Question

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 1

`rho=kr^(2)`
`E(r-R/2)=1/8E(r-R)`
`(q_("enclosed"))/(4piepsilon_(0)(R//2)^(2))=1/8Q/(4piepsilon_(0)R^(2))`
or `32 q_("enclosed")=int_(0)^(R//2) kr^(a) 4pir^(2) dr =(4pik)/((a+3))(R/2)^((a+3)`
or `Q=(4pik)/((a+3)) R^((a+3))`
or `Q/(q_("enclosed"))=2^((a+3))`
or `2^(a+3)=32`
or a=2