Electrostatics and Magnetostatics
Maxwell's equations describe electrodynamics as:
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:
Since the electric field E is the gradient of the electric potential V, , the first equation yields this 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 this PDE:
For magnetostatic problems, Dirichlet boundary conditions specify the magnetic potential A on the boundary.