Ziggurat
Penultimate Amazing
- Joined
- Jun 19, 2003
- Messages
- 61,768
In other words, and on the face of it, the attraction or repulsion of two like charges should depend upon the speed the frame of reference in which they are observed. I don't see how that works - not that I want to make a big deal out of it.
Well, you can think of magnetism as the consequence of sonsidering Coulomb attraction and repulsion plus relativistic length contraction.
So first let's consider two parallel wires, with parallel currents. We can consider each wire as if it were composed of a line of positive charges (the nuclei) and a line of negative charges (the electrons), where the positive charges are stationary relative to the lab and the negative charges are moving relative to the lab. First, let's consider the forces on the positive nuclei in the wires. In the reference frame of the nuclei, there is no net charge on either wire, so there is no net Coulomb force on the nuclei in either wire. There is a magnetic field, but the charges aren't moving, so there is no force from this magnetic field on the nuclei. So nothing should happen to the nuclei.
OK, now what about the electrons? Well, in the lab reference frame, the moving electrons have the same charge density as the nuclei. But they're moving, which means that using relativity, they have been length contracted. So if we switch to the electron's reference frame (where the electrons are stationary and the nuclei are moving), then the charge density of the electrons decreases since they are no longer length contracted. Furthermore, the nuclei now are length contracted, so their charge density increases. Which means that in the electron's reference frame the two wires are not charge neutral. Each wire is positively charged. Which means that the electrons in each wire feel an attraction to the nuclei in the other wire. There is also a magnetic field in this reference frame, but the electrons aren't moving, so in this reference frame there is no magnetic force on the electrons.
So it's possible to figure out the electromagnetic forces in your system by only considering Coulomb interactions, and ignoring magnetic fields. Do do that, we need to switch reference frames so that we examine each charge in a reference frame where it isn't moving, and that change of reference frames causes different length contraction for other charges in the system. So if we have different charges moving differently (as is common), then we need to use multiple reference frames, each of which sees a different charge distribution. But if we're willing to use magnetic fields, then we can get the exact same total forces, all calculated from a single reference frame.
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