Thu 5 Apr, 2012 09:14 pm
I have a question:
A common demonstration of the relationship between magnetic fields, electric fields, and reletivity involes a wire with a current, and an electron moving at the same speed and direction as the electrons in the wire (parellel to the wire). In the reference frame of the protons in the wire the electron outside of the wire experiences a force in the direction of the wire because of the magnetic field created by the moving electrons. But in the reference frame of the electron outside of the wire, a magnetic field ca not be use to explain this force because its velocity in its own reference frame is 0, and therefore any magnetic field will not cause a force (force being equal to velocity cross product magnetic field, so if velocity is 0 force is 0). The force can be explain, however, with electrostaric forces and reletivity as follow: in the electron's frame of reference the electrons in the wire are also stationary, but the protons are moving. Therefore reletivity (or a lorentz transform really) tells us that the protons contract in the direction of their motion with that electron, ie they all get a bit closer together. This causes an electric charge imbalance because the proton are just abit denser in the wire than the electron, therefore the electron outside the wire experience a force in the direction of the wire.
That's all fine and good, but I run into trouble when the electron is moving towards the wire. In this case, I try to do the same thing, in the reference frame of the protons in the wire the electron outside the wire experiences a force directed parrellel to the wire, as a result of the cross product of the magnetic field created by the electrons in moving in the wire and the velocity of the electron outside of the wire. As above, when we look at things from the reference frame of the electron outside the wire its velocity is 0, and we cannot use magnetic fields to explain the force it experiences. We are left with electrostatic forces and relativity, but I cant get a satisfactory answer as I could above, following is the best I can do:
In the reference frame of the electron outside the wire there are wire protons and wire electrons. we'll stay in the reference frame of the electron ouside the wire for this whole explanation, and first lets look at the wire protons. each wire proton has a velocity vector that is perpendicular to its direction vector to all the other protons. Therefore it has no component of its velocity vector that is directed towards the other protons, so from the perspective of the electron outside the wire, the reletivistic contraction really doesn't do much to the protons, they dont contract with respect to eachother. Now for the electrons in the wire: it gets a bit hard to desscribe without a picture, but each electron has a velocity vector with respect to the electron ouside that wire that is made up of two parts; one part is exactly the same as the one for the protons, and the other is in the direction of the other "upstream" electrons in the wire. These of course combine to make one velocity vector, and all of the wire electrons contract together along this velocity vector. Now staying in the reference frame of the electron outside of the wire this causes the "downstream" electrons to leave the wire slightly in such a way that they are on the side of the wire closer to the electron outside the wire, and the "upstream" electrons to leave the wire in such a way that they are ouside the wire on the opposite side of the wire from the electron outside the wire (what a mouth full). Without really going through all the math, I believe this could cause a force parellel to the wire on the electron outside the wire, and it was of course the goal to demonstrate this with only relativity and electrostatics, ie no mention of magnitism.
However this explanation is not satisfactory to me because it seems to cause all sorts of problems with causality and leads me to believe that I am not correctly applying relativity. For example lets say the electrons in the wire are going really fast (a good fraction of the speed of light), and I am in the reference frame of the protons with my head right next to the wire. All is fine, the electrons are contained by the wire, and I'm safe. But now a relativistic space ship flys right towards me, and from the spaceships perspectice the electrons leave the wire (just like they did for the electron ouside the wire) and actually pass through my head, killing me. So I end up alive in one reference frame an dead in another, after I see the ship pass I could get on a radio and send them a message which they would recieve, but they couldn't becasue for them I would be dead... so that just doesn't make any sense. So what gives, what am I missing, how can the force on the electron outside the wire be explained from the reference frame of the electron outside the wire, without magnitism, and without causing all kinds of rediculous disagreements in the state of things from the various reference frames.