JEE Main 2025 Jan 22 Shift 1, Physics Q2: Gauss's Law

JEE Main2025Jan 22, Shift 1Physics

A line charge of length $\frac{a}{2}$ is kept at the center of an edge BC of a cube ABCDEFGH having edge length 'a' as shown in the figure. If the density of line is $λ$ C per unit length, then the total electric flux through all the faces of the cube will be ____. (Take, $ϵ_{0}$ as the free space permittivity)

A
$\frac{λa}{8 ϵ _{0}}$
B
$\frac{λa}{16 ϵ _{0}}$
C
$\frac{λa}{2 ϵ _{0}}$
D
$\frac{λa}{4 ϵ _{0}}$

Solution

The line charge of length $a /2$ is centred on the midpoint of edge BC. Hence only half of its length, $a /4$ , lies inside the cube; the other $a /4$ projects outside the cube.

However, since edge BC is shared between four cubes when space is tiled, the charge lying on the edge is shared equally by the surrounding cubes. The portion of charge effectively enclosed by this one cube is $\frac{1}{4}$ of $λ \cdot \frac{a}{2}$ :

$q_{in} = \frac{λ a /2}{4} = \frac{λa}{8}$ .

By Gauss's law, $ϕ = \frac{q _{in}}{ϵ _{0}} = \frac{λa}{8 ϵ _{0}}$ .

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JEE Main 2025 Jan 22 Shift 1, Physics Q2: Gauss's Law

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