The intensity of an electric field depends only on the coordinates x and y as follows,
`E = (a(xhati +y hatj))/(x^(2)+y^(2))` where, a is constant and `hati` and `hatj` are the unit vectors of the x and y axes. Find the charges within a sphere of radius R with the centre at the origin.

Answer

A, D

Explanation

At any point `P(x,y,z)` on the sphere a unit vector perpendicular to the sphere radially outwards is
`hatn=x/(sqrt(x^2+y^2+z^2)) hati+y/(sqrt(x^2+y^2+z^2))hatj+z/(sqrt(x^2+y^2+z^2))hatk`
`x/Rhati+y/Rhatj+z/Rhatk` as `x^2+y^2 +z^2=R^2`
Let us find the electric flux passing through a small area `dS` at point P on the sphere
`dphi=E hatn dS={(ax^2)/(R(x^2+y^2))+(ay^2)/(R(x^2+y^2)}dS`
`=(a/R)aS`
here we not that `dphi` is independent of the coordinates x,y, and z. Therefore,total flux passing through the sphere
`phi=int dphi=a/R int dS=(a/R)(4piR^2)`
`=4piR`
From Gauss's law, `
`hi=q_("in")/epsilon_0` or `(4piaR)=q_("in")/epsilon_0`
`:. q_("in")=4piepsilon_0aR`