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Electrostatics and Magnetostatics

Maxwell's equations describe electrodynamics as follows:

D=ρB=0×E=Bt×H=Dt+J

The electric flux density D is related to the electric field E, D=εE, where ε is the electrical permittivity of the material.

The magnetic flux density B is related to the magnetic field H, B=μH, 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:

(εE)=ρ×E=0

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

B=0×H=J

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

B=×A×(1μ×A)=J

Using the identity

×(×A)=(A)2A

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

2A=A=μJ

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