A square loop of sides a = 1 m is held normally in front of a point… | JEE Main 2025 (24 Jan, Shift 1)

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Electric Flux & Gauss's Law

A square loop of sides $a = 1\text{ m}$ is held normally in front of a point charge $q = 1\text{ C}$ at a distance $\frac{a}{2}$ . The flux of the electric field through the shaded region is $\frac{5}{p} \times \frac{1}{\varepsilon_0}\,\frac{\text{Nm}^{2}}{\text{C}}$ , where the value of p is ________.

Numerical answer type. Work it out, then open the answer.

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Answer: 48

The square is one face of a cube of side $a$ with $q$ at its centre: whole-square flux $= \frac{q}{6\varepsilon_0}$ , each quarter $= \frac{q}{24\varepsilon_0}$ .
The shaded region is the lower half (2 quarters) plus one triangle cut by the diagonal of an upper quarter through the centre (half a quarter).
$\phi = \frac{2q}{24\varepsilon_0} + \frac{q}{48\varepsilon_0} = \frac{5q}{48\varepsilon_0}$ , with $q = 1$ C
So $p = 48$ .

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