I once did a calculation on an electromagnetic wave propagating between two parallel plates to see which was greater: the electrostatic attraction between the opposite charges induced on the plates, or the magnetic repulsion between the parallel currents induced in the same plates. You can almost see where this is going if you have any intuition for these things. It turns out that in Nature, these two forces are perfectly balanced. It’s just one of those things.
I was actually trying to answer the question of “why do like currents attract?” instead of repelling, the way like charges do. I wanted to draw some cosmic conclousion about the fact that if they repelled, then there would be a net repulsion between the two parallel plates; and that the only way the universe made sense was that if there was no net force of any kind between those plates. I was able to verify by direct calculation that the net force indeed went to zero for a freely propagating e-m wave, but I fell short of my ultimate goal. I never came up with a reason why the net force had to go to zero for reasons that were self-evident.
Someone posted a problem the other day on a website I go to called stackeckchange.com . The problem was a parallel-wire transmission line connected to a resistor. The poster had recognized that when you connect a battery, opposite charges appear on the two wires, so they attract; and also, when current flows, the currents are opposite so they repel. The amount of charge hardly depends on the resistance, so it (and the attraction) are virtually constant. The amount of repulsion depends on the amount of current, or the resistance. So the poster askes: for what value of resistance does the net force go to zero?
You can see that this is similar to the problem I analyzed with the parallel plate. So I posted an answer in the form of a conjecture: that if you treat this as a transmission line, the condition for zero net force ought to be a freely propagating wave; and the way to get this was to have your load impedance matched to your line impedance, so there is no reflected wave.
This answer generated a lot of ridicule from the experts in the group, and we’re still arguing about it. But it looks to me very much like I’m turning out to be right. Of course no one will admit it, but what else is new? In the meantime, the process of arguing has solidified some ideas I’ve had floating around for a while about how to calculate the impedance of different transmission lines, and I’ve come up with some pretty cool tricks that I think I’m going to post one of these days.
You can check out the discussion at http://physics.stackexchange.com/questions/3306/when-is-the-force-null-between-parallel-conducting-wires
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One of the annoying things about that website is that guys can go back and change their comments afterwards. One of the guys who was ridiculing me has gone back and deleted all his comments, so the discussion thread doesn't exactly make sense anymore. But I'd say that's a pretty good sign I was winning the argument.
Hi!
Thank you for this interesting post. I take it that the intention is to show that the two parts of the Lorentz force f=q(E+(vxB)) perfectly balance out to zero for the two parallel plates. If so, I'd be very interested to see how one would prove this!
Kind regards
I'm glad you liked the post. But I was actually doing almost the opposite of what you said. Instead of showing that the two parts of the Lorentz force balance out to zero, I think I was showing that IF they balance out to zero, you can calculate the relative forces. What I couldn't quite come up with was a convincing physical argument as to why the Lorentz forces OUGHT TO balance out to zero.
Have you tried out switching the frame of references from the lab's perspective (Where you showed the plates to be static) and the frame of the moving electrons in the induced current (From where the plates seem to be moving).?
this is purely a random thought but since Lorentz force ought to be invariant in both cases maybe that's why the forces precisely balance each other.
(Or in other the physical argument that the two plates having balanced force is another way of of stating the Lorentz invariance of the Lorentz force?)
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