Question

Twelve infinite long wires of uniform linear charge density (λ) are passing along the twelve edges of a cube. Find electric flux through any face of cube. ​

Answer 1

Calculating flux due to a single wire. In the figure x = l tan θ x=ltan⁡θ. Or, d x = l sec 2 θ d θ dx=lsec2⁡θdθ. Electric field due to this wire, → E = λ cos θ 2 π ϵ o l E→=λcos⁡θ2πϵol Value of the differential area is → d A = ( l ) ( d x ) = l 2 sec 2 θ d θ dA→=(l)(dx)=l2sec2⁡θdθ Flux is the dot product of the electric field vector and area vector. d ϕ = → E ⋅ → d A dϕ=E→⋅dA→ d ϕ = λ cos θ 2 π ϵ o l ⋅ l 2 sec 2 θ d θ ⋅ cos ( 180 − θ ) dϕ=λcos⁡θ2πϵol⋅l2sec2⁡θdθ⋅cos⁡(180−θ) ϕ = ∫ π / 4 0 λ l 2 π ϵ o d θ ϕ=∫0π/4λl2πϵodθ Solving this, the flux due to a single wire is ϕ = λ l 8 ϵ o ϕ=λl8ϵo Last step is to multiply it by 4. I think that there is a much easier way instead of doing the math. Most of the questions I have done based on finding flux through cube or its faces have some kind of symmetry and I guess that this question too involves some trick which I can't figure.