Question

A sphere of radius R has a volume density of charge $? = kr$, where r is the distance from the centre of the sphere and k is constant. The magnitude of the electric field which exists at the sur ace o the sphere is given by ($e_0$ = permittivity of the free space)

• A. $\frac{4\pi kR^{4}}{3\varepsilon_{0}}$
• B. $\frac{kR}{3\varepsilon_{0}}$
• C. $\frac{4\pi kR}{\varepsilon_{0}}$
• D. $\frac{kR^{2}}{4\varepsilon_{0}}$

Answer 1

The Correct Option is D

Given, $\rho=K \cdot r$
By Gauss's theorem
$E\left(4 \pi r^{2}\right) =\frac{\int \rho \times 4 \pi r^{2} d r}{\varepsilon_{0}} $
$=\frac{\int K r \times 4 \pi r^{2} d r}{\varepsilon_{0}} $
$\Rightarrow E =\frac{K r^{2}}{4 \varepsilon_{0}} $
Here $ r =R$
So, $E=\frac{K R^{2}}{4 \varepsilon_{0}}$