Charge in a conducting spherical shell ?

A conducting spherical shell of inner radius a and outer radius b carries a net charge Q. A point charge q is placed at the center of this shell. Determine the surface charge density on (a) the inner surface of the shell and (b) the outer surface of the shell.

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1 Answer
Nov 26, 2017

Inner Surface: #\quad \sigma_a = q_a/(4\pia^2) = -q/(4\pia^2)#
Outer Surface:#\quad \sigma_b = q_b/(4\pib^2) = (Q+q)/(4\pib^2)#


Since the electric field must necessarily vanish inside the volume of the conducting sphere, the charges must drift in such a way as to cancel the electric field due to the charge #q# at the centre.

Inner Surface: Consider an imaginary sphere enclosing the inner surface of radius #a#, lying just outside this surface and inside the volume of the conducting sphere. By Gauss's Law the electric flux through this surface is related to the total charge enclosed by this surface,

By Gauss Law, #\Phi_E = \oint vec E.vec(ds) = (q+q_a)/\epsilon_0#
where #q# is the charge at the centre and #q_a# is the total induced charge on the inner surface.

Since the Electric field vanishes everywhere inside the volume of a good conductor, its value is zero everywhere on the Gaussian surface we have considered. So the surface integral is zero.

#\Phi_E = \oint vec E.vec(ds) = (q+q_a)/\epsilon_0 = 0; \qquad \rightarrow q_a = -q# ...... (1)
This is the total charge induced on the inner surface. Because the electric field from the centra;l charge is spherically symmetric, this induced charge must be distributed uniformly distributed too.

So the charge density on the inner sphere is : #\sigma_a = q_a/(4\pia^2) = -q/(4\pia^2)#

Outer Surface: The net charge on the outer surface has two components - free charge #q_b^{"free"} = Q# and induced charge #q_b^{"ind"}#

#q_b = q_b^{"ind"} + q_b^{"free"} = q_b^{"ind"}+Q#

Because the induced charges are a result of polarization due to the electric field of the central charge, the net induced charge on the inner and outer surfaces of the good conductor must be zero :

#q_a + q_b^{"ind"} = 0; \qquad q_b^{"ind"} = -q_a#

Writing #q_a# in terms of #q# using (1), #\quad q_b^{"ind"} = -q_a = q#

Thus the total charge on the outer surface is : #q_b = Q + q#

So the charge density on the outer sphere is : #\sigma_b = q_b/(4\pib^2) = (Q+q)/(4\pib^2)#