Three concentric metal shells A, B and C of respective radii $a, b$ and $c$ ($a<b<c$) have surface charge densities $+\sigma,-\sigma$ and $+\sigma$ respectively. The potential of shell B is :

(C) $\frac{\sigma}{\epsilon_{o}}\left[\frac{a^{2}-b^{2}}{b}+c\right]$

(A) $\frac{\sigma}{\epsilon_{o}}\left[\frac{b^{2}-c^{2}}{c}+a\right]$

(B) $\frac{\sigma}{\epsilon_{o}}\left[\frac{a^{2}-b^{2}}{a}+c\right]$

(D) $\frac{\sigma}{\epsilon_{o}}\left[\frac{b^{2}-c^{2}}{b}+a\right]$

JEE Main · 2018 · 08 Apr 2018 · Offline · Electrostatics

Three concentric metal shells A, B and C of respective radii

a, b

and

c

(

a < b < c

) have surface charge densities

+ σ, - σ

and

+ σ

respectively. The potential of shell B is :

Answer: (C)

The potential at B is the sum of what all three shells produce there. A shell contributes

\frac{k Q}{r}

at points inside it, using its own radius, and

\frac{k Q}{d}

at points outside it, using the distance

d

. Point B lies outside A, on B itself, and inside C, so with charges

Q_{A} = 4 π a^{2} σ

Q_{B} = - 4 π b^{2} σ

and

Q_{C} = 4 π c^{2} σ

the contributions are

\frac{k 4 π a ^{2} σ}{b}

\frac{k 4 π b ^{2} ( - σ )}{b}

and

\frac{k 4 π c ^{2} σ}{c}

. Writing

k = \frac{1}{4 π ϵ _{0}}

and adding,

V_{B} = \frac{σ}{ϵ _{0}} (\frac{a ^{2}}{b} - b + c) = \frac{σ}{ϵ _{0}} (\frac{a ^{2} - b ^{2}}{b} + c)

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option