Question

A solid spherical region, having a spherical cavity whose diameter ${}^{\prime }{R}^{\prime }$ is equal to the radius of the spherical region, has a total charge ${}^{\prime }{Q}^{\prime }$. Find the electric field at a point $P$ as shown.

Answer 1

Volumetric charge density of given structure,

$\rho =\frac{Q}{\frac{4}{3}\pi R^3-\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3}\Rightarrow \rho =\frac{8}{7}\frac{Q}{\frac{4}{3}\pi R^3}$

$Q_1=\rho \frac{4}{3}\pi R^3=\frac{8}{7}Q,Q_2=\rho \frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3=\frac{1}{7}Q$

$E_p=E_1-E_2=\frac{1}{4\pi {ϵ}_0}\frac{Q_1}{x^2}-\frac{1}{4\pi {ϵ}_0}\frac{Q_2}{{(x-\frac{R}{2})}^2}$

$\Rightarrow E_p=\frac{Q}{28\pi {ϵ}_0}[\frac{8}{x^2}-\frac{1}{{(x-\frac{R}{2})}^2}]$.