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.

Answer

B

Explanation

Let us consider a spherical shell of radius x and thickness `dx`. The volume of this shell is `4pix^2(dx)`. The charge enclosed in this spherical shell is
`dq=(4pir^2)dxxxkx^a`
`:.` `dq=4pikx^(2+a)dx`.

For `r=R`:
The total charge enclosed in the sphere of radius R is
`Q=int_0^R4pikx^(2+a)dx=4pik(R^(3+a))/(3+a)`
`:.` The electric field at `r=R` is
`E_1=(1)/(4piepsilon_0)(4pikR^(3+a))/((3+a)R^2)=(1)/(4piepsilon_0)(4pik)/(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)4pikx^(2+a)dx=(4pik(R//2)^(3+a))/(3+a)`
`:.` The electric field at `r=R//2` is
`E_2=(1)/(4piepsilon_0)(4pik)/(3+a)(R//2)^(3+a)/(R//2)^2=(1)/(4piepsilon_0)(4pik)/(3+a)(R/2)^(1+a)`
Given, `E_2=1/8E_1`
`:.` `(1)/(4piepsilon_0)(4pik)/((3+a))(R/2)^(1+a)=(1)/(2^3)xx(1)/(4piepsilon_0)(4pik)/(3+a)R^(1+a)`
`implies 1+a=3impliesa=2`