Let `P(r)=(Q)/(piR^4)r` be the charge density distribution for a solid sphere of radius R and total charge Q. For a point 'p' inside the sphere at distance `r_1` from the centre of the sphere, the magnitude of electric field is:

Options

A. `(Qr_1^2)/(4piepsilon_0R^4)`
B. `(Qr_1^2)/(3piepsilon_0R^4)`
C. 0
D. `Q/(4piepsilon_0r_1^2)`

Answer

A

Explanation

Let us visualise a thin spherical shell like the region of radius x and radial thickness dx. The magnitude of charge in this spherical shell segment can be written as follows :
`dQ_"in"=rho(4pix^2dx)=(Q/(piR^4)x)(4pix^2dx)=(4Q)/R^4 x^3dx`
On integrating the above relation, we can calculate charge within radius r by using integration limits of x from 0 to r . We get the following result :
`Q_"in"=(4Q)/R^4 int_0^r x^3dx=(Qr^4)/R^4` ...(i)
`phi=Exx4pir^2` ...(ii)
Gauss.s law can be written as follows :
`phi=Q_"in"/epsilon_0` ...(iii)
Now substituting equation (i) and (ii) in equation (iii), we get the following :
`Exx4pir^2=((Qr^4)/(R^4))/epsilon_0`
`rArr E=(Qr^2)/(4piepsilon_0R^4)`