Determine the vector potential due to a short wire running from (−L/2, 0, 0) to (L/2, 0, 0) carrying a current I = I0cos(ωt) (consider only points at distance r >> L from the origin). (You may neglect the effect of the return circuit.) Now determine the corresponding scalar potential for large r.
V(x, t) = 1/4πε0 ∫d3x' ρ(x' , t - |x' - x|/c) / |x' - x|
A(x, t) = μ0/4π ∫d3x' j (x' , t - |x' - x|/c) / |x' - x|
Where A is the vector potential and V is the scalar potential.
The Attempt at a Solution
I managed to solve the first part for the vector potential by using the fact that r>>L and therefore |x' - x| ≈ r
and using that j = I/c x(hat) (in the x direction, and I let C= the cross sectional area of the wire)
this gave A(x, t) = (μ0L I0 cos(ω(t-r/c)))/4πr
When trying to determine the vector potential I ran into an issue, I'm not entirely sure what ρ is explicitly. I had a guess that is was ρ = Q/LC, and then using the fact that I=dQ/dt and integrating to find Q, but this was a stab in the dark and I have no idea if it's right.
Any help would be greatly appreciated, thank you!