A beutifully simple interpretation of Vector Potential

[shanesworld]

This is not my theory, or even new, rather pertaining to established physical knowledge, but I simply find it fascinating. It pertains to several areas of physics, and/or variational mathematics, so I've posted it here in the General Physics area. My reasons for posting is because it is one of my favorite interpretations Vector Potential (I find it interesting and gratifying) So here we go, it's time to share.

Vector potential, which is often perceived as a somewhat abstract idea to the layman, and even those knowledgeable about physics, has a very simplifying interpretation, something that makes it appear quite intuitive. The path of deriving this from items such as Maxwell's equations isn't necessarily that simple...the result, however is awfully pleasing.

Quite simply, the vector potential ,(\vec{A} in TeX), is constantly proportional to the momentum contained within the fields of a system. Explicitly, for example in classical electrodynamics, vector potential=c/charge*(field momentum).

The notion of field momentum sometimes confuses people, however it is relatively (no pun intended)approachable from the stance of conservation of momentum. For example one may consider a system in which energy and momentum are each conserved. A field acting on a charge may cause the charge to gain kinetic momentum, however that momentum did not just spontaneously come into existence...No... No... Rather it came from somewhere, namely the field itself.

So next time you are looking at something like max well's equations and you see something like the magnetic field = the curl of the vector potential, it might be interesting to realize in amazement that this also means the magnetic field is proportional to the curl of "the momentum contained in the electromagnetic fields". Personally, my brain likes that interpretation because the concept of momentum is more graspable to me in some ways than "Vector potential", but that is besides the point.

 

[Bill_K]

This is not my theory, or even new, rather pertaining to established physical knowledge, but I simply find it fascinating. It pertains to several areas of physics, and/or variational mathematics, so I've posted it here in the General Physics area. My reasons for posting is because it is one of my favorite interpretations Vector Potential (I find it interesting and gratifying) Quite simply, the vector potential ,(\vec{A} in TeX), is constantly proportional to the momentum contained within the fields of a system. Explicitly, for example in classical electrodynamics, vector potential=c/charge*(field momentum).

Interesting idea, shanesworld, but I'm afraid it is not even close to being true. For one thing, the vector potential is not unique. One might claim it is true in a particular gauge, but that cannot be either. The electomagnetic momentum density is (1/4πc)(E x B). Take a situation where E = 0. Then the momentum is zero, but clearly A does not have to be.

 

[xlines]

This is not my theory, ...

Quite simply, the vector potential ,(\vec{A} in TeX), is constantly proportional to the momentum contained within the fields of a system... the point.

Hello!

Yes, I remember something of the kind and I agree that it would be better interpretation if it is true. BUT, as far as I can recall, Griffits mentioned this in his Introduction to Electrodynamics, and there was condition for it. I don't remember what. Also, vector potential was associated with "flow" of momentum, not the momentum itself.

Do note that I could be easily wrong, my memory is somewhat vague on the subject. This is going to be interesting to clear up, I'm glad you brought up the subject.

Cheers!

 

[xlines]

... BUT, as far as I can recall, Griffits mentioned this in his Introduction to Electrodynamics, and there was condition for it ... vector potential was associated with "flow" of momentum, not the momentum itself.

Ok, here is a follow-up: Griffiths really does mention this in "Introduction to Electrodynamics", 3rd ed., p. 236 says: "... A does not admit a simple physical interpretation in terms of potential energy per unit charge. (In some contexts it can be interpreted as momentum per unit charge)" citing M. D. Semon and J. R. Taylor, Am. J. Phys. 64, 1361 (1996). I can't fetch that, so if someone can help ... So, I was wrong on the flow part, Bill K explained why such interpretation can take place only for a specific gauge along with possible additional conditions. Since this interpretation is not widely known, I guess it is valid only on a specific and uncommon physical regime.

 

[Bill_K]

Ah Ok, sorry, what he is talking about is the *canonical* momentum. The equations of motion of a charged particle in an electromagnetic field can be written nicely in terms of the combination P = p + (e/c) A, where p is the mechanical momentum. The extra term (e/c) A looks formally like a momentum but it is not really the momentum of anything. p is the momentum. As we said before, A is not uniquely defined anyway.

