An analogy would be speed of wind, Vs. speed of the molecules that make up air. In stationary air, the RMS speed of the molecules is about 500 m/s. If you imagine air blowing at 10 m/s, you can see that that’s just a small average trend on top of the random velocity.
The question I had was whether the analogy actually holds for electrons in metal. Do electrons in metals have a reasonably well defined speed, the way air molecules do? But apparently they do, the Fermi velocity, about 1570 km/s.
Has someone messed up here or am I an idiot?
Moving charges created an magnetic field only when they are accelerating. This is why DC current does not give you a magnetic field and AC current does.
Wait, I’m wrong there too! I can make an electromagnet from DC current but not a transformer.
HELP!!
I think you’re convolving in your mind “creating a magnetic field”, which moving charges do, and “emitting electromagnetic radiation”, which accelerating charges do.
Magnitudes of forces transform between frames; see here. So the force as measured in the frame where the charges are at rest is different from the force measured in the frame where the charges are moving.
Not if the force is zero (or orthogonal to the lorentz boost), as it is in this case.
Yes, you could. But what MikeS was saying is also correct. In one frame of reference you se a magnetic field that is exactly cancelled by the change in the transformed electric field. In another frame of reference (the one where they are at rest) there is just the electric field and no magnetic field.
Yes. I was riffing off “In fact, the electrons would experience no net force in the limit as v approaches c” with, “there is a net force, which is the same as the electrostatic force in the rest frame, and you don’t have to wait for the limit as v -> c, it’s constant all the time”.
There’s no such thing as magnetic force or magnetic field. Magnetism is just the manifestation of relativistic effects bearing upon electrical interactions. That is, it’s not quite correct to say that the magnetic force “opposes” the electric force, rather it would be more precise to talk about distortions of the patterns of electrical interactions due to relativistic length contraction and time dilation.
Yeah, I hope he just meant that the force is the same “for all v, even for v near c”, rather than “only in the limit v->c”
That’s not quite right either. There is only the “electromagnetic field”. The electric field is no more real than the magnetic field, and vice-versa. That’s part of the point of relativity: one frame of reference is just as “real” as any other.
Take a look at the site I linked to. The simplified version at the bottom suffices for us; the S frame is the “rest frame” of the charges and the S’ frame is the “lab frame”, moving in the -x direction relative to the rest frame. The component of the force parallel to the motion is the same in both frames; the components perpendicular to the direction of motion vary between frames.
No, the total electromagnetic force goes smoothly to zero as the speed of the charges in the lab frame approaches c. The magnitude of charge #1’s electric field (as measured in the lab frame) at the position of charge #2 is proportional to the Lorentz factor γ = (1-v[sup]2[/sup]/c[sup]2[/sup])[sup]-1/2[/sup]; the magnitude of the magnetic field at the position of charge #2 is equal to v/c[sup]2[/sup] times the electric field magnitude. (See Griffiths’s or Purcell’s E&M texts for a derivation.) The two forces are in opposite directions, so you subtract one from the other, and you end up with a total force that depends on the speed as √(1-v[sup]2[/sup]/c[sup]2[/sup]) = 1/γ — which is exactly what’s predicted from the force-transformation law in the link I gave above.
Could someone tell me (using small words please) if a magnetic field produced and measured in one reference frame would measure differently in another reference frame?
I’m afraid I have to disagree on this one. Electric field is an independent entity just as an electric charge. It exists in all frames of reference, and it doesn’t matter from what frame of reference you are observing electromagnetic interactions the electric field is always there, whereas when you observe magnetic field it’s existence depends on the frame of reference. That is, there are frames of reference in which magnetic field doesn’t exist. This fact, as I see it, belies the assumed notion that magnetic field is as real as electric one.
Yes, both electric and magnetic fields change when you look at them in different reference frames. You can see a magnetic field in one reference frame, and not see any at all in a second reference frame.
This isn’t true. If you have both an electric field and a magnetic field in one frame, it turns out that there (almost) always exists another reference frame where the magnetic field vanishes or the electric field vanishes (but not both.) In particular, you can cook up situations in which the electric field is non-zero in one frame and zero in another frame.
As an example: consider a current-carrying loop of wire moving through space at uniform speed. As the loop of wire passes by a given point, the changing magnetic field at that point creates an electric field there (by Faraday’s Law.) But we could equally well transform into a frame where the loop is at rest; in this frame, there’s a magnetic field but zero electric field. So you can’t say that “the electric field exists in all frames of reference, while the magnetic field’s existence depends on your frame.”
MikeS, perhaps there is a little miscommunication here. I thought we were referring to any difference from the force caused by the static electric field of two stationary electrons, granting the usual gamma factor in the relativistic force transformations. The difference is zero in the frame of reference in which the charges are stationary; therefore the difference is zero in any other frame. It doesn’t matter whether or not v approaches c. The difference is always zero. This is distinct from cases in which the electrons have differing velocities, where no such argument can be made. In that case there is a frame in which there is no magnetic field, or frames in which there is a combination of magnetic and electric fields. In each case the force transforms as you point out, but originate from differing combinations of electric and magnetic fields. I guess I see now that you were responding to leahcim saying “…show that the total force is constant at all speeds”; there I was taking for granted the lorentz force transformation, and thought he was also.
The only reason why the electric field in the setup you propose has zero effect is that it is cancelled out in the electric wire which is electrically neutral.
Ah, I see. Way up in post #14, I was trying to say that “each electron, individually, experiences no net force in the frame where v -> c.” But when I said “the electrons experience no net force”, it was taken to mean “the sum of the total force on electron #1 and the force on electron #2 is zero.” Both statements are, of course, true; but the sum of the total forces between the electrons is exactly zero in all frames (in this particular situation, at least), not just when v -> c.
I can’t make heads or tails of this response. MikeS’s point wasn’t concerning whether or not the loop itself felt any “effect”; he correctly gave an example where an electric field can be transformed away (and where the magnetic field cannot!).
If it helps turn you around: if you asks most theorists, they assume that magnetic monopoles exist. It’s true that we haven’t discovered them (they must be very massive), but lifting Polchinski’s quote from the wiki article (which you may find informative): the existence of monopoles is “one of the safest bets that one can make about physics not yet seen.”
I like the way my old advisor used to phrase it: “We know that monopoles exist; there might just be a very small number of them, like zero”. In other words, the laws of physics are such that monopoles are possible, and would be formed under the right conditions, but the right conditions to form them are extraordinarily rare, such that it’s plausible that there are none at all anywhere in the observable Universe.
iamnotbatman, MikeS is correct that forces transform even from transverse boosts. I think your mistake is in only considering the length contraction, and not the time dilation. It’s related to the relativistic transverse Doppler shift.