Flaw in the integral form of faradays law with large loops

[ThomGunn]

The situation is as such. you have a magnetic passing through a loop(loop 1)of wire as some time t, say you also have a larger loop of say one light year in radius surrounding loop 1, call this loop 2. at some time t the magnetic field is shut off. when this happens would loop 2 instantaneously respond via Faraday's law? The integral form of Faraday's law makes no distinction about the radius of the loop?

Is this possible? what am I missing, these type of paradoxical statements come up with relativity but Maxwell's equations are generally paradox friend and play well with relativity.

 

[UltrafastPED]

Your surface is no longer "simply connected" - this leads to complications with the integration!

Simply connected: http://www.solitaryroad.com/c262.html

I couldn't find a direct discussion of the problem you are posing, but here is a general discussion of some of the issues, but in the context of quantum mechanics:
http://books.google.com/books?id=7d...ntegral on non-simply connected space&f=falseNote: I think I mis-read the situation; see post #4.

 

[ThomGunn]

hmmm. I see, could you help me understand why this isn't simply connected? I don't see anything peculiar about the space so that it isn't simply connected, but I'm not exactly sure.

 

[Dale]

when this happens would loop 2 instantaneously respond via Faraday's law?

Faradays law applies, in its integral form, for both loop 1 and loop 2 at all times. Faradays law would not predict an instantaneous change in the EMF for loop 2. For any set of fields which is a solution to Maxwells equations the change in flux will never happen superluminally.

 

[vanhees71]

Maxwell's equations are not paradoxical at all and they are perfectly in accordance with relativity. That's how relativity was discovered: The Maxwell equations, although known to be very successful in discribing electromagnetic phenomena, where not Galilei invariant. Many physicists (including Voigt, FitzGerald, Lorentz, and Poincare) thought one had to introduce a preferred frame of reference (called the "ether rest frame") to accommodate this. The more they thought about it, the more complicated the material named ether became, and finally it was Einstein's fresh point of view that one has to adapt the space-time description to the invariance group of the Maxwell Equations, now called Poincare symmetry (invariance of the natural laws under proper orthochronous Lorentz transformations and space-time translations).

The usual treatment of macroscopic electrodynamics is, however, mostly plagued by tacitly making non-relativistic approximations to the treatment of matter and thus the consitutive equations. This is a pity, because it makes a lot of unnecessary trouble with interesting phenomena like the homopolar generator and energy-momentum bilance, leading to complicated explanations with "hidden momenta" and all that. If one treats everything relativistically, no such oddities are necessary.

Another source of confusion is that many textbooks state the Maxwell equations in integral form making (again often tacitly) special assumptions. Particularly Faraday's Law is plagued from these sins. When dealing with time dependent surfaces and its boudaries, one has to include the magnetic force in the electromotive force, and everything is fine. It's very nicely explained in the Wikipedia:

http://en.wikipedia.org/wiki/Faraday's_law_of_induction#Proof_of_Faraday.27s_law

The most simple form, is naturally the differential (local) form of the equations, because classical Maxwell theory is the paradigmatic example of a relativistical (classical) field theory.

Now to your problem: When dealing with this problem, you have to solve first the problem of the electromagnetic fields produced by the time-dependent current in the larger loop. Here you have to use the full time-dependent Maxwell equations since you are dealing with a situation where the relevant parts of the setup are much larger than the typical wavelengths of the produced em. waves. The electromagnetic fields cannot propagator faster than the speed of light in vacuo, and thus the front of the signal reaches the smaller loop only after the time this em. wave needs to reach it (i.e., one year in your example). So as long as you use the full Maxwell equations there cannot be any contradiction with relativity!

 

[Delta2]

First of all the magnetic field cannot be shut off instanteneously, cause it has to be a continuous function of time(since we know it is differientiable with respect to time). Also the wave that is produced by shutting it down will NOT contribute to the total magnetic flux of the outer loop, until its front hits the outer loop.