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Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as:

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

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:

$\begin{array}{l} \nabla \cdot D = \nabla \cdot (ε E) = ρ, \\ \nabla \times E = 0, \end{array}$

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:

$- \nabla \cdot (ε \nabla V) = ρ .$

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:

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

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

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

Using the identity

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

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

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

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:

$B = μ H + μ_{0} M .$

Here, $μ = μ_{0} μ_{r}$ , 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,

$\begin{array}{l} H = \frac{1}{μ_{0} μ_{r}} B - \frac{1}{μ_{r}} M, \\ \nabla \times H = \nabla \times (\frac{1}{μ_{0} μ_{r}} \times A - \frac{1}{μ_{r}} M) = J . \end{array}$

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

$\nabla \times (\frac{1}{μ_{r} μ_{0}} \nabla \times A) = J + \nabla \times (\frac{1}{μ_{r}} M) .$