Magnetic vector potential

The calculation of magnetic field in BiotSavart is based on the magnetic vector potential A. The magnetic flux density B is the curl of the vector potential:

B = curl A

The magnetic vector potential is defined up to a gauge transformation (the addition of the gradient of an arbitrary scalar function). BiotSavart uses the so-called Coulomb gauge, in which the vector potential has no divergence:

div A=0

In this gauge the vector potential satisfies the following equation of the Poisson type:

laplacian A = J

where J is the known current density. The solution of the equiation is

A = integral ...

This fundamental relation gives rise to closed-form expressions for the magnetic field of loops and line segments. These expressions are used to calculate the magnetic field of solenoids (approximated as a collection of loops) and wires (approximated as a collection of straight line segments).