Electrodynamics/Magnetic Potential

Gauss' Law of Magnetostatics

Gauss's Law for electrostatics states that

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$

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:

$\nabla \cdot \mathbf {B} =0$

Vector Potential

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:

$\mathbf {B} =\nabla \times \mathbf {A}$

Calculation of Vector Potential.

The vector potential is given by

$\mathbf {A} (\mathbf {r} )=\int {\frac {\mathbf {j} (\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'} |}}dV$