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The region between two concentric sphere...

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

`Q/(2pi a^2)`

B

`Q/(2pi (b^2-a^2))`

C

`2Q/(pi (a^2-b^2))`

D

`2Q/(pi a^2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Draw, Gaussian surface at distance r from centre,
Using Gauss law `(Q+ int_a^r A/r 4pie^2 dr)/e_0=E 4 pi r^2`
`E4 pi e_0 r^2=Q+A 4 pi [(r^2-a^2)/2)`
`E=1/(4 pi e_0) [Q/r^2+A2 p ((r^2+a^2)/r^2))], E=1/(4pi e_0) [Q/r^2 +A2 pi -(A2 pi a^2)/r^2))`
For E to be constant (ie independent of r)
`Q/r^2- (2A pi a^2)/r^2=0 .......(i) therefore E=1/(4 pi e_0) times A times 2pi `.........(ii)
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 (using equation (i)] :
As `Q=2 pi Aa^2 i.e., A=Q/(2 pi a^2)`
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