There is one extra ##\omega## factor in both but I can no longer edit the post. It all worked out anyway, except that I thought there was a general strategy to obtain dispersion relations for generic waves, instead it looks like there are ad hoc methods, so to speak.
Oh, ok I asked my instructor and I was told to set the reference point not at infinity but at a finite distance from the wire. Do you think that can help?
It was not a deliberate sandbag, it is meant to be answered correctly, so as things stand I am failing to answer it properly, and yet I communicated it to you correctly, and I am persuaded the discussion is correct.
Ok so you stand by your analysis that it seems not well defined, but what did you mean when you said that ##C\log(|r_2-r_1|)## can be deduced from the configuration? How?
You are seeing it in terms of energy to get a certain configuration, but the wires are fixed in a constant position. There are no assumptions on how close or far they are, not infinitely far though, one intersects the xy plane in a point r_1 and the other in a point r_2. What the question seems...
The answer is given: it is ##-\frac{2\lambda^2}{4\pi\epsilon_0} \log(|r_2-r_1|)##. What I am asked is to prove that it can be written like that. How is it you got that expression naively? If I get right the ##\log(|r_2-r_1|)## bit I am basically done.
It was simply suggested in class I have literally 0 reference on it. And the energy density was defined as the energy in the field between the plane z=a and z=a+1. But I cannot find a way to calculate that that doesn't end up being infinite.
But that gives something like ##C\cdot \log\frac{a + \sqrt{x^2+a^2}}{-a+\sqrt{x^2+a^2}}## assuming a symmetric rod, times some constant, ##1/2 \lambda## and that limit is still infinite for ##a \rightarrow \infty##...
Actually, one last thing. Here I somehow convinced myself I would get ##\log (|r_2-r_1|)## but I actually don't, do I? $$ -\frac{\lambda}{2\pi\epsilon_0}\left(\log\frac{r_1}{a}+\log\frac{r_2}{a}\right) = -\frac{\lambda}{2\pi\epsilon_0}\left(\log(r_2) - \log(r_1) \right)$$ which is quite...
Thank, appreciate your help. At this point I am convinced that it's just about getting those differential equations for the generic fields. But I think that this question could also shed light on the matter: if we double the charge density on the rods (##2\lambda## instead of ## \lambda ##) how...
Ok, and equation-wise what can one say? Taking the curl of one of Maxwell's equations I get
$$ \nabla \times \nabla \times \mathbf{E} = -\frac{\partial}{\partial t} \nabla \times \mathbf{B}$$
from which
$$ \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 E = - \mu_0 \frac{\partial \mathbf{J}}{\partial...