A line charge of length a/2 is kept at the center of an edge B C of a cube ABCDEFGH having edge length ' a ' as shown in the figure. If the density of line charge is λ C per unit length, then the total electric flux through all the faces of the cube will be ___________ . (Take, ε₀ as the free space permittivity)

Solution

Physics sir · 3 steps

The charge sits on an edge, and an edge is shared by four cubes, so this cube gets only a quarter of the flux.

1. Total charge. The line charge has length a/2 and density λ, so it carries q = (λ a)/2. If it were fully enclosed, Gauss's law would give a flux of q/ε₀.
2. Edge sharing. The charge lies along edge BC, not inside the cube. Imagine identical cubes stacked around that edge: exactly 4 cubes meet at it and together enclose the charge completely. By symmetry, each cube receives an equal share of the flux.

   φ = 1/4×q/ε₀ = 1/4×(λ a)/2ε₀ = (λ a)/8ε₀

3. So. The total flux through all the faces of the cube is (λ a)/8ε₀, which is option (b).

The idea to keep. For a charge on the boundary of a cube, build the full set of cubes around it. Inside gives the full q/ε₀, on a face half, on an edge a quarter, at a corner one-eighth.

Answer from the NTA final answer key. Solution by PrepWiser.