There are three co-centric conducting spherical shells A, B and C of radii a, b and c respectively (cba) and they are charged with charge q_ 1 , q_ 2 and q_ 3 respectively. The potentials of the spheres A, B and C

Question

There are three co-centric conducting spherical shells A, B and C of radii a, b and c respectively (c>b>a) and they are charged with charge q_ 1 , q_ 2 and q_ 3 respectively. The potentials of the spheres A, B and C

Quantrex Academy · Accepted Answer

Correct answer: C. 1 4 π _ o ( q_ 1 a + q_ 2 b + q_ 3 c ), 1 4 π _ o ( q_ 1 +q_ 2 b + q_ 3 c ), 1 4 π _ o ( q_ 1 +q_ 2 +q_ 3 c )

The potential at any point due to a charged spherical shell of radius

R

and charge

Q

is

V = \frac{1}{4\pi\epsilon_o} \frac{Q}{R}

for points inside or on the surface (

r \le R

), and

V = \frac{1}{4\pi\epsilon_o} \frac{Q}{r}

for points outside (

r > R

).
Let

k = \frac{1}{4\pi\epsilon_o}

. The potentials of the three shells

A, B, C

with radii

a < b < c

and charges

q_1, q_2, q_3

are calculated by superposition.
For shell

A

(

r = a

):
Shell

A

is on its own surface, while it is inside shells

B

and

C

.

V_A = k\frac{q_1}{a} + k\frac{q_2}{b} + k\frac{q_3}{c} = \frac{1}{4\pi\epsilon_o} \left( \frac{q_1}{a} + \frac{q_2}{b} + \frac{q_3}{c} \right)

For shell

B

(

r = b

):
Shell

B

is outside shell

A

, on its own surface, and inside shell

C

.

V_B = k\frac{q_1}{b} + k\frac{q_2}{b} + k\frac{q_3}{c} = \frac{1}{4\pi\epsilon_o} \left( \frac{q_1 + q_2}{b} + \frac{q_3}{c} \right)

For shell

C

(

r = c

):
Shell

C

is outside shells

A

and

B

, and on its own surface.

V_C = k\frac{q_1}{c} + k\frac{q_2}{c} + k\frac{q_3}{c} = \frac{1}{4\pi\epsilon_o} \left( \frac{q_1 + q_2 + q_3}{c} \right)

Comparing these expressions with the given options, option (3) matches correctly.

There are three co-centric conducting spherical shells A, B and C...

Options

Correct answer

Step-by-step solution

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