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.

 

[paranormal]

The concept of a vector potential can be confusing, but it is an important concept in the study of electromagnetism. Simply put, the vector potential is a mathematical tool used to describe the behavior of electric and magnetic fields. It is related to the electric and magnetic fields through the equations B = del cross A and E = -del phi - dA/dt, where A is the vector potential and phi is the scalar potential.

The physical interpretation of the vector potential can be understood by considering the behavior of a charged particle in an electromagnetic field. The electric field, E, is responsible for the motion of the particle, while the magnetic field, B, affects the direction of the particle's motion. The vector potential, A, represents the "hidden" or "non-observable" part of the magnetic field that influences the motion of the charged particle.

In other words, while the magnetic field, B, is directly observable, the vector potential, A, is not. It is a mathematical construct that helps us understand the behavior of charged particles in an electromagnetic field. So, while it may seem abstract, the vector potential is a crucial concept in understanding the physical behavior of electricity and magnetism.

I hope this helps clarify the concept of vector potential for you. If you have any further questions, don't hesitate to reach out for clarification. Keep up your studies and good luck with your homework.