Apparent paradox in electrodynamics problem

Question

A few days ago I saw on Veritasium a video about electrodynamics and relativity Veritasium video and it got me thinking a bit about the nature of light.

Between minutes 1:10 and 2:40 Derek from Veritasium describes how the magnetic field transforms in electric field if you change the frame of refference. But let's suppose that his "positively charged cat" moves near the wire with a non-relativistic speed. If this would be the case , then we could say that the charge densities stay the same in the refference frame of the cat since the distances between the charges didn't change significantly. So the electric field would be 0, and so the cat shouldn't move ( but this would contradict the behaviour in the 2 frames).

My question is , why in this case it is necessary to use special relativity since it's a classical problem? I mean , in classical mechanics we use relativity to get a better approximation of the answer ( at low speeds), but here without relativity the answer it's just wrong, even at low velocities. I think it has to do with the fact that the light is actually made of magnetic and electric fields, and so you can't have one without the other even at low speeds, but this answer doesn't really satisfy me. ( I did the math but I can't find a reasonable intuitive explanation on why this is so)

Thank you!

Answer 1

In the video the "positive cat" has a velocity $v<<c$ and it is equal to that of electrons. The extremely small effect in the space-restriction is compansated by the high number of electrons in the metal (usually $10^{23} electrons/cm^3$).

The reason of using the special-relativity is hidden in the mathematical structure of this elegant theory:

the special-relativity is based on the Faraday tensor, which is a double antisymmetric covariant tensor, defined as

$F_{ik} = \begin{pmatrix} 0 & B_3 & -B_2 & E_1 \\ -B_3 & 0 & B_1 & E_2 \\ B_2 & -B_1 & 0 & E_3 \\ -E_1 & -E_2 &-E_3 & 0 \end{pmatrix}$

Changing the frame of reference, this tensor transforms its components in the following way:

$F^{i'k'}= A_{i}^{i'}A_k^{k'}F^{ik}$,

where $A_{i}^{i'}$ and $A_k^{k'}$ are Lorentz matrices.

Calculating, for example, $B_3'$, you can find that

$B_3'= \gamma(v)(B_3 - \frac{v}{c} E_2)$.

The importance of it is that changing frame of reference, a $purely$ electric (or magnetic) field transforms in a combination of electric $and$ magnetic fields, like in the video.

This is possible only by using the special relativity formalism, and it is why it is used.