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Consider a homogeneous dielectric with the coefficient of dielectricity ε, the magnetic
permeability *µ*, and no charge at any point. The fields must satisfy a
special set of general Maxwell's equations:

$$\begin{array}{c}\nabla \times E=-\mu \frac{\partial H}{\partial t}\\ \nabla \times H=\epsilon \frac{\partial E}{\partial t}+J\end{array}$$

Here, **E** is the electric field, **H** is the magnetic field, and **J** is the
current density. In the absence of a current, you can eliminate **H** from the first set and **E** from the second
set and see that both fields satisfy wave equations with the wave speed $$\sqrt{\epsilon \mu}$$:

$$\begin{array}{c}\Delta E-\epsilon \mu \frac{{\partial}^{2}E}{\partial {t}^{2}}=0\\ \Delta H-\epsilon \mu \frac{{\partial}^{2}H}{\partial {t}^{2}}=0\end{array}$$

Consider the case of a charge-free homogeneous dielectric with the coefficient of
dielectricity *ε*, magnetic permeability *µ*, and
conductivity *σ*. The current density is

$$J=\sigma E$$

and the waves are damped by the Ohmic resistance,

$$\Delta E-\mu \sigma \frac{\partial E}{\partial t}-\epsilon \mu \frac{{\partial}^{2}E}{\partial {t}^{2}}=0$$

Equations for **H** are similar.

For time-harmonic fields, use the complex form of the equations, replacing **E** with

$${E}_{c}{e}^{j\omega t}$$

For a plane, the electric field is $${E}_{c}=\left(0,0,{E}_{c}\right),\text{}J=\left(0,0,J{e}^{j\omega t}\right)$$, and the magnetic field is

$$H=\left({H}_{x},{H}_{y},0\right)=\frac{-1}{j\mu \sigma}\nabla \times {E}_{c}$$

The scalar equation for *E _{c}* becomes

$$-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla {E}_{c}\right)+\left(j\omega \sigma -{\omega}^{2}\epsilon \right){E}_{c}=0$$

The PDE Modeler app uses this equation when it is in the **AC Power
Electromagnetics** application mode . The equation is a complex Helmholtz
equation that describes the propagation of plane electromagnetic waves in imperfect
dielectrics and good conductors (*σ* » *ωε*). A complex
permittivity *ε _{c}* can be defined as

The boundary conditions associated with this mode are:

The Dirichlet boundary condition specifying the value of the electric field

*E*on the boundary_{c}The Neumann boundary condition specifying the normal derivative of

*E*, which is equivalent to specifying the tangential component of the magnetic field_{c}**H**:$${H}_{t}=\frac{j}{\omega}n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla {E}_{c}\right)$$

The solution is the electric field **E**. Using the solution,
you can compute the current density **J** =
*σ***E** and the magnetic flux
density

$$B=\frac{j}{\omega}\nabla \times E$$

Using the PDE Modeler app, you can plot the electric field **E**, the current density **J**, the magnetic
field **H**, and the magnetic flux density **B**. You also can plot the resistive heating rate

$$Q={E}_{c}^{2}/\sigma $$

You can plot the magnetic field and the magnetic flux density as vector fields by using arrows.

[1] Popovic, B. D.,
*Introductory Engineering Electromagnetics*. Reading, MA:
Addison-Wesley, 1971.