Consider a hemi-spherical closed Gaussian surface of radius $R = \sqrt{2} c m$ . Two point charges $+ 2 q$ and $- q$ are placed at a distance X from the plane surface as shown in the figure on the line passing through the centre and perpendicular to the plane surface . If the electric flux on the plane and curved surface are equal, find the value of X (in cm).

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Detailed Solution

Question Image
Flux on the plane surface $ϕ_{1} = \frac{3 q}{2 ε_{0}} (1 - \cos θ)$

Flux on the curved surface
$ϕ_{2} = \frac{2 q}{ε_{0}} - \frac{3 q}{2 ε_{0}} (1 - \cos θ)$
$∵ ϕ_{1} = ϕ_{2}$
$\begin{array}{l} \Rightarrow \frac{3 q}{2 ε_{0}} (1 - \cos θ) = \frac{2 q}{ε_{0}} - \frac{3 q}{2 ε_{0}} (1 - \cos θ) \\ \cos θ = \frac{1}{3} \Rightarrow \frac{x}{\sqrt{R^{2} + x^{2}}} = \frac{1}{3} \\ \Rightarrow x = \frac{R}{2 \sqrt{2}} \end{array}$