Electrodynamics/Magnetic Potential

Vector Potential

Similarly to the potential of the electric field, we can define a vector field to represent the potential energy of the magnetic field. The magnetic potential field, A, can be a useful tool for determining the magnetic field, if the former is known and the latter is not.

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

Relation to Magnetic Field

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:

$𝐁=\mathrm{\nabla }\times 𝐀$

Gauss' Law of Magnetostatics

Gauss' law of magnetostatics can be written as such:

$\mathrm{\nabla }\cdot 𝐁=0$

We know that this must be true because B is the curl of A, and the divergence of any curl is always zero.