# I Can the Lorentz Force be understood non-relativistically?

1. Dec 23, 2017

### RiemannLebesgueLemma

I'm not sure if this belongs in special/general relativity or in this subforum.

I'm currently trying to refresh and strengthen my E&M, and I remembered that one thing that bugged me when I first learned about magnetism was the velocity in the Lorentz force,
$$\vec{F} = q\vec{v} \times \vec{B}$$.
It wasn't really clarified in the lecture video that I was watching, nor in my first-year E&M class, which reference frame this is valid in. I asked my professor at the time, and he gave me quite an unsatisfactory answer. My thinking was that, since E&M was developed before relativity, we should be able to understand this velocity in a non-relativistic context; this leads to some paradoxes, since, for example, we could entirely null the effect of a neutral wire on a moving charge just by boosting to the frame with the velocity of the charge carriers in the wire. If we don't consider Lorentz transformations, then the moving charge in the new frame sees only a neutral wire with no current, and thus should feel no force.

This led me to come to the kind of independent conclusion that the force can only be understood in a relativistic context, and that the full force,
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$
measures velocity in a frame relative to that in which we measure E and B.

This makes sense to me, but it doesn't seem like it is really highlighted in either Jackson or Griffiths. Purcell, after stating the force law, does stress that F, E, B and v must all be measured in the same inertial frame. Purcell also mentions in its introduction that Lorentz was close to the theory of relativity in his work on moving charges, which seems to suggest that this problem of velocity-dependence was one of the motivating factors for relativity.

Is it true that we can only understand the Lorentz force within the framework of special relativity? Also, if so, how did they account for magnetic forces before relativity?

2. Dec 23, 2017

### Staff: Mentor

In the pre-relativity days, physicists assumed the existence of a medium (the luminiferous aether) that defined a preferred reference frame for electromagnetism. My understanding is that $\vec v$ was assumed to be measured relative to the aether.

3. Dec 23, 2017

### Staff: Mentor

Maxwell's equations did come before relativity. One can say special relativity was Einstein's solution to the riddles created by Maxwell's Equations.