How is velocity measured in magnetism?

Question

In magnetism, we can apply two basic equations which for the most part, explain the forces on currents in wires, namely $ F=qB \times{v}$ and $B = \frac{\mu_0 I}{2\pi r}\hat{r}\times\hat{v}$.

The problem is, the current of the object creating the magnetic field on our particle is dependent on its 'velocity'. Moreover, the force itself depends on the velocity of the particle. I had always thought that '$v$' meant the relative velocity between both the particle in question and the source of the field, this seems like a natural interpretation of the equation because otherwise we obviously break some symmetry laws, like newton's first law. So I came to the conclusion that $I$ in this case could be expressed in terms of $v$ and vice versa.

This should work in the case of two wires with some current of some magnitude going through them, because no matter what the frame, relative magnetic fields of the positive and negative charges cancel out to the same value.

However, say we have two electron beams, one observer travelling 'with' the beams and other still as the source of the gun. The moving observer shall only see electrical repulsion between the two beams, through Gauss's law. Surely the observer at the source sees the same thing. I've been told this isn't true, that they observe different forces through an effect that is easily explained by special relativity. But wouldn't it solve the problem just as easily if we had relative velocity as a measure?

My question essentially: I think there are two mathematically consistent ways to express the laws of magnetism in classical mechanics, either through special relativity or using $v$ as relative velocity. Thus, the latter is only not true because we observed it as such. Is this the case or is there also something theoretically wrong with the second interpretation?

Answer 1

Velocity is measured in the frame that we choose it to be.

You've found two of the infinite number of ways to get the right answer. The fact is, velocity is always relative to something - this is the fundamental principle of special relativity. In this case, it's relative to the observer that we choose.

When doing these calculations, we first figure out whose perspective we observe from. In the case of the electron beams, you can calculate the repulsion from the moving observer's perspective – we use Gauss' law and see the beams be repulsed away from each other due to electrostatic repulsion. For simplicity, let's say that electrons are perfectly aligned, separated by distance $d$. Then, in the electron frame, we see only an electric field that causes some given force on each electron.

In the lab frame, things are a little more complicated - there is a repulsive electric force as well as a weak, attractive, magnetic force caused by the magnetic field of the other electron. If you do the calculations, you get the same value for the force between them diminished by a factor of $\frac{1}{\gamma} = \sqrt{1-v^2/c^2}$, which turns out to cause exactly the same motion because of the way forces transform in special relativity. (here, v represents the velocity of the electrons in the lab frame, and also the relative velocity between the two frames.

Two equally valid methods for determining the force on the electrons, but one happens to be easier in this case.

To deal with more complicated scenarios involving both electric and magnetic fields, we use the following transforms:

Specifically,

$E'_{||} = E_{||}$ (parallel to the direction of motion of the frame)
$B'_{||} = B_{||}$ (parallel to the direction of motion of the frame)
$E_\perp' = \gamma(E_\perp + v \times B)$ (perpendicular to the direction of motion of the frame)
$B_\perp' = \gamma(B_\perp - \frac{v \times E}{c^2})$ (perpendicular to the direction of motion of the frame)
Where v is the velocity between the 'primed' and 'unprimed' frames (e.g. you measure $E$ and $B$ on the ground, but on a rail car moving with velocity v you measure $E'$ and $B'$.

There are deeper explanations - in the classical world, it can be said that

A magnetic field is just an electric field moving in a different frame.

This idea is explained here in great detail.

The fact is that electricity and magnetism are two facets of the same fundamental force, electromagnetism. E and B fields change between each other depending on the velocity of the observer, so it's hard to say that one is 'real' and the other is not.