Question

consider a charged sphere of radius R containing charge q,completely enclosed by a spherical cavity of inner radius a and outer radius b.calculate the charge density on all surface and potential everywhere

Answer 1

Given that,

Inner radius = a

Outer radius = b

Charge = q

Suppose, The charge Q is placed at center of the sphere

We need to calculate the charge density on inner surface

Using formula of charge density

$\sigma_{in}=\dfrac{q}{4\pi r_{in}^2}$

Put the value into the formula

$\sigma_{in}=\dfrac{q}{4\pi a^2}$

The charge on outer surface is

$q'=q+Q$

We need to calculate the charge density on outer surface

Using formula of charge density

$\sigma_{out}=\dfrac{q'}{4\pi r_{out}^2}$

Put the value into the formula

$\sigma_{out}=\dfrac{q+Q}{4\pi b^2}$

We need to calculate the potential on inner surface

Using formula of potential

$V_{in}=\dfrac{kq}{r_{in}}$

Put the value into the formula

$V_{in}=\dfrac{kq}{a}$

We need to calculate the potential on outer surface

Using formula of potential

$V_{out}=\dfrac{kq'}{r_{out}}$

Put the value into the formula

$V_{out}=\dfrac{k(q+Q)}{b}$

Hence, This is the required solution.