A solid sphere of radius R has a charge Q distributed in its volume with a charge density `rho=kr^a`, where k and a are constants and r is the distance from its centre. If the electric field at `r=(R)/(2)` is `1/8` times that `r=R`, find the value of a.

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

Let us visualise one segment of charged in the form of spherical shell of radius x and thickness dx.
`dp=rho(4pix^2)dx=(kx^a)(4pix^2)dx=4pikx^(2+a)dx`
Integrating the above relation
`q=4pik int_0^r x^(2+a) dx rArr q =4pik [x^(3+a)/(3+a)]_0^r`
`rArr q=(4pik)/(3+a) r^(3+a)`...(ii)
Using Gauss.s theorem
`phi=q/epsilon_0 rArr E(4pir^2)=(4pikr^(3+a))/((3+a)epsilon_0)`
`rArr E=(kr^(1+a))/((3+a)epsilon_0)` ...(ii)
Here, `(kR^(1+a))/((3+a)epsilon_0)=8(k(R//2)^(1+a))/((3+a)epsilon_0)`
`rArr R^(1+a) =8 (R//2)^(1+a)`
`rArr 2^(1+a)=8 rArr 1 + a = 3 rArr ` a=2