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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
$\frac{{\pi {R^2}\sigma }}{{{\varepsilon _0}}}$
$\frac{{2\pi {{\left( {{R^2} - {x^2}} \right)}^{}}\sigma }}{{{\varepsilon _0}}}$
$\frac{{\pi {{\left( {R - x} \right)}^2}\sigma }}{{{\varepsilon _0}}}$
$\frac{{\pi {{\left( {{R^2} - {x^2}} \right)}^{}}\sigma }}{{{\varepsilon _0}}}$
Solution
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, $\phi=\frac{Q_{\text {enclosed }}}{\epsilon_{0}}$
here, $Q_{\text {enclosed}}=\pi a^{2} \sigma=\pi\left(R^{2}-x^{2}\right) \sigma$
thus, $\phi=\frac{\pi\left(R^{2}-x^{2}\right) \sigma}{\epsilon_{0}}$
