Question

A region in space has a total charge $Q$ distributed spherically such that the volume charge density $\rho \left(r\right)$ is given by
$\rho \left(r\right)=3ar\left(2R\right)$ for $r\le \frac{R}{2}$
$\rho \left(r\right)=a[1-{\left(\frac{r}{R}\right)}^2]$ for $\frac{R}{2}\le r\le R$
$\rho \left(r\right)=0$ for $r>R$
Find the electric field at $\rho (\frac{R}{2}<x<R)$

Answer 1

Charge on hypothetical sphere of radius $x$ is given by

$Q_{enclosed}={\int }_0^{R/2}\frac{3\alpha r}{2R}\left(4\pi r^{2}dr\right)+{\int }_{R/2}^x\alpha [1-{\left(\frac{r}{R}\right)}^2]4\pi r^{2}dr$
$Q_{enclosed}=\frac{3\alpha }{2R}\left(4\pi \right){\int }_0^{R/2}{r}^{3}dr+4\pi \alpha {\int }_{R/2}x[1-\frac{r^2}{R^2}]r^{2}dr$
$=\frac{3\alpha }{2R}\left(4\pi \right)\times \frac{{\left(\frac{R}{2}\right)}^4}{4}-4\pi \alpha \frac{R^3}{3\times 8}+\frac{4\pi \alpha x^3}{3}+\frac{4\pi \alpha x^5}{5R^2}-\frac{4\pi \alpha R^3}{32\times 5}$
$=\frac{3\alpha \pi R^3}{32}-\frac{4\pi \alpha R^3}{24}+\frac{4\pi \alpha x^3}{3}+\frac{4\pi \alpha x^5}{5R^2}-\frac{4\pi \alpha R^3}{160}$
$=\alpha \pi (\frac{3R^3}{32}-\frac{R^3}{6}+\frac{4x^3}{3}-\frac{4x^5}{5R^2}-\frac{1}{40}{R}^3)$
$\oint E.ds=Q_{enclosed}E(4\pi x^2=\alpha \pi (\frac{3R^3}{32}-\frac{R^3}{6}+\frac{4x^3}{3}-\frac{4x^5}{5R^2}-\frac{R^3}{40})$
$E=[\frac{3\alpha R^3}{128x^2}+\frac{\alpha R^3}{24x^2}+\frac{\alpha x}{3}+\frac{\alpha x^3}{5R^2}-\frac{R^3}{160\pi x^2}]$