Question

A solid spherical conducting shell has inner radius $a$ and outer radius $2a$. At the centre of the shell a point charge $+Q$ is located. What must be the charge of the shell be in order for the charge density on the inner and outer surfaces of the shell to be exactly equal?

• A. $-5Q$
• B. $+3Q$
• C. $-4Q$
• D. $+4Q$

Answer 1

The correct option is A $-5Q$

Inner radius $=a$

outer radius $=2a$

point charge $=+Q$

The total charge enclosed for the inner sphere is $q$. Hence, the surface charge density for the inner sphere is

${\sigma }_{inner}=\frac{q}{4\pi R_1^2}=\frac{q}{4\pi a^2}$

The total charge enclosed for the outer sphere is $=Q+q$

Hence, the surface change density for the outer sphere is -

${\sigma }_{inner}=\frac{q}{4\pi {\left(a\right)}^2}$

The electric field at a point at a distance $x$ from the sphere is given by Gauss's law,

by imaging a sphere of radius $2a$$E\times (4\pi .{4a}^2)=\frac{Q_{enclosed}}{{ϵ}_0}$

or, $E=\frac{Q+q}{4\pi {ϵ}_0\times {4a}^2}=\frac{Q+q}{16\pi {ϵ}_0{a}^2}=-5Q$