Magnetic vector potential
The magnetic vector potential is a vector field that serves as the potential for the magnetic field. The curl of the magnetic vector potential is the magnetic field.
The magnetic vector potential is preferred when working with the Lagrangian in classical mechanics and quantum mechanics.
Calculating magnetic vector potential
The magnetic vector potential contributed by a length with current running through it is
What is the magnetic vector potential a distance from a long straight current element?
Assume the wire is placed on the z-axis. Hence the distance from a differential element to a point in space is
Hence,
Now it is necessary to integrate over the length of the rod. Since the rod is arbitrarily said to be on the z-axis, it can be said, for simplicity, to extend from to This simply requires multiplying by 2. Also, use the substitution
Magnetic field
The magnetic field is the curl of the vector potential.
Find the magnetic field in a region with magnetic vector potential
Since is in spherical coordinates, use the spherical definition of the curl.
The only part that will survive for the given is
Electric field
The partial derivative of the magnetic vector potential contributes partially to the induced electric field according to faraday's law.
Recall Faraday's law: