A charge Q is distributed over three concentric spherical shell of radii $\mathrm{a}, \mathrm{b}, \mathrm{c}(\mathrm{a}<\mathrm{b}<\mathrm{c})$ such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where $\mathrm{r}<\mathrm{a}$, would be:

(D) $\frac{\mathrm{Q}(\mathrm{a}+\mathrm{b}+\mathrm{C})}{4 \pi \epsilon_{0}\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right)}$

(A) $\frac{Q}{12 \pi \epsilon_{0}} \frac{a b+b c+c a}{a b c}$

(B) $\frac{Q\left(a^{2}+b^{2}+c^{2}\right)}{4 \pi \epsilon_{0}\left(a^{3}+b^{3}+c^{3}\right)}$

(C) $\frac{Q}{4 \pi \epsilon_{0}(a+b+c)}$

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Electrostatics

A charge Q is distributed over three concentric spherical shell of radii

a, b, c (a < b < c)

such that their surface charge densities are equal to one another.

The total potential at a point at distance r from their common centre, where

r < a

, would be:

(A) $\frac{Q}{12 π ϵ _{0}} \frac{ab + b c + c a}{ab c}$
(B) $\frac{Q ( a ^{2} + b ^{2} + c ^{2} )}{4 π ϵ _{0} ( a ^{3} + b ^{3} + c ^{3} )}$
(C) $\frac{Q}{4 π ϵ _{0} ( a + b + c )}$
(D) $\frac{Q ( a + b + C )}{4 π ϵ _{0} ( a ^{2} + b ^{2} + c ^{2} )}$ ✓ Correct

Answer: (D)

Equal surface charge density ties the shell charges to their areas,

σ = \frac{Q _{a}}{4 π a ^{2}} = \frac{Q _{b}}{4 π b ^{2}} = \frac{Q _{c}}{4 π c ^{2}}

, so

Q_{a} : Q_{b} : Q_{c} = a^{2} : b^{2} : c^{2}

and each is

Q_{i} = \frac{Q i ^{2}}{a ^{2} + b ^{2} + c ^{2}}

. For a point inside all three shells,

r < a

, every shell contributes its own surface potential

\frac{k Q _{i}}{i}

, the same everywhere inside it. Adding the three,

V = \frac{k}{a ^{2} + b ^{2} + c ^{2}} (Q \frac{a ^{2}}{a} + Q \frac{b ^{2}}{b} + Q \frac{c ^{2}}{c}) = \frac{Q ( a + b + c )}{4 π ϵ _{0} ( a ^{2} + b ^{2} + c ^{2} )}

. The

i^{2}

from the charge and the

1/ i

from the inside-shell potential combine to leave a clean

a + b + c

on top.

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option