## Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as:

`$\begin{array}{c}\nabla \cdot D=\rho ,\\ \nabla \cdot B=0,\\ \nabla ×E=-\frac{\partial B}{\partial t},\\ \nabla ×H=J+\frac{\partial D}{\partial t}.\end{array}$`

Here, E and H are the electric and magnetic field intensities, D and B are the electric and magnetic flux densities, and ρ and J are the electric charge and current densities.

### Electrostatics

For electrostatic problems, Maxwell's equations simplify to this form:

`$\begin{array}{l}\nabla \cdot D=\nabla \cdot \left(\epsilon \text{ }E\right)=\rho ,\\ \nabla ×E=0,\end{array}$`

where ε is the electrical permittivity of the material.

Because the electric field E is the gradient of the electric potential V, $E=-\nabla V.$, the first equation yields this PDE:

`$-\nabla \cdot \left(\epsilon \text{ }\nabla V\right)=\rho .$`

For electrostatic problems, Dirichlet boundary conditions specify the electric potential V on the boundary.

### Magnetostatics

For magnetostatic problems, Maxwell's equations simplify to this form:

`$\begin{array}{l}\nabla \cdot B=0,\\ \nabla ×H=J+\frac{\partial \left(\epsilon E\right)}{\partial t}=J.\end{array}$`

Because $\nabla \cdot B=0$, there exists a magnetic vector potential A, such that $B=\nabla ×A$. For non-ferromagnetic materials, , where µ is the magnetic permeability of the material. Therefore,

`$\begin{array}{l}H={\mu }^{-1}\nabla ×A,\\ \nabla ×\left({\mu }^{-1}\nabla ×A\right)=J.\end{array}$`

Using the identity

`$\nabla ×\left(\nabla ×A\right)=\nabla \left(\nabla \cdot A\right)-{\nabla }^{2}A$`

and the Coulomb gauge $\nabla ·A=0$, simplify the equation for A in terms of J to this PDE:

`$-{\nabla }^{2}A=-\nabla \cdot \nabla A=\mu J.$`

For magnetostatic problems, Dirichlet boundary conditions specify the magnetic potential A on the boundary.

### Magnetostatics with Permanent Magnets

In the case of a permanent magnet, the constitutive relation between B and H includes the magnetization M:

`$B=\mu H+{\mu }_{0}M.$`

Here, $\mu ={\mu }_{0}{\mu }_{r}$, where μr is the relative magnetic permeability of the material, and μ0 is the vacuum permeability.

Because $\nabla \cdot B=0$, there exists a magnetic vector potential A, such that $B=\nabla ×A$. Therefore,

The equation for A in terms of the current density J and magnetization M is

`$\nabla ×\left(\frac{1}{{\mu }_{r}{\mu }_{0}}\nabla ×A\right)=J+\nabla ×\left(\frac{1}{{\mu }_{r}}M\right).$`