 

[shanesworld]

Ah Ok, sorry, what he is talking about is the *canonical* momentum. The equations of motion of a charged particle in an electromagnetic field can be written nicely in terms of the combination P = p + (e/c) A, where p is the mechanical momentum. The extra term (e/c) A looks formally like a momentum but it is not really the momentum of anything. p is the momentum. As we said before, A is not uniquely defined anyway.

Ok...first I appreciate the comments, it anything some of the varying points sparked some curiosity in me. It's kind of funny that you said this Bill_K, because I was just about to write you about thiis...but I was in part referring to cannonical momentum...I was reasonably sure about what I posted when I did, but these are definitely some questions I will have to think about. Someone mentioned Griffiths introduction too...

Here's a bit about what I was referring to and some differences that spark some interest:


To quote the result of section 2.17, of a book by Oxford Graduate Texts titled Analytical Mechanics for Relativity and Quantum Mechanics...section describes classical system with electromagnetic field acting on point charges

The generalized moment of particles in an electromagnetic field are not simply the particle momentum. They are

P_n=the partial derivative of the Lagrangian with respect to velocity=m_n*v_n+q_n/c*{A}

…which might be considered as the vector sum of the particle momentum (m_nv_n) and the field momentumq_ q_n/c*\vec{A}. It is this generalized momentum that is conserved…”

Griffiths:. The relation between "momentum density" and the pointing vector is discussed in chapter 8 of Griffiths “Introduction to Electrodynamics.” Equation 8.30. …momentum density=c^2*S…basically what you wrote.


But you raised some interesting questions. it would be interesting to see how these concepts tie into each other. I don't necessarily see the gauge freedom being a problem in itself, but I would have to give it some more consideration.

 

[Dale]

Also, that is the cannoical momentum of a particle moving in an external field. I can't remember if the external field is assumed to be static. It is not the canonical momentum of the field itself. I.e. in this formulation A is evaluated only at the point where the charge is located. To get the momentum of the field itself you would need to integrate over space.

 

[Bill_K]

…which might be considered as the vector sum of the particle momentum (m_nv_n) and the field momentum q_ q_n/c*\vec{A}. It is this generalized momentum that is conserved…”

That's the right idea in general, but the canonical momentum is mostly just a convenient quantity in this case because it is *never* conserved. The equation of motion is dP/dt = - ∂H/∂x, and the right hand side will always be nonzero if there's a B field. (And if there's not a B field, you don't need A!)

 

[shanesworld]

That's the right idea in general, but the canonical momentum is mostly just a convenient quantity in this case because it is *never* conserved. The equation of motion is dP/dt = - ∂H/∂x, and the right hand side will always be nonzero if there's a B field. (And if there's not a B field, you don't need A!)

Actually, that is a direct contradition to the texts that I have. Cannoncial moments is exatctly what IS conserved. In fact this is what makes it possible to derive such things as relativistic wave equations i.e. dirac and klein-gordon types. I have more confidence in the Oxford book currently.

 

[shanesworld]

Also, that is the cannoical momentum of a particle moving in an external field. I can't remember if the external field is assumed to be static. It is not the canonical momentum of the field itself. I.e. in this formulation A is evaluated only at the point where the charge is located. To get the momentum of the field itself you would need to integrate over space.

The field is external to the particle(s) but still part of the system being modeled...I don't think this is a "self interaction" with the particle...so yeah,..I agree if that is what you mean. It is still a vector function though. "A" acts on the particle but still multi-dimensional field. For point like particles one must sum over partilces, essentially the analogy to integrating. My instinct is with you though for general charge densities, integrating makes sense to me..of course that doesn't guaruntee much. The field is not static...but is time dependent. Static felds would not be conducive conservation laws. It is considered to be a function of the (3 )spacial dimensions, and of time. There is also a four vector analogy to the three vector "A" where the 4-vector A=A^0\hat{e}_0+A^1\hat{e}_1+..., and in this case A^0 is prop to "scalar potential". Actually, I think this (4-vector) version DOES assume a particular gauge choice, the Lorentz Guage, but enables maxwells equations to be compressed into a single equation... ie , where the 4-divergence of the 4-vector potential is zero. That is a continuity equation as well.
Well that is how I see it, ...I think we are close to the same page,
I'm glad that some are familiar with this...part of my reason for posting is I don't hear it enough...thanks.