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Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as:

εE=ρ,H=0,×E=μHt,×H=εEt+J.

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:

(εE)=ρ,×E=0.

Since the electric field E is the gradient of the electric potential V, E=V, the first equation yields this PDE:

(εV)=ρ.

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

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

H=0,×H=J.

Since H=0, there exists a magnetic vector potential A, such that

H=μ1×A,×(μ1×A)=J.

Using the identity

×(×A)=(A)2A

and the Coulomb gauge ·A=0, simplify the equation for A in terms of J to this PDE:

2A=A=μJ.

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