Exploring the Role of the Vector Potential in Faraday's Law of Induction

[bob012345]

Suppose there is an almost infinitly long but narrow solenoid with an AC current surrounded by a much larger loop such that there is no magnetic field except in the solenoid. I had always thought it didn't matter what part of the outer loop the flux changed in, there would be an induced electric field and thus a current in the outer loop. Now, I'm reading about the Maxwell-Lodge controversy which makes a big deal over the fact that there is no magnetic field outside the solenoid and none at the outer loop. So, 1) Does Faraday's Law still give the correct result by itself? 2) Is the issue mainly about the role of the Vector potential in mediating Faraday's law? Thanks.

 

[Dale]

I'm reading about the Maxwell-Lodge controversy which makes a big deal over the fact that there is no magnetic field outside the solenoid and none at the outer loop.

I tend not to get worked up about results of physically impossible configurations. The Maxwell-Lodge concept is based on an infinitely long solenoid. Sometimes you can get away with infinite configurations of one sort or another, but this is a case where you cannot. For a finite solenoid the issue does not arise, and since only finite solenoids exist, that is all that is needed.

 

[bob012345]

Ok, forget the word infinite. Just make the flux outside the solenoid very very small compared to that inside. That should be doable.

 

[Dale]

Yes, as long as you have a finite current configuration then Faraday’s law holds just fine.

 

[bob012345]

Yes, as long as you have a finite current configuration then Faraday’s law holds just fine.

Thanks. I take it then that implies Faraday's law would always give the correct result even as the solenoid approached infinity in the limit. But I'm still curious with why Faraday's law needs some field at the outer loop. Is the heart of the issue something like "how does the outer loop know about the flux change if there is no field there?" or just action at a distance? If so, why then would the specific value at the loop tell one everything about the total flux change inside?

Does using the Vector Potential get rid of the issue? Is the VP more fundamental? Thanks.

 

[Dale]

Is the heart of the issue something like "how does the outer loop know about the flux change if there is no field there?" or just action at a distance? If so, why then would the specific value at the loop tell one everything about the total flux change inside?

To my understanding the issue is mostly a mathematical one that arises when trying to do integration in a space with a non-trivial topology. An infinitely long solenoid partitions space into an inside and an outside. There is no such topological distinction for a finite solenoid.

 

[bob012345]

Thanks. I pulled out Feynman Volume 2 15-7 where he discusses the Vector Potential. He thought the VP was more fundamental because if it's central role in quantum mechanics and QED.