The flat base of a hemisphere of radius a with no charge inside it lies in a horizontal plane. A uniform electric field $\vec{E}$ is applied at an angle $π / 4$ with the vertical direction. The electric flux through the curved surface of the hemisphere is

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$(π + 2) {πa}^{2} E / (2 \sqrt{2})$

$π a^{2} E / (2 \sqrt{2})$

$π a^{2} E / \sqrt{2}$

$π a^{2} E$

answer is D.

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Detailed Solution

Given Data: Electric field is applied to the hemisphere as shown below.

Diagram, engineering drawing

Description automatically generated

Concept Used: Gauss Law

Detailed Solution:

By Gauss law,

$ϕ = \oint E . d S = E \oint d S \cos 45 °$

Here, for the hemisphere, $ϕ_{curved} = ϕ_{circular}$

Thus, $ϕ_{curved} = E π a^{2} \cdot \frac{1}{\sqrt{2}} = \frac{E π a^{2}}{\sqrt{2}}$

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