## Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as follows:

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

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:

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

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

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

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

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

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

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

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

Using the identity

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

and the Coulomb gauge $\nabla ·A=0$, simplify the equation for A in terms of J to 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.

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