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.

Explanation

Let us consider as sphereical shell of radius x and thickness dx. The volume of this shell is `4pix^(2)(dx)`.
The charged enclosed in this spherical shell is
`dq=(4pi x^(2))dxxxkx^(a)`
`:.dq=4pi kx^(2+a)dx`.

For r=R
The total charge enclosed in the sphere of radius R is
`Q=int_(0)^(R)4pi k x^(2+a)dx=4pi d(R^(3+a))/((3+a))`
`:.` The electric field at r=R is
`E_(1)=1/(4 pi epsilon_(0))(4 pi kR^(3+a))/((3+a)R^(2))=1/(4 pi epsilon_(0))(4 pi k)/((3+a))R^(1+a)`
For r=R/2
The total charge enclosed in the sphere of radius R/2 is
`Q.=int_(0)^(R//2)4 pi kx^(2+a)dx=(4pik(R//2)^(3+a))/((3+a))`
`:.` The electric field at r=R/2 is
`E_(2)=1/(4 pi epsilon_(0))(4pik)/(3+a)((R//2)^(3+a))/((R//2)^(2))=1/(4 pi epsilon_(0))(4pik)/((3+a))(R/2)^(1+a)`
Given `E_(2)=1/8E_(1)`
`:.1/(4 pi epsilon_(0))(4pi k)/((3+a))(R/2)^(1+a)=1/(2^(3))xx1/(4 pi epsilon_(0))(4pik)/((3+a))R^(1+a)`
`implies1+a=3impliesa=2`