Harmonic Electromagnetics Equations

Maxwell's equations describe electrodynamics as:

$\begin{matrix} ε \nabla \cdot E = ρ, \\ \nabla \cdot H = 0, \\ \nabla \times E = - μ \frac{\partial H}{\partial t}, \\ \nabla \times H = J + ε \frac{\partial E}{\partial t} . \end{matrix}$

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.

The time-harmonic electric and magnetic fields can be represented using these formulas:

$E = \hat{E} e^{i ω t},$

$H = \hat{H} e^{i ω t} .$

Accounting for the electric conductivity of the material and the applied current separately, you can represent the total electric current density as the sum of the current density $σ E$ due to the electric field and the current density of the applied current: $J = σ E + J_{a}$ . Here, σ is the conductivity of the material. For a time-harmonic problem, the applied current can be defined as:

$J_{a} = \hat{J} e^{i ω t} .$

Maxwell’s equations for the electric field yield this equation:

$- \nabla \times (μ^{- 1} \nabla \times E) = ε \frac{\partial^{2} E}{\partial t^{2}} + σ \frac{\partial E}{\partial t} + \frac{\partial J_{a}}{\partial t} .$

For the time-harmonic electric field and applied current, the derivative $\frac{\partial}{\partial t} = i ω$ , and the resulting equation is:

$\nabla \times (μ^{- 1} \nabla \times \hat{E}) + (i σ ω - ε ω^{2}) \hat{E} = - i ω \hat{J} .$

Given an incident electric field E_i and a scattered electric field E_s, you can compute the total electric field E. Due to linearity, it suffices to solve the equation for the scattered field with the boundary condition for the scattered field along the scattering object determined by

$n \times E_{i} = - n \times E_{s} .$

For the time-harmonic magnetic field and applied current, Maxwell's equations can be simplified under the assumption of zero conductivity to this form:

$\nabla \times (ε^{- 1} \nabla \times \hat{H}) - μ ω^{2} \hat{H} = \nabla \times (ε^{- 1} \hat{J}) .$

For the time-harmonic magnetic field, it suffices to solve the equation for the scattered field with the boundary condition for the scattered field along the scattering object determined by

$n \times H_{i} = - n \times H_{s} .$

Here, H_i is an incident magnetic field, and H_s is a scattered magnetic field.