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

Options

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

Answer

C

Explanation

Total charge
`Q=underset(0)overset(tau)(int)rho dV=underset(0)overset(tau)(int)rho_(0)(5/4-r/R)4pi r^(2)dr`
`=4pi rho_(0) underset(0)overset(tau)(int)((5r^(2))/4-r^(3)/R)dr=4pi rho_(0)[(5r^(3))/12-r^(4)/(4R)]`
`E=(KQ)/r^(2)=1/(4pi in_(0)r^(2))4pi rho_(0)[5/12 r^(3)-r^(4)/(4R)]`
`=(rho_(0)r)/(4 in_(0))[5/3-r/R]`