Question

A sphere of radius R carries a charge of volume charge density ρ ar, where a is a constant and r denotes the distance from the centre of the sphere. Calculate the total charge enclosed by the sphere and the electric field at points lying inside and outside the sphere. 

Answer 1

$\rho=ar,$

$\rho=\frac QV,$

$Q=\int_0^Rar(4\pi r^2)dr=4a\pi\int_0^Rr^3dr=4a\pi\frac{R^4}4=\pi aR^4,$

$E=\frac{Q}{4\pi r^2\varepsilon_0}=\frac{aR^4}{4\varepsilon_0r^2}.$