The region between two concentric spheres of radii a and b (gt a) contains volume charge The I where C is a constant and r is the radial distance as shown in figure. A point dens') barge q is placed at the origin, `r = 0`. c find the value of C for which the electric field in the region between the spheres is constant (i.e. r independent).

Options

A. `Q/(2pi a^2)`
B. `Q/(2pi (b^2-a^2))`
C. `2Q/(pi (a^2-b^2))`
D. `2Q/(pi a^2)`

Answer

A

Explanation

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)`