What Is the Physical Meaning of Vector Potential in Electromagnetism?

[Gary Roach]

Homework Statement

The concept of a scalar potential is reasonably straight forward. It is the energy needed to move to a point from some arbitrary reference point, the reference point being the origin for most mechanical problems and infinity for most electromagnetic problems.And of course this will produce a scalar field.

The physical meaning of a vector potential, on the other hand, is alluding me. All of my texts seem to be very vague at this point. Mathematically I can say that B = del cross A where A is a vector potential but what does that mean physically.

Any clarification of this point will be sincerely appreciated.

Gary R

Homework Equations

The Attempt at a Solution

 

[diazona]

It's true that the physical significance of a vector potential is not at all as clear as it is for a scalar potential. Just as the scalar potential describes a difference between points, the vector potential describes a difference between paths. For example, when you move a charge from one point to another, it gains a certain amount of energy, and the scalar potential let's you figure out how much. Similarly, when you move a current from one path to another (imagine you have a wire carrying current from point A to point B, and you bend it into a different shape), the current gains a certain amount of momentum, and the vector potential let's you figure out how much.

In practice, it's more useful to talk about the vector potential around a loop, rather than along an arbitrary open path. In the example above, if you take the original path and the new path (in reverse), you form a loop. In more advanced physics, this ties into the interpretation of the vector potential (and the scalar potential) as the connection of a gauge covariant derivative: essentially it describes the transformations you have to make on a quantum field as you go from one point in space to another.

 

[Gary Roach]

Thanks diazona

Just what I needed. It's nice to know that I'm not just dense.

Gary R

 

[vela]

In volume two of the Feynman Lectures, Feynman notes that in magnetostatics the energy of currents in a magnetic field is given by$$U= \frac{1}{2}\int \vec{j}\cdot\vec{A}\,dV$$In comparison, for electrostatics, you have$$U = \frac{1}{2}\int \rho\phi\,dV$$But then he points out the idea of the vector potential as potential energy for currents doesn't turn out to be very useful.

He also discusses how A fits into quantum mechanics. It's worth a read if you get a chance.