Question

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

Answer 1

Correct Answer - A
Due to spherically symmetric charge distribution, the charge on the spherical surfacec fo radius `r` in Fig (when `r lt R`) is

`q = int rho. dV =int_(0)^(r ) rho_(0) ((5)/(4) - (x)/(R ) 4 pi x^(2) dx)`
`= rho_(0) ((5)/(4) 4pi (x^(3))/(3) -(4pi)/(R ) (x^(4))/(4))_(0)^(r ) = rho_(0) ((5)/(3) pi r^(3) -(pi r^(4))/(R ))`
`= (rho_(0)r)/(4 in_(0)) ((5)/(3) - (r )/(R ))`