A solid sphere of radius R has a charge Q distributed in its volume with a charge density ρ = kra, 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 at r = R, find the value of a.

Using Gauss's law : electric field at r=R/2 is E1.
4π(R/2)2 = Qen/ε0
or E1 = Qen/4πε0(R/2)2
Here Qen = ∫0R/2 ρ 4πr2dr = 4π∫0R/2 krar2dr = 4πk∫0R/2ra+2dr = 4πk(R/2)a+3/a+3
now E1 = 4πk(R/2)a+3/a+3 / 4πε0(R/2)2 = k(a+3)-1(R/2)a+1/ε0
Similarly electric field at r=R is E2 = Qen/4πε0R2
where Qen = ∫0R 4πkra+2dr = 4πkRa+3/a+3
now E2 = k(a+3)-1Ra+1/ε0
As E1 = (1/8)E2 ⇒ k(a+3)-1(R/2)a+1/ε0 = (1/8)k(a+3)-1Ra+1/ε0
or (1/2)a+1 = (1/8)1
or a+1 = 3 ⇒ a = 2