Recent content by Karl86

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.

I have no idea how this dispersion relation was deduced, and also what's the meaning of including plus and minus in the formula.

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.

Yes, I'm convinced. The discussion that arose from a seemingly innocent problem was remarkable though :D

The field is constant and pushes it away from the wire, the charge never stops, so the KE is never 0. IS that right?

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.

Those integrals were infinite too :(

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...