Electrodynamics/Magnetic Potential

Gauss' Law of Magnetostatics

Gauss's Law for electrostatics states that

$\mathrm{\nabla }\cdot 𝐄=\frac{\rho }{{ϵ}_0}$

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:

$\mathrm{\nabla }\cdot 𝐁=0$

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:

$𝐁=\mathrm{\nabla }\times 𝐀$

Calculation of Vector Potential.

The vector potential is given by

$𝐀(𝐫)=\int \frac{𝐣({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }|}dV$