Magnetism results from relativistic effects on a moving charge?

[Leon31415]

Hi

Does anyone have a link to a proof that magnetism is a result of relativistic effects on a moving charge? I am sure that I have seen one but I can't find it.

Thanks

John

 

[StatusX]

I don't have a link, and I don't know what you mean by a proof, but here's a rough argument. Because of length contraction, the charge density of an object increases as $\gamma$ with velocity, and so it's electric field changes as well. In order for different, relatively moving observers to observe the same force, there must be an extra term coming from the movement of the object, which is the magnetic field.

For example, say there are two uniform, parellel line charges of opposite signs. Then they attract with a certain force. Now if we transform to a frame moving parellel to the wires, their lengths contract, so the charge density increases (since total charge is invariant under lorentz transformations). Thus we should expect the wires to pull on each other more strongly, which would contradict the prediction of the first frame. This effect is counteracted by fact that the magnetic fields of two parellel wires of oppostie current are such as to repel the two wires, and so the predictions come out the same in both frames.

This is an argument that what appears as an electric field to one observer will appear as a magnetic field to another. This together with a proof that Maxwell's equations are lorentz invariant are really the closest you can come to the proof you want.

 

[pervect]

I'd suggest the wikipedia article http://en.wikipedia.org/w/index.php?title=Relativistic_electromagnetism&oldid=145189916

and it's reference
http://physics.weber.edu/schroeder/MRR/mrr.html

If you want a well-known peer reviewed textbook source, get a hold of Purcell's book on electromagnetism.

I believe Feynman talks about this in his lectures somewhere as well.

There is a "simplified" version of the derivation also available at http://www.chip-architect.com/news/2007_02_27_Magnetism_as_a_Relativistic_side_effect.html

You might have run across this earlier on PF, it's written by one of our regulars (though I haven't seen him around for a while).

One comment I would make - gravity serves as a counterexample for the statement that the the complete equations for electromagnetism can be derived just from Coulomb's law.

The fundamental issue is that Coulomb's law can't be complete, because the forces it gives don't transform properly. Magnetism is one of the simplest ways of adding a force to the Coulomb law force that makes the resulting force behave in a properly covariant manner (i.e. in a manner in which the forces and 4-forces transform according to the requirements of relativity).