A solid ball of radius R has a charge density `rho` given by `rho = rho_(0)(1-(r )/(R )) for 0le r le R` The electric field outside the ball is :

Options

A. `(rho_(0)R^(3))/(epsilon_(0)r^(2))`
B. `(2 rho_(0)R^(3))/(3 epsilon_(0) r^(2))`
C. `(3 rho_(0)R^(3))/(4 epsilon_(0) r^(2))`
D. `(rho_(0) R^(2))/(12 epsilon_(0) r^(2))`

Answer

D

Explanation

Charge density, `rho = rho_(0)(1 - (r)/(R))`
` dq = rho dv`
`q_(i n ) = int dq = rho dv = rho_(0) (1 - (r)/(R)) 4 pi r^(2) dr` (`:. dv = 4 pi r^(2) dr`)
`= 4 pi rho_(0) int_(0)^(R) (1 - (r)/( R)) r^(2) dr = 4 pi rho _(0) [ (R^(3))/( 12)]`
`q = (pi rho_(0) R^(3))/( 3) so, E . 4 pi r^(2) = ((pi rho_(0) R^(3))/( 3 in _(0)))`
`:. ` Electric field outside the ball, ` E = ( rho_(0) R^(3))/( 12 in _(0) r^(2))`