qv x B force in electron's proper frame

[Swamp Thing]

Can we describe / explain the B x v force in the electron's own reference frame without reference to relativistic invariants, 4-vectors, tensors et al?

The aim would be to explain things like the following video without the notion of "field lines" that electrons and wires move through. But the target audience would be people who understand non-relativistic electromagnetics including Maxwell's equations.

 

[pines-demon]

The Lorenz force
$$\mathbf F = q(\mathbf E +\mathbf v \times \mathbf B)$$
for a particle (charge $q$) in an electric field $\mathbf E$ and magnetic field $\mathbf B$, looks like
$$F'=q\mathbf E'$$
in the particle's frame. Where $\mathbf E'$ is the electric field in the particle frame.

This means that in the particle's perspective, there is no $\mathbf v \times\mathbf B$ term, only electric fields (as its own velocity is null in that frame).

Edit: I do not think this helps much to solve the Faraday paradox. As (1) you also have to transform the magnetic fields (2) the frame is rotating (3) it is a metal with currents not a single electron.

 

[Orodruin]

people who understand non-relativistic electromagnetics including Maxwell's equations

There is no such thing. Maxwell’s equations describe a relativistic field theory by construction. In fact, it is the field theory that led Einstein to develop relativity because it was not consistent with non-relativistic physics.

 

[Swamp Thing]

There is no such thing. Maxwell’s equations describe a relativistic field theory by construction. In fact, it is the field theory that led Einstein to develop relativity because it was not consistent with non-relativistic physics.

In our EE course (late 1970s) we learnt Maxwell's equations and how to apply them in a single reference frame, but not how to transform the field properties to other frames. When dealing with charges or wires in a magnetic field we would do the sums in the magnets reference frame. I don't remember thinking about how it would look in the electron or wire frame.

So from the replies, it seems that that kind of target audience is stuck with moving lines of force cutting a stationary wire or moving charge (naive pseudo relativistic intuition) ?

 

[Dale]

Can we describe / explain the B x v force in the electron's own reference frame without reference to relativistic invariants, 4-vectors, tensors et al?

Sure. $v=0$ so $B \times v=0$ regardless of $B$

 

[Vanadium 50]

Beat me to it!

 

[Orodruin]

Sure. $v=0$ so $B \times v=0$ regardless of $B$

This reminds me of that time I discovered the question "How far does the muon travel in its rest frame?" in a modern physics course with the correction template unironically suggesting something different from zero ...

It is now a favourite question of mine to pose to students (obviously with the correct answer) as it illustrates a key concept that way too many just don't grasp ...