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

Answer

B

Explanation

Apply shell theorem, the total charge upto distance r can be calculated as follows :
`=4pir^(2)*dr*rho`
`=4pir^(2)*dr*rho_(0)[(5)/(4)-(r)/(R)]`
`=4pirho_(0)[(5)/(4)r^(2)dr-(r^(3))/(R)dr]`
`intdq =q=4pirho_(0)int_(0)^(t)((5)/(4)r^(2)dr-(r^(2))/(R)dr)`
`=4pirho_(0)[(5)/(4)(r^(3))/(3)-1/R(r^(4))/(4)\]`
`E=(kq)/(r^(2))`
`=(1)/(4piepsi_(0))(1)/(r^(2))*4pirho_(0)[(5)/(4)((r^(3))/(3))-(r^(4))/(4R)]`
`E=(rho_(0)r)/(4epsi_(0))((5)/(3)-(r)/(R))`