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. $\frac{\alpha}{2\varepsilon_{0}}$
• B. $\frac{\alpha}{\alpha\varepsilon_{0}}$
• 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 $q$
$\int\limits_{{s}}$$E.ds=\frac{Q_{enc}}{\varepsilon_{0}}$
$E4\pi r^{2}=\frac{1}{\varepsilon_{0}}\left(q+\int\limits^r_{{R}}\frac{\alpha}{r}\left(4\pi r^{2}\right)dr\right)$;
$E4\pi r^{2}=\frac{\left(q-2\pi\alpha R^{2}\right)}{\varepsilon_{0}}+\frac{4\pi\alpha r^{2}}{2\varepsilon_{0}}$
The intensity $E$ does not depend on $R$ if
$\frac{q-2\pi\alpha R^{3}}{\varepsilon_{0}}=0$ or $q=2\pi\alpha R^{2}$