An electricity and magnetism problem by Jaswinder Singh

Consider a Thin Spherical Gaussian Surface centered around the given sphere with radius $r , ~ r > R$ and thickness $dr$

Denote the Electric Flux through that Gaussian surface by $\phi$ . Due to symmetry, the electric field is of uniform magnitude and normal to th egaussian surface

$\phi = \frac{\text{Charge Contained inside Gaussian Surface}}{\epsilon_0} = E \cdot \oint dA$

Charge Contained inside Gaussian Surface = $Q + \int \limits _R ^r { \rho 4 \pi r^2 dr} = Q + \int \limits _R ^r{4 A \pi r dr} = Q + [2 A \pi r^2] _R ^r = Q + 2 A \pi (r^2 - R^2)$

Hence, $\phi = \frac{Q + 2A(r^2 - R^2)}{\epsilon_0} = E \cdot 4\pi r^2 \Rightarrow E = \frac{Q}{4 \pi \epsilon _0 r^2} - \frac{A \cdot R^2}{2 \epsilon_0 r^2}$

To make E independant of $R$ , $Q = \boxed{2 \pi A R^2}$