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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 q1, q2 and q3 respectively. The potentials of the spheres A, B and C respectively, are :

14 π o ( q1a+ q2b+ q3c ), 14 π o ( q1+q2b+ q3c ), 14 π o ( q1+q2+q3c )

14 π o ( q1+q2+q3a ), 14 π o ( q1+q2+q3b ), 14 π o ( q1+q2+q3c )

14 π o ( q1a+ q2b+ q3c ), 14 π o ( q1+q2+q3b ), 14 π o ( q1+q2+q3c )

14 π o ( q1+q2+q3a ), 14 π o ( q1+q2b+ q3c ), 14 π o ( q1a+ q2b+ q3c )

Electrostatics

JEE Main level, single correct, medium

153 of 311

Question 153 of 311

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

Show answer and solution

Answer: (C) $\frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_{1}}{a}+\frac{q_{2}}{b}+\frac{q_{3}}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_{1}+q_{2}}{b}+\frac{q_{3}}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_{1}+q_{2}+q_{3}}{c}\right)$

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

R

and charge

Q

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 \gt R

).
Let

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

. The potentials of the three shells

A, B, C

with radii

a \lt b \lt 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.

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