Gauss' Law of Magnetostatics
Gauss's Law for electrostatics states that
This tells us that the source of electric fields are charges. However, experiments show that there are no corresponding "charges"(monopoles) for magnetic field. The magnetic field do not have a source, and so always forms closed loops.
Gauss' law of magnetostatics is an expression of the fact. It can be written as such:
Since B is divergence-free, B must be the curl of some vector A. This vector is called the vector potential, the direct analog of the electric potential, also known as the scalar potential.
The Biot-Savart Law can be difficult to compute directly, but if we know the magnetic potential field, we can find the magnetic field easily:
Calculation of Vector Potential.
The vector potential is given by