Applications involving electrostatics include high voltage apparatuses, electronic
devices, and capacitors. In electrostatics, the time rate of change is slow, and the
wavelengths are very large compared to the size of the domain of interest. The electrostatic
scalar potential *V* is related to the electric field *E*
by *E* = –∇*V*. Using the Maxwell's equation ∇ · *D* = *ρ* and the relationship *D* = *εE*, you can write the Poisson equation

–∇ · (*ε*∇*V*) =
*ρ*,

where *ε* is the dielectric permittivity and *ρ* is the
space charge density.

For electrostatics problems, you can use Dirichlet boundary conditions specifying the
electrostatic potential *V* on the boundary or Neumann boundary conditions
specifying the surface charge **n** · (*ε*∇*V*) on the boundary.

Applications involving magnetostatics include magnets, electric motors, and transformers. In magnetostatics, the time rate of change is slow.

Maxwell's equations for steady cases are $$\nabla \times H=J$$ and $$\nabla \cdot B=0$$. Here, $$B=\mu H$$, where **B** is the magnetic flux density,
**H** is the magnetic field intensity, **J** is the current density, and *µ* is the
material's magnetic permeability.

Since $$\nabla \cdot B=0$$, there exists a magnetic vector potential **A** such that $$B=\nabla \times A\text{and}\nabla \times \left(\frac{1}{\mu}\nabla \times A\right)=J$$.

If the current flows are parallel to the *z*-axis, then $$A=\left(0,0,A\right)\text{and}J=\left(0,0,J\right)$$. Using the common gauge assumption $$\nabla \xb7A=0$$, simplify the equation for **A** in terms of
**J** to the scalar elliptic PDE:

$$-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla A\right)=J$$, where $$J=J\left(x,y\right)$$. For the 2-D case, $$B=\left(\frac{\partial A}{\partial y},-\frac{\partial A}{\partial x},0\right)$$.

For subdomain borders between regions of different material properties, **H** x **n** must be continuous. This
implies the continuity of the derivative $$\frac{1}{\mu}\frac{\partial A}{\partial n}$$. Also, in ferromagnetic materials, *µ* usually depends on
the field strength |*B*| = |∇*A*|. The Dirichlet boundary
condition specifies the value of the magnetostatic potential *A* on the
boundary. The Neumann condition specifies the value of the normal component of $$n\cdot \left(\frac{1}{\mu}\nabla A\right)$$ on the boundary. This is equivalent to specifying the tangential value of
the magnetic field **H** on the boundary.