Electrostatics and Magnetostatics Equations

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. By default, the toolbox uses the zero Neumann boundary condition and assumes that the boundary is insulated, so there is no electric current through 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. By default, the toolbox uses the zero Neumann boundary condition and assumes that:

• For 2-D magnetostatic analysis, there is no magnetic flux through the boundary.

• For 3-D magnetostatic analysis, $${}^{{ϵ}^d\frac{\partial c_e}{\partial t}-\frac{\partial }{\partial x}\left[D_{eff}^d\frac{\partial c_e}{\partial x}\right]=a^d\left(1-t_{+}\right)j^d,d\in \left\{a,c\right\}}$$, where u is a magnetic vector potential, and c represents properties of the material, such as permittivity , permeability, or conductivity.

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

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