Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as:

`$\begin{array}{c}\epsilon \text{\hspace{0.17em}}\nabla \cdot E=\rho ,\\ \nabla \cdot H=0,\\ \nabla ×E=-\mu \frac{\partial H}{\partial t},\\ \nabla ×H=\epsilon \frac{\partial E}{\partial t}+J.\end{array}$`

Here, E and H are the electric and magnetic fields, ε and µ are the electrical permittivity and magnetic permeability of the material, and ρ and J are the electric charge and current densities.

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 this 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:

`$\begin{array}{l}\nabla \cdot H=0,\\ \nabla ×H=J.\end{array}$`

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

`$\begin{array}{l}H={\mu }^{-1}\nabla ×A,\\ \nabla ×\left({\mu }^{-1}\nabla ×A\right)=J.\end{array}$`

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 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.