What's the Deal with Magnetic Fields?
lcamtuf.substack.com
lcamtuf.substack.com
In the 16 intervening years, physics classes must have been rather confusing: how did people feel about transforming coordinates into a moving frame in a manner that pretty much required length contraction, time dilation, and simultaneity violation before anyone really believed that the universe actually worked that way?
I often wonder about the “luminiferous aether”— it sounds a bit silly. If there were message boards form physicists back then, would there be a bunch of writing: “well, we think this is a bit dumb, we’re just using it to make the math work out” for us to read? I’m sure somebody’s checked their personal letters…
I would think it's at least worth mentioning that Maxwell's equations did unify the electric and magnetic forces prior to relativity.
They aren't always taught as completely separate things, even in classical physics.
I would also say, this is a great example of how once we have a word, or a an explanation for a thing, it pretty much becomes invisible to us.
The magnetic phenomenon still amazes me, at 65 years old. It's what as a child, and a youth, inspired me to study electrical engineering.
Holding two permanent magnets near each other, and physically feeling the force between them, even though they don't touch each other, really frikin amazes me! Still!
Just drop everything you've ever learned or thought you knew, and hold those two magnets, and wonder at how do they know the other one is near?
Reality is just so much more amazing than make believe. It's so sad to see so much of the modern world lost in make believe...
So on average, relativistic effects are negligible. Then how can relativistic effects then be used in the later arguments for where magnetic effects come from?
> The effect is negligible in most contexts unless the objects are moving at very high (“relativistic”) speeds, but electric fields are a notable exception — even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute.
The point of the text you’ve quoted is to set up the seeming contradiction or unexpected idea. Then the hope is that the author explains it in the rest of the article.
It is fine to say that the article doesn’t follow through with the implied promise of the quote, but the deficiency is in the answering if that is the case, not the quoted text.
lcamtuf is analyzing a somewhat odd scenario, and he seems to have carefully omitted all the math. In the lab frame, there's a stationary test charge and an uncharged current-carrying wire. Ignoring relativity for a bit, the magnetic field around the wire is proportional to the current I (and I is proportional to v_drift/c). The Lorentz force [0] on the charge is q·v_charge·B, where v_charge is the velocity of the test charge, which is, drumroll please, exactly zero. There's no force, at least to first order! A current carrying wire does not attract charges of one particular charge, nor could it without rather odd symmetry violations.
If you want to analyze this in a moving frame, even at first order in the boost velocity v_boost (the velocity of the moving frame, which does not need to equal v_drift), it gets more complicated: the test charge is moving, so it will experience a Lorentz force. But the fields themselves transform too, like this:
https://en.wikipedia.org/wiki/Classical_electromagnetism_and...
And, of course, to first order everything will cancel out and there is still no force! (Otherwise there would be a massive inconsistency: a current-carrying wire would attract negative charges from the perspective of someone walking one way and would repel them from the perspective of someone walking the other way, and they would end up rather dramatically disagreeing with each other, and special relativity is not that weird.)
The actual phenomenon that lcamtuf is talking about only shows up at higher order in the velocities involved, and one might argue that relativity does affect electromagnetic fields even at first order (see that link above!).
FWIW, a useful heuristic for thinking about relativity is that the simultaneity-violating term in the Lorentz transformation (the -γβx term in t') implies that almost everyone's intuition for what's going on with multiple moving objects that are not in the same place (like multiple different electrons in a wire from the perspective of someone moving relative to the wire) is likely to be wrong. I haven't considered lcamtuf's explanation that carefully, and I'm not entirely convinced that it's right. But even that is a higher order effect, and lcamtuf's explanation seems like it has to be wrong even at first order! After all, at first order, there is no force on the test charge. But in a moving frame, as above, there is a Lorentz force on the charge unless the magnetic field disappears entirely, and v_drift is not going to be the boost velocity that makes the B field disappear. So I would apply large grains of salt to this model.
[0] Wow, Lorentz got his last name all of this stuff!
1) Lots of small things that interact constructively.
IIRC, the conduction electrons in copper have something like -13600 coulombs of charge per cubic centimetre, but that all cancels out with the positive ions right next to them; the magnetic fields from each electron's motion don't cancel out*, but the motion of the ions is ~ isotropic*, so they do cancel each other*.
2) The strength of the electric (and by extension magnetic) force is also strong, which in turn also helps make this easier to notice.
* in the lab frame
https://en.wikipedia.org/wiki/Electromagnetic_four-potential
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.
Took us a few conversational turns to realize we had different capitalization. (This was in the old days when courting was mostly by speaking in person.)