An infinitely long uniform line charge distribution of charge per unit length λ lies parallel to the y-axis in the y-z plane at z = √32a. If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is λLn/ε₀ (ε₀ = permittivity of free space), then the value of n is :

Let the perpendicular from O to the line charge intersects the line charge at point Q. OQ = √3a²
Angle subtended by AB at Q is ∠AQB = 2∠AQO
tan∠AQO = AO/OQ = a/√3a² = 1/√3 = tan30°
∴∠AQB = 60°
Let us consider a cylindrical gaussian surface with the axis along line charge of length L and radius a (since AQ = a).
Φ_cylinder,curved = q_enc/ε₀ = λL/ε₀
Flux through the rectangle is same as the flux through 1/6th of the cuboid since it subtends 60° at Q and field lines are symmetrically radially outwards.
Hence, Φ_rectangle = (1/6)λL/ε₀
∴n = 6