Vector Potential

Any field with 0 divergence can be written as a curl of another vector. $A$ is also not unique to form $\vec{B}$ since we could always add the gradient of some vector to it, as the curl of the gradient of a vector is 0, and curl is distributive

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

• $\vec{A}$ is the vector potential
• $\vec{B}$ is the magnetic field

$\vec{A}=\frac{{\mu }_0}{4\pi }\int \frac{\vec{J}}{r}\mathrm{d}{\tau }^{\prime }$

• $\vec{J}$ is the current density

$\nabla \cdot \vec{A}=0$

• In the case of magnetostatics

$\vec{A}=\frac{\vec{B}\times \vec{r}}{2}$

• For constant $\vec{B}$

${\nabla }^{2}A=-{\mu }_0\vec{J}$