Let there be a spherically symmetric charge distribution with charge density varying as `rho(r)=rho(5/4-r/R)` upto `r=R`, and `rho(r)=0` for `rgtR`, where r is the distance from the origin. The electric field at a distance r(rltR) from the origin is given by

Options

A. `(4pirho_(0)r)/(3epsilon_(0)) (5/4-r/R)`
B. `(rho_(0)r)/(4epsilon_(0)) (5/3-r/R)`
C. `(4rho_(0)r)/(3epsilon_(0))(5/4-r/R)`
D. `(rho_(0)r)/(3epsilon_(0)) (5/4-r/R)`

Answer

B

Explanation

Apply shell theorem, the total charge upto distance r can be calculated as followed
`dq=4pr^(2). dr .r`
`=4pir^(2).dr . rho_(0)[5/4-r/R]`
`=4pi rho_(0)[5/4r^(2)dr - (r^(3))/R dr]`
`int dq=q=4 pi rho_(0) int_(0)^(r)(5/4r^(2) dr - (r^(3))/R dr)`
`=4pi rho_(0)[5/4(r^(2))/3 - 1/R (r^(4))/4]`
`E=(kq)/(r^(2))`
`=1/(4 pi epsilon_(0)) . 1/(r^(2)) . 4 p rho_(0)[5/4((r^(2))/3) - (r^(4))/(4R)]`
`E=(rho_(0)r)/(4 epsilon_(0)) (5/3-r/R)`