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

Answer 1

The correct option is C $2\pi \alpha R^2$

Using Gauss theorem for spherical surface of radius $r$ outside the sphere with a uniform charge density $\rho$ and a charge $q$

${\int }_{S}E.ds=\frac{Q_{enc}}{{\epsilon }_0}$

$E4\pi r^2=\frac{(q-2\pi \alpha R^2)}{{\epsilon }_0}+\frac{4\pi \alpha r^2}{2{\epsilon }_0}$

The intensity $E$ does not depend on $R$ if

$\frac{q-2\pi \alpha R^2}{{\epsilon }_0}=0orq=2\pi \alpha R^2$