Question

An infinite, uniformly charged sheet with surface charge density $\sigma$ cuts through a spherical Gaussian surface of radius $R$ at a distance $x$ from its center, as shown in the figure. The electric flux $\Phi$ through the Gaussian surface is :

• A. $\frac{\pi R^2\sigma }{{\epsilon }_0}$
• B. $\frac{2\pi {(R}^2-x^2)\sigma }{{\epsilon }_0}$
• C. $\frac{2\pi {(R-x)}^2\sigma }{{\epsilon }_0}$
• D. $\frac{\pi {(R}^2-x^2)\sigma }{{\epsilon }_0}$

Answer 1

The correct option is D $\frac{\pi {(R}^2-x^2)\sigma }{{\epsilon }_0}$

Here the infinite sheet inside the spherical Gaussian surface is a circular sheet of radius $a=\sqrt{R^2-x^2}$

using Gauss's law, the electric flux, $\varphi =\frac{Q_{enclosed}}{{ϵ}_0}$

here, $Q_{enclosed}=\pi a^2\sigma =\pi (R^2-x^2)\sigma$

thus, $\varphi =\frac{\pi (R^2-x^2)\sigma }{{ϵ}_0}$