Maxwell's equations describe electrodynamics as follows:
The electric flux density D is related to the electric field E, , where ε is the electrical permittivity of the material.
The magnetic flux density B is related to the magnetic field 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:
Since the electric field E is the gradient of the electric potential V, , the first equation yields the following PDE:
For electrostatic problems, Dirichlet boundary conditions specify the electric potential V on the boundary.
For magnetostatic problems, Maxwell's equations simplify to this form:
Since , there exists a magnetic vector potential A, such that
Using the identity
and the Coulomb gauge , simplify the equation for A in terms of J to the following PDE:
For magnetostatic problems, Dirichlet boundary conditions specify the magnetic potential on the boundary.