Question

A system consists of a uniformly charged sphere of radius $R$ and a surrounding medium filled by a charge with the volume density $\rho =\frac{\alpha }{r}$, where $\alpha$ is a positive constant and $r$ is the distance from the centre of the sphere. Find the charge of the sphere for which the electric field intensity $E$ outside the sphere is independent of $R$.

• A. $\frac{\alpha }{2{\epsilon }_0}$
• B. $\frac{\alpha }{\alpha {\epsilon }_0}$
• C. $2\pi \alpha R^2$
• D. None of these

Answer 1

Answer: (C)

Using Gauss theorem for spherical surface of radius $r$ outside the sphere with a uniform charge density $\rho$ and a charge $q$
$\underset{s}{\int }$$E.ds=\frac{Q_{enc}}{{\epsilon }_0}$
$E4\pi r^2=\frac{1}{{\epsilon }_0}\left(q+\underset{R}{\overset{r}{\int }}\frac{\alpha }{r}\left(4\pi r^2\right)dr\right)$;
$E4\pi r^2=\frac{\left(q-2\pi \alpha R^2\right)}{{\epsilon }_0}+\frac{4\pi \alpha r^2}{2{\epsilon }_0}$
The intensity $E$ does not depend on $R$ if
$\frac{q-2\pi \alpha R^3}{{\epsilon }_0}=0$ or $q=2\pi \alpha R^2$