A line charge of length $\frac{'\mathrm{a}'}{2}$ 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 $\lambda \mathrm{C}$ per unit length, then the total electric flux through all the faces of the cube will be ______. (Take, $\in_{0}$ as the free space permittivity)

(A) $\frac{\lambda \mathrm{a}}{8 \epsilon_{0}}$

(B) $\frac{\lambda \mathrm{a}}{16 \epsilon_{0}}$

(C) $\frac{\lambda a}{2 \epsilon_{0}}$

(D) $\frac{\lambda \mathrm{a}}{4 \epsilon_{0}}$

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Electrostatics

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,

\in_{0}

as the free space permittivity)

(A) $\frac{λ a}{8 ϵ _{0}}$ ✓ Correct
(B) $\frac{λ a}{16 ϵ _{0}}$
(C) $\frac{λa}{2 ϵ _{0}}$
(D) $\frac{λ a}{4 ϵ _{0}}$

Answer: (A)

Solution

The total charge on the segment is

Q = λ \cdot \frac{a}{2} = \frac{λa}{2} .

The segment sits at the midpoint of an edge of the cube. An edge of a cube is shared symmetrically among 4 identical cubes meeting along that line, the way four books share a common spine, so only one quarter of the flux emanating from this charge threads through any one of those cubes, including ours. The effective enclosed charge for Gauss's law is therefore

q_{in} = \frac{1}{4} \cdot \frac{λa}{2} = \frac{λa}{8} .

By Gauss's law, the total flux through all faces of the cube is

ϕ = \frac{q _{in}}{ϵ _{0}} = \frac{λa}{8 ϵ _{0}} .

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option

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JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Electrostatics

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