A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric 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 $- 3 Q$ , the new potential difference between the same two surfaces is :

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Answer 1

The electric field in between the shell and sphere is
$E . π x^{2} = \frac{Q _{e n}}{ϵ _{0}} = \frac{Q}{ϵ _{0}}$ (using Gauss's law)
$E = \frac{Q}{4 π ϵ _{0} x ^{2}}$
The potential difference between the shells is $d V = V_{r} - V_{R} = \int_{r}^{R} E d x = \int_{r}^{R} \frac{Q}{4 π ϵ _{0} x ^{2}} d x = \frac{Q}{4 π ϵ _{0}} (1 / r - 1 / R)$
Thus, $V = \frac{Q}{4 π ϵ _{0}} (1 / r - 1 / R)$
As the potential difference between solid sphere and hollow shell depends on the radii of two spheres and charge on the inner sphere, Since the two values have not changed, potential difference does not change. Hence the potential difference remains $V$ .

Answer 2

The electric field in between the shell and sphere is

E . π x^{2} = \frac{Q _{e n}}{ϵ _{0}} = \frac{Q}{ϵ _{0}}

(using Gauss's law)

E = \frac{Q}{4 π ϵ _{0} x ^{2}}

The potential difference between the shells is

d V = V_{r} - V_{R} = \int_{r}^{R} E d x = \int_{r}^{R} \frac{Q}{4 π ϵ _{0} x ^{2}} d x = \frac{Q}{4 π ϵ _{0}} (1 / r - 1 / R)

Thus,

V = \frac{Q}{4 π ϵ _{0}} (1 / r - 1 / R)

As the potential difference between solid sphere and hollow shell depends on the radii of two spheres and charge on the inner sphere, Since the two values have not changed, potential difference does not change. Hence the potential difference remains

V

.

$V$

$2 V$

$4 V$

$- 2 V$

A conducting shell $S_{1}$ having a charge Q is surrounded by an uncharged concentric conducting spherical shell $S_{2}$ . Let the potential difference between $S_{1}$ and that $S_{2}$ be V. If the shell $S_{2}$ is now given a charge -3Q, the new potential difference between the same two shells is :

A solid conducting sphere of radius $a$ has a net positive charge $2 Q$ . A conducting spherical shell of inner radius $b$ and outer radius $c$ is concentric with the solid sphere and has a net charge $- Q$ . The surface charge density on the inner and outer surfaces of the spherical shell will be

Revise with Concepts

Electric Field and Potential Due to Conductor

V

2V

4V

−2V

A conducting shell S1​ having a charge Q is surrounded by an uncharged concentric conducting spherical shell S2​. Let the potential difference between S1​ and that S2​ be V. If the shell S2​ is now given a charge -3Q, the new potential difference between the same two shells is :

A solid conducting sphere of radius a has a net positive charge 2Q. A conducting spherical shell of inner radius b and outer radius c is concentric with the solid sphere and has a net charge −Q. The surface charge density on the inner and outer surfaces of the spherical shell will be