Horace Heffner wrote:
At 10:00 AM 12/8/4, Stephen A. Lawrence wrote:
I had three comments on this analysis...
First, watch out for Shadowitz -- I've seen an instance where he messed up an analysis by using the "motion" of the EM field relative to a particle, which has no role in relativistic EM. Rindler, Jackson, and Griffiths seem more reliable, to name some I'm aware of. I don't know any reason to doubt Shadowitz's formula for pancaking, but you should definitely double check any general assertions he makes about how fields transform.
I've seen it various other places too. No refs handy at the moment.
Oh, the pancaking is fine. The caveat is with respect to Shadowitz -- I'm looking at a scan of p. 124 from his "Electricity and Magnetism" in which he concludes that, in a particular case, moving a magnet past a wire produces no EMF in the wire, while moving the wire past the magnet does produce such an EMF. Someplace in there he seems to have suffered a breakdown in intuition which goes pretty deep. After seeing this particular analysis I'd tend to avoid him in favor of other authors. (I don't know the edition and don't have the book, just a scan of a few pages someone sent me during a conversation about homopolar generators. I suppose it's even possible that the text wasn't actually by Shadowitz, but the person who sent it to me is generally pretty reliable.)
[ ... ]
Finally, let's do just that. For simplicity, assume a rotating ring of uniform negative charge density, with a fixed positive charge in the middle of the ring. Let's look at the axial field.
Since the ring is uniform, the 4-current density is not varying in time, and we can forget about the "retarded" part.
This analysis bothers me. It says the whole is not the sum of the parts.
Well, if the parts are accelerating, then perhaps it's not. Rindler, in his misnamed "Introduction to Special Relativity" (if that's an "introduction" then I'm the Pope) goes through the derivation of the pancaking for a single charge in uniform motion, but I got bogged down at the start of that section and went off to study French. (Call me a dilettante, I won't object...) Just before that, he covers the retarded integrals used to obtain the 4-vector potential in the general case, and there were some very tricky bits in there for accelerating charges.
Here's the same argument I already gave, in slightly more detail (I've left out the epsilons and mus on general principles).
The basic formula for A at a particular point, from Rindler, 2nd edition, p. 111, or Griffiths, 3rd edition, p. 423 is just
A = (1/4pi)integral([J]dV/r)
where the integral is taken over all space, [J] is the retarded value of the 4-current density, and r is the distance from the point where one is evaluating A. Since J is time invariant in this case, [J] = J. Each component of J is integrated separately, which means
phi = (1/4pi) integral(rho dV/r)
where phi = electric potential and rho = charge density.
To look at it yet one more way, if you're looking at a case where the current is not varying, then you're in the domain of magnetostatics and you don't need anything beyond simple E&M to analyze it. Fancier approaches, such as the pancaking model, must agree with a simple analysis in simple cases.
I showed that if pancaking is valid for an individual particle, then the sum of such individual pancaking effects does not cancel at all points.
But again, the formula you started with was for a point charge in uniform motion.
However, I must admit I had the nagging feeling I probably left other important effects out of my analysis, like abberation, which might negate field pancaking. I have the impression that aberration applies to photons though, and pancaking to fields. There should be a simple way to visualize this situation. (Beign a rank amateur, I don't consider tensor analysis simple.)
Huh. I agree, there should. But I sure don't know what it is, either -- wish I did. Consider this:
There are two charged rings, one positive and one negative, with equal total charge quantities, arbitrarily close together. In the lab frame one is spinning and the other isn't. Hence, in the lab frame, there is a magnetic field present, but the E field is negligible. (You can replace the two charged rings with a simple loop of wire carrying a current, if you prefer -- the point is that the net charge density is zero when averaged over any finite volume.)
Now, look at it in a frame of reference which is rotating with the rotating ring. A moment's gedanken experimentation with a "test charge" moving tangentially to the rotating ring (which will feel a force due to the B field in the lab frame) should convince you that there's an E field in the rotating frame of reference. But charge is conserved -- just moving into a different frame of reference doesn't create or destroy it. So where's the E field in the rotating frame coming from? The divergence of E must be zero everywhere unless there's some nonzero charge density somewhere. Did somebody repeal Gauss's law, or what?
If you think the answer's obvious then you may need to think about it some more :-) (The discussion of this one accumulated about 200 posts on sci.physics.relativity.)
[snip conclusions]
This field is well understood and it's certainly conservative.
I am curious as to just why it is thought the huge polar jets of material fly out of black holes and neutron stars. An analagous (and additional) polar gravitational field should develop in the vicinity of black holes, if the analysis is done according to the gravimagnetic isomorphism I proposed on this list anyway.
I think I just fell into the deep end of the pool here...
'Way past me, Horace! :-)
