I have a question about the direction of the magnetic vector potential

[MacOfficial]

Suppose my reference system is x coming out of the page toward you, y is in the plane of the page going left and right and z is in the plane of the page going up and down. Further suppose that the magnetic field is parallel to the x-axis and the electric field is parallel to the z axis. Finally, assume the direction of propagation of the electric and magnetic fields is along the positive y axis, as would appear to be dictated by the Poynting Vector, E X B.

If I now wanted to represent the magnetic vector potential (from which the B field is generated) on a diagram as described above, how would I draw it? Is it parallel to the z axis or parallel to the y-axis or would it be at some other arbitrary direction?

I am not sure I am even asking the right question yet but I am struggling to visualize the magnetic vector potential with respect to the propagating electric and magnetic fields. Thanks in advance.

 

[chrisbaird]

It depends on what gauge you choose to use, what the charges and currents are doing, and what your boundary conditions are, in order to concretely determine the potentials in electrodynamics. In general $\textbf{B}=\nabla\times\textbf{A}$ and $\textbf{E}=-\nabla\Phi-\partial\textbf{A}/\partial t$. If you choose the Coulomb gauge, and you are in free space (no charges, currents, or boundary conditions) like for a plane wave, then you can make the scalar potential $\Phi$ go away. In that case, then the vector potential A just points in the same direction as the electric field E, but is out of phase. In your setup, A would be pointing in the z direction.

 

[MacOfficial]

Chris:

Thanks so much for your reply. I have two, additional questions, if you don't mind.

1. Can you suggest a good reference which discusses this at length? I have several texts on electromagnetic theory but none of them seems to cover this point directly. The closest I could come was the Feynman lecture series but I was still left a little mystified. Any thoughts you might have would be appreciated. I would like to study the specifics of how the direction is deduced so I can be sure I understand.

2. Suppose we take the general case where there are free charges and currents (but allow me to skip the particular boundary conditions for a moment). In that case, and given the general equations you have specified, can you make a generalized statement about the direction of the vector potential?

Once again, thank you very much.

Sean

 

[chrisbaird]

Jackson's book is the standard.

 

[MacOfficial]

Chris:

Once again, thanks.

Just to be sure, do you mean: Jackson, John D., (1999)., Classical Electrodynamics., New York, NY: John Wiley & Sons?

Sean