## Harmonic Electromagnetics Equations

Maxwell's equations describe electrodynamics as:

$$\begin{array}{c}\epsilon \text{\hspace{0.05em}}\nabla \cdot E=\rho ,\\ \nabla \cdot H=0,\\ \nabla \times E=-\mu \frac{\partial H}{\partial t},\\ \nabla \times H=J+\epsilon \frac{\partial E}{\partial t}.\end{array}$$

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:

$$E=\widehat{E}{e}^{i\omega t},$$

$$H=\widehat{H}{e}^{i\omega t}.$$

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 $$\sigma E$$ due to the electric field and the current density of the applied current: $$J=\sigma E+{J}_{a}$$. Here, σ is the conductivity of the material. For a time-harmonic problem, the applied current can be defined as:

$${J}_{a}=\widehat{J}{e}^{i\omega t}.$$

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

$$-\nabla \times \left({\mu}^{-1}\nabla \times E\right)=\epsilon \frac{{\partial}^{2}E}{\partial {t}^{2}}+\sigma \frac{\partial E}{\partial t}+\frac{\partial {J}_{a}}{\partial t}.$$

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

$$\nabla \times \left({\mu}^{-1}\nabla \times \widehat{E}\right)+\left(i\sigma \omega -\epsilon {\omega}^{2}\right)\widehat{E}=-i\omega \widehat{J}.$$

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:

$$\nabla \times \left({\epsilon}^{-1}\nabla \times \widehat{H}\right)-\mu {\omega}^{2}\widehat{H}=\nabla \times \left({\epsilon}^{-1}\widehat{J}\right).$$

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.