A sphere of radius R has a volume density of charge `rho= kr`, where r is the distance from the centre of the sphere and k is constant. The magnitude of the electric field which exists at the surface of the sphere is given by (`epsi_(0)` = permittivity of free space)

Options

A. `(4pi kR^4)/(3 epsilon_0)`
B. `( kR)/(3 epsilon_0)`
C. `(4pi kR)/( epsilon_0)`
D. `( kR^2)/(4 epsilon_0)`

Answer

D

Explanation

Volume density `(rho)=kr`
Total charge `(q) = int rhodv = int_0^(R) kr 4pir^2 dr`
`q=4k[r^4/4]_0^R=(4kR^4)/(4)=kR^4`
According to Gaussian law
`ointvecE.vec(ds)=(q_("in"))/(epsilon_0)impliesExx4piR^2=(kR^4)/epsilon_0implies E=(kR^2)/(4 epsilon_0)`