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. `(rho_(0)r)/(3epsilon_(0))((5)/(4) - (r )/(R ))`
B. `(4pi rho_(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

Consider a thin spherical shell of radius x and thickness dx as shown in the figure.
Volume of the shell, `dV = 4pi x^(2)dx`
Let us draw a Gaussian surface of radius `r(r lt R)` as shown in the figure. Total charge enclosed inside the Gaussian surface is
`Q_(i n) = underset(0)overset(r )int rho dV = underset(0)overset(r )int rho_(0)((5)/(4) - (x)/(R )) 4pi x^(2) dx = 4pi rho_(0) underset(0)overset(r )int((5)/(4)x^(2) - (x^(3))/(R )) dx`
`= 4pi rho_(0)[(5)/(12)x^(3) - (x^(4))/(4R)]_(0)^(r ) = 4pi rho_(0) [(5)/(12)r^(3) - (r^(4))/(4R)]`
`= (4pi rho_(0))/(4)[(5)/(3)r^(3) - (r^(4))/(R )] = pi rho_(0)[(5)/(3)r^(3) - (r^(4))/(R )]`
According to Gauss.s law,
`E4pi r^(2) = (Q_(i n))/(epsilon_(0)) rArr E4pi r^(2) = (pi rho_(0))/(epsilon_(0))[(5)/(3)r^(3) - (r^(4))/(R )]`
`E = (pi rho_(0)r^(3))/(4pi r^(2) epsilon_(0))[(5)/(3) - (r )/(R )], E = (rho_(0)r)/(4 epsilon_(0))[(5)/(3) - (r )/(R )]`