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

The Correct Option is 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 $q=2\pi\alpha R^{2}$

Answer 2

Given:
A uniformly charged sphere of radius R with charge Q.
Surrounding medium with charge volume density \(\rho = \frac{\alpha}{r}\), where \(\alpha\) is a positive constant and r is the distance from the center of the sphere.

Electric Field Due to Sphere:
The electric field \(E_{\text{sphere}}\) outside a uniformly charged sphere of radius R and charge Q is:
\(E_{\text{sphere}} = \frac{Q}{4 \pi \epsilon_0 r^2}\)

Electric Field Due to Surrounding Medium:
The electric field \(E_{\text{medium}}\) due to the surrounding charge density \(\rho = \frac{\alpha}{r}\) for r \(\gt\) R is:
\(E_{\text{medium}} = \frac{\alpha}{2 \epsilon_0}\)

Total Electric Field Outside the Sphere:
The total electric field \(E_{\text{total}}\) outside the sphere is the sum of \(E_{\text{sphere}}\) and \(E_{\text{medium}}\):
\(E_{\text{total}} = E_{\text{sphere}} + E_{\text{medium}}\)
\(E_{\text{total}} = \frac{Q}{4 \pi \epsilon_0 r^2} + \frac{\alpha}{2 \epsilon_0}\)

Condition for E to be Independent of R:
For \(E_{\text{total}}\) to be independent of R, the term \(\frac{Q}{4 \pi \epsilon_0 r^2}\) must equal \(\frac{\alpha}{2 \epsilon_0}\):
\(\frac{Q}{4 \pi \epsilon_0 r^2} = \frac{\alpha}{2 \epsilon_0}\)
\(Q = 2 \pi \alpha R^2\)

So, the correct option is (C): \(2 \pi \alpha R^2\)