How Maxwell's equations explain the Lorentz contraction?

[marcosschiavi]

I did more than one course of classical electromagnetism in college. Recently, however, after reading "How Relativity Connects Electric and Magnetic Fields" (http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html) I was astounded to realize how little I knew about it! In college (if I remember well) never was mentioned the relationship between Maxwell's equations and relativity.

If we have two charges A and B that move at the same speed v, I always thought that there would be no magnetic field between them because the relative velocity between A and B is zero. I never noticed that the velocity v in the Maxwell equations were ABSOLUTE. However after reading the article I realized that from the viewpoint of an observer at rest there is a magnetic attraction force, but from the viewpoint of an observer that also moves with velocity v, there is none!

What I don’t understand is how to explain such obvious discrepancy without resorting to the relativity theory (that came much later) and how can someone teach electromagnetism without relativity ...

 

[Dr Lots-o'watts]

That is quite exactly the problem that Einstein tackles (and solves) in his first 1905 paper on relativity.

 

[Born2bwire]

Maxwell's Equations satisfy special relativity though. Part of this is inherent in the fact that the wave velocity has a maximum of c. Out of this we can deduce the Lorentz transformations of the fields. No extra physics is required though this wasn't fully fleshed out until Lorentz, Poincare and Einstein.

The Lorentz transformation of the electromagnetic fields allows for a lab frame of pure electric (or magnetic fields) to be converted into a moving frame with both electric and magnetic fields. Thus, if you have your two charges that are moving along with the same velocity relative to each other, then yes you will have a magnetic field excited by the two charges (calculated via say Biot-Savart). If the magnetic field exerts a force on the other charge, then in the rest frame of the charge we will find that the transformed fields give rise to an electric field. This electric field of course exerts a force on the charge regardless of its velocity. Done properly, one finds that the force in the rest frame and lab frame come out to be the same.