Electromagnetic Boundary Problems

1    Maxwell’s Equations and Sources

1.1    MAXWELL EQUATIONS IN FREE SPACE

The Maxwell equations describing a time-dependent electromagnetic field E, B existing in the presence of an electric charge density ρ_tot, and an electric current density J_tot, are

$\nabla \cdot \left({ϵ}_0\mathbf{E}\right)={\rho }_{\text{tot}}$

(1.1)

$\nabla \cdot \mathbf{B}=0$

(1.2)

$\nabla \times \frac{\mathbf{B}}{{\mu }_0}={\mathbf{J}}_{\text{tot}}+{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}$

(1.3)

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

(1.4)

where E is the electric field and B is the magnetic induction. The constants μ₀ = 4π × 10⁻⁷ H/m and ϵ₀ = 8.854… × 10⁻¹² F/m are the permeability and permittivity, respectively, of free space. Their presence is associated with the choice of SI (MKS) units for the fields, charges, and currents.

The fields E and B have direct physical meaning: the force on an infinitesimal (point) charged particle of ...