has any irregular shape. take a closed loop such taht 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, electric field is zero, whatever is the shape.