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Maxwell's equations describe electrodynamics as follows:

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

The electric flux density **D** is related to the electric
field **E**, $$D=\epsilon E$$, where *ε* is the electrical permittivity of the
material.

The magnetic flux density **B** is related to the magnetic
field **H**, $$B=\mu H$$, where *µ* is the magnetic permeability of the
material.

Also, here **J** is the electric current density, and
*ρ* is the electric charge density.

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

$$\begin{array}{l}\nabla \cdot \left(\epsilon \text{\hspace{0.05em}}E\right)=\rho \\ \text{\hspace{0.17em}}\nabla \times E=0\end{array}$$

Since the electric field **E** is the gradient of the
electric potential *V*, $$E=-\nabla V$$, the first equation yields the following PDE:

$$-\nabla \cdot \left(\epsilon \text{\hspace{0.05em}}\nabla V\right)=\rho $$

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

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

$$\begin{array}{l}\nabla \cdot B=0\\ \nabla \times H=J\end{array}$$

Since $$\nabla \cdot B=0$$, there exists a magnetic vector potential **A**, such that

$$\begin{array}{l}B=\nabla \times A\\ \nabla \times \left(\frac{1}{\mu}\nabla \times A\right)=J\end{array}$$

Using the identity

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

and the Coulomb gauge $$\nabla \xb7A=0$$, simplify the equation for **A** in terms of
**J** to the following PDE:

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

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