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. (a) `(rho_0r)/(4epsilon_0)(5/3-r/R)`
B. (b) `(4pirho_0r)/(3epsilon_0)(5/3-r/R)`
C. (c) `(4rho_0r)/(4epsilon_0)(5/4-r/R)`
D. (d) `(rho_0r)/(3epsilon_0)(5/4-r/R)`

Answer 1

Correct Answer - A
Let us consider a spherical shell of radius x and thickness dx.
Charge on this shell
`dq=rho.4pix^2dx=rho_0(5/4-x/R)*rpix^2dx`
`:.` Total charge in the spherical region from centre to r (rltR) is
`q=int dq=4pirho_0int_0^r(5/4-x/R)x^2dx`

`=4pirho_0[5/4*r^3/3-1/R*r^4/4]=pirho_0r^3(5/3-r/R)`
`:.` Electric field at r, `E=(1)/(4piin_0)*q/r^2`
`=(1)/(4piin_0)*(pirho_0r^3)/(r^2)(5/3-r/R)=(rho_0r)/(4in_0)(5/3-r/R)`