This is really interesting as an explanation. It's not usually taught this way, because the formalization of Maxwell's Equations pre-dates General Relativity.
Is it actually theoretically sound?
Is it actually theoretically sound?
F = (p / 2πrε) * q
If we switch to a moving reference frame (up at velocity v), the charge density decreases to
p' = p*sqrt(1-v^2/c^2) = p - pv^2/2c^2 + O(v^4/c^4) (from Taylor series)
The force should stay the same, so
(p / 2πrε) * q = (p' / 2πrε) * q + F'
where F' is some other magnetic force created by the current. Solving, to second-order we have
F' = pqv^2/4πrεc^2
If we introduce a new constant
µ = 1 / εc^2,
we get
F' = (µ / 4π) * pqv^2 / r
Now, we know
F' = B * qv
where B is the magnetic field, so
B = dF'/d(qv) = µpv / 2πr = µI / 2πr (where I is the current in the wire).
This agrees with the Biot-Savart Law for an infinitely long wire.
There are situations in which you have both fields and you can't attribute the magnetic field to a purely electrostatic field in another frame. It turns out that all observers agree on the value of k=E²/c² - B², so if you have k<0, you can't possibly find a frame with no magnetic fields, because that would imply k≥0.
So, you can use relativity to motivate the need for introducing a B field, instead of "pulling it out of the hat", but in general you need both E and B. In the modern formalist these are the components of a larger object called F, the field strength tensor.