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) Write the expression for electric field at a point x r2 from the centre of the shell.

(a) Charge at the outer surface of the spherical shell is Q

Since, charge q is placed at center, thus the charge induced at the inner surface is -q and outer surface is +q.

Therefore, total charge on inner surface will be -q and on outer surface will be Q+q.

Now,

Surface charge density on inner surface = Total charge/Inner surface area = -q/4πr12

Surface charge density on the outer surface  = Total charge/Outer surface area = Q+q/4πr22

(b) The electric field at a distance x from the center, using Gauss law is:

E(x) = 1/4πε0(Q+q/x2)

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