Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as:

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

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:

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

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:

$$-\nabla \cdot \left(\epsilon \nabla V\right)=\rho .$$

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:

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

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

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

Using the identity

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

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 μ_r 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 $$B=\nabla \times A$$. Therefore,

$$\begin{array}{l}H\text{ =}\frac{1}{{\mu }_0{\mu }_r}B-\frac{1}{{\mu }_r}M,\\ \nabla \times H=\nabla \times \left(\frac{1}{{\mu }_0{\mu }_r}\times A-\frac{1}{{\mu }_r}M\right)=J.\end{array}$$

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

$$\nabla \times \left(\frac{1}{{\mu }_r{\mu }_0}\nabla \times A\right)=J+\nabla \times \left(\frac{1}{{\mu }_r}M\right).$$