# Relativity and Electromagnetism

1. Oct 23, 2007

### Himanshu

Situation

A positive test charge q is moving parallel to a current-carrying wire with velocity v relative to the wire in frame S. It is assumed that the net charge on the wire is zero and that the electrons in the wire also move with velocity v in a straight line. The leftward current in the wire produces a magnetic field that forms circles around the wire and is directed into the page at the location of the moving test charge. Therefore, a magnetic force directed away from the wire is exerted on the test charge. However, no electric force acts on the test charge because the net charge on the wire is zero when viewed in this frame. Now consider the same situation as viewed from another frame S', where the test charge is at rest. In this frame, the positive charges in the wire move to the left, the electrons in the wire are at rest, and the wire still carries a current. Because the test charge is not moving in this frame, there is no magnetic force exerted on the test charge when viewed in this frame. However, if a force is exerted on the test charge in frame S, the frame of the wire, as described earlier, a force must be exerted on it in any other frame. What is the origin of this force in frame S', the frame of the test charge? The answer to this question is provided by the special theory of relativity.

A Possible Solution

When the situation is viewed in frame S the positive charges are at rest and the electrons in the wire move to the right with a velocity v. Because of length contraction, the electrons appear to be closer together than their proper separation. Because there is no net charge on the wire this contracted separation must equal the separation between the stationary positive charges. The situation is quite different when viewed in frame S'. In this frame, the positive charges appear closer together because of length contraction, and the electrons in the wire are at rest with a separation that is greater than that viewed in frame S. Therefore, there is a net positive charge on the wire when viewed in frame S'. This net positive charge produces an electric field pointing away from the wire toward the test charge, and so the test charge experiences an electric force directed away from the wire. Thus, what was viewed as a magnetic field (and a corresponding magnetic force) in the frame of the wire transforms into an electric field (and a corresponding electric force) in the frame of the test charge.

Problem

How in the world is length contraction feasible? In Electric Current the electrons move with Drift Velocity.For ordinary currents, this drift velocity is on the order of millimeters per second in contrast to the speeds of the electrons themselves which are on the order of a million meters per second. At such low speeds the effect of length contraction would be negligible. How is relativity a possible solution to the above situation.

Last edited: Oct 23, 2007
2. Oct 23, 2007

### Ich

No, the speed of the electrons really is comparable to drift velocity.
The fact that these effects are observable just show the incredible strength of electromagnetism. It only needs a minuscule shift from neutrality to generate them.
Just imagine: an excess charge of a few Coulombs is enough to keep all the protons in a white dwarf from collapsing. (At least, if I made no error when I calculated this, back in the last millennium).

3. Oct 27, 2007

### Staff: Mentor

My favorite explanation of this is at

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

"it's remarkable that we can measure magnetic forces at all, since the average drift velocity in a household wire is only a snail's pace: v/c is typically only 10^-13, so the Lorentz factor differs from 1 only by about one part in 10^26. We can still measure this effect because the total charge of all the conduction electrons in a meter-long wire is tens of thousands of coulombs; two such charges separated by only a few millimeters would exert enormous electrostatic forces on each other."

This goes along with Ich's explanation: not only is the electrostatic force constant incredibly large, due to the large number of charge carriers there is also a relatively large net charge from even a small length contraction.

Last edited: Oct 27, 2007
4. Oct 27, 2007

### pmb_phy

Sounds okay so far.
I disagree. The test charge has a net force acting on it due to the field it is in. You had stated that the charge density in this frame is zero and therefore there can be no electric field on this charge. Therefore force on the test charge directed radially outward from the wire, the magnitude of the force being given by F = qvxB (The magnetic part of the Lorentz force).
I disagree. In S' there is still a current. The current consists of the motion of the positive charges while the negative charges are at rest.
That is correct.
You could have done a Lorentz transformation on a general static magnetic field with no electric field an you woul have got the answer you're seeking for. However I applaud your effort for seeking the mechanism in this case. Bravo!

In principle, more people should do this kind of thing. Recall this nice little principle from Spacetime Physics - 2nd Ed., Taylor and Wheeler
I love to collect these little treasures so I put them in a file and put the file on my web site. For more see - http://www.geocities.com/physics_world/ref/quotes.htm
Enjoy!
It is more than an appearance since its something which is physicall measureable and has a definite effect which can also be measured. I recommend that you think of it as real. Just my humble opinion of course.
That is exactly correct.
You are 100% quite correct!
Uh oh! :tongue:
You make an invalid assumption here. You assert that the length contraction is negligible where in fact it is not. A quote which applies here is from Special Relativity, by A.P. French, page 259. French writes
Words of wisdom to remember! This was a quote from the end of the same derivation that you just went through, although French does the math and its the math that shows you that the speed is not neglegible in this case.

I created a web page to describe the electric field generated by a rotating magnet which is rotating about its center of symmetry. It is located at

http://www.geocities.com/physics_world/em/rotating_magnet.htm

The part you just discussed and that of French, which is also described in described in

The Feynman Lectures on Physics - Volume II, Feynman, Leighton and Sands, Addison Wesley, 1977, pages 13-7 to 13-12

If you have anymore questions or if you disagree with what I've said above please let me know and I'd be more than happy to do whatever I can to clarify this issue. In the meantime I recommend that you follow the derivation as close as possible up until Eq. (8). If you accept the derivation up until that point then work out the actual numbers. Notice that only the positive charge density and the speed of the wire (through $\gamma\beta^2$ is used to calulate the charge density creating the field!

Best regards

Pete

Last edited: Oct 27, 2007