Question

Find the magnitude of the electric field strength as a function of the distance r both inside and outside the ball

Answer 1

Charge within radius r is

$q={\int }_0^r{\rho }_0[1-\frac{r}{R}]4\pi r^{2}dr$

$=4\pi {\rho }_0{\int }_0^r(r^2-\frac{r^3}{R})dr$

$=4\pi {\rho }_0(\frac{r^3}{3}-\frac{r^4}{4R})$

For r < R,

$E=\frac{kq}{r^2}=\frac{{\rho }_{0}r}{3{\epsilon }_0}[1-\frac{3r}{4R}]$