A solid non conducting sphere of radius R has a non-uniform charge distribution of volume charge density, `rho=rho_(0)r/R`, where `rho_(0)` is a constant and r is the distance from the centre of the sphere. Show that : (i) the total charge on the sphere is `Q=pirho_(0)R^(3)` (ii) the electric field inside the sphere has a magnitude given by, `E=(KQr^(2))/R^(4)`

Answer

`0, (2Klambda)/r, 0`

Explanation

(i) `q=int dq=underset(0)overset(R)(int) rho_(0) r/Rxx4pi r^r=(rho_(0)x4pi R^(4))/R=pi rho_(0)R^(3)`
(ii) `E=(KQ')/r^(2)=K/r^(2)xx(rho_(0)pir^(4))/R=(KQr^(2))/R^(4)` where
`Q'=underset(0)overset(tau)(int)rho_(0)x/Rxx4pix^(2)dx=rho_(0)/(4R)xx4pixxr^(4)=(rho_(0)pi r^(4))/R`