JEE Main · 2016 · 03 Apr 2016 · Offline · Electrostatics

The region between two concentric spheres of radii 'a' and 'b', respectively (see figure), has volume charge density

ρ = \frac{A}{r}

, where

A

is a constant and

r

is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is :

(A) $\frac{Q}{2 π ( b ^{2} - a ^{2} )}$
(B) $\frac{2 Q}{π ( a ^{2} - b ^{2} )}$
(C) $\frac{2 Q}{π a ^{2}}$
(D) $\frac{Q}{2 π a ^{2}}$ ✓ Correct

Answer: (D)

Solution

Take a Gaussian sphere of radius

r

between the shells. It encloses the central charge

Q

plus the shell charge from

a

r

\int_{a}^{r} \frac{A}{r} 4 π r^{2} d r = 2 π A (r^{2} - a^{2})

. Gauss's law gives

E 4 π r^{2} = \frac{1}{ε _{0}} [Q + 2 π A (r^{2} - a^{2})]

, so

E = \frac{Q - 2 π A a ^{2}}{4 π ε _{0} r ^{2}} + \frac{A}{2 ε _{0}}

. Only the first term depends on

r

, so for

E

to be constant it must vanish, requiring

Q - 2 π A a^{2} = 0

. Hence

A = \frac{Q}{2 π a ^{2}}

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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JEE Main · 2016 · 03 Apr 2016 · Offline · Electrostatics

Solution

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