Question

A solid conducting sphere, having a charge Q, is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of -4Q, the new potential difference between the same two surfaces is ______.

Options

• -2V

• 2V

• 4V

• V

Answer 1

A solid conducting sphere, having a charge Q, is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of -4Q, the new potential difference between the same two surfaces is V.

Explanation:

Given: The surrounding conducting spherical shell has no charge, the solid conducting sphere has a charge of Q, and there is a potential difference of V between the two spherical surfaces.

To find: If the shell is given a charge of -4Q, what would the new potential difference be between the two spherical surfaces. Assume that b is the outer radius of the surrounding conducting spherical shell and that an is the radius of the solid conducting sphere.

(i)

(ii)

About diagram (i):

The potential at a solid conducting sphere's surface equals

`V_s = (kQ)/a` `(k = 1/(4piε_0) "is a constant.")`

The surface potential of the surrounding conducting spherical shell:

`V_{sh} = (kQ)/b`

As a result, the potential difference between two spherical surfaces is as follows:

`V = V_s - V_{sh} = kQ(1/a - 1/b)`

About diagram (ii)

The potential at a solid conducting sphere's surface equals

`V_s^' = (kQ)/a - (k(4Q))/b`

The surface potential of the surrounding conducting spherical shell:

`V_{sh}^' = (kQ)/b - (k(4Q))/b`

As a result of this, the potential difference between two spherical surfaces is:

`V_s^' - V_{sh}^' = ((kQ)/a - (k(4Q))/b) - ((kQ)/b - (k(4Q))/b)` = V