magnetic vector potential

[AriAstronomer]

Hey everyone,
So in my E&M class we've started to talk about magnetic vector potential. I understand mathematically how it relates to the magnetic field and charge density, but I'm still a bit unsure conceptually what it means. Is it similar to electric potential in the sense that it measures differences in...energy? Since it's a vector, the pathway you choose to get from a to b matters? A point in the right direction would be marvelous.
Thanks,
Ari

[macaw]

Hey everyone,
So in my E&M class we've started to talk about magnetic vector potential. I understand mathematically how it relates to the magnetic field and charge density, but I'm still a bit unsure conceptually what it means.
Thanks,
Ari

Hi Ari,

The vector potential arises from physical and mathematical reasoning.

1. The physical reasoning. Maxwell says:

div(B)=0

that is, there are no magnetic "monopoles"

2. The math reasoning:

div(curl(A))=0

Comparing 1 with 2 we obtain:

B=curl(A)

where A is the vector potential describing B.

Is it similar to electric potential in the sense that it measures differences in...energy? Since it's a vector, the pathway you choose to get from a to b matters? A point in the right direction would be marvelous.

No, it isn't similar. In the case of the electric field, the fact that in a STATIC magnetic field (i.e. dB/dt=0) we have

curl(E)=0

produces immediately:

E=grad(phi)

where phi is a scalar potential.

It is only in THIS case that the integral of E along an arbitrary path depends only on the endpoints, not on the path taken. The theorem is valid only for gradients, not for arbitrary fields. So, it isn't valid for B.

[tusenfem]

There is also a real meaning to the magnetic potential, if I could only find the description in Physics Today ....
Okay, it is the Ahoronov-Bohm effect.

If you take two cylinders inside eachother and drive currents through them such that the outer cylinder negates the field of the inner cylinder, i.e. outside there is no field, and you send a beam of electrons perpendicular to the axis of the cylinders then you find that the electrons are influenced through the vector potential A.

Ohhh, the wiki is rather complicated in its explanation, but the physics today version is much more readable. Unfortunately, I seem to have lost my passwd. I will request a new one and get the pdf for you later.

ETA: I now have the article in pdf, if you would like to have it send me a pm with your email address.

[ShinAce]

Hey everyone,
So in my E&M class we've started to talk about magnetic vector potential. I understand mathematically how it relates to the magnetic field and charge density, but I'm still a bit unsure conceptually what it means. Is it similar to electric potential in the sense that it measures differences in...energy? Since it's a vector, the pathway you choose to get from a to b matters? A point in the right direction would be marvelous.
Thanks,
Ari

I'm a little rusty in this regard, but I believe you are correct about the path dependence. It would only be path independent in a uniform magnetic field.