## Saturday, March 19, 2016

### Plane waves: Basic Knowledge

I want to start with a brief look at the basic knowledge to understand the others future articles about this topic. This is mostly a reference for people that are already familiar with electromagnetism (I'll basically just write down some equations).
Like I said I will make only a superficial description (especially for the Maxwell equations), but If you want some clarification or you just want to correct me on something, write it in the comment below.

Maxwell's equations

Maxwell's equations are a set of partial differential equations that relate the fields with the sources.

$\color{white} \nabla \cdot \underline{d}(\underline{r},t) =\rho(\underline{r},t)$
$\nabla \cdot \underline{b}(\underline{r},t) =0$
$\nabla \times \underline{h}(\underline{r},t)=\frac{\partial \underline{d}(\underline{r},t)}{\partial t} + \underline{j}(\underline{r},t)$
$\nabla \times \underline{e}(\underline{r},t)=-\frac{\partial \underline{b}(\underline{r},t)}{\partial t}$

Electric and magnetic phenomena are connected together with another equation: lorentz equation.
$$\underline{f}=q(\underline{e}+\underline{v}\times \underline{b})$$
Note: the magnetic field doesn't have effect on stationary charges, or rather a charge that is initially at rest in a magnetic filed it stay at rest.

We can also write the equations in the domain of frequency, the vantage is that an operation of integration in the domain of frequency is simply a multiplication for ${j\omega}^{-1}$ and a derivative is a multiplication for $j\omega$.

$\nabla \cdot \underline{D}(\underline{r},\omega) =\rho(\underline{r},\omega)$
$\nabla \cdot \underline{B}(\underline{r},\omega) =0$
$\nabla \times \underline{H}(\underline{r},\omega)=j\omega \underline{D}(\underline{r},\omega) + \underline{J}(\underline{r},\omega)$
$\nabla \times \underline{E}(\underline{r},t)=-j\omega\underline{B}(\underline{r},\omega)$

Note that the maxwell equation don't have any information about the propriety of the material , but we know that a material under the effect of an electromagnetic is polarized, it is magnetized and, if it is a conductor, is traversed by the conduction current. Those effects are represented in the constitutive relations:
$$\underline{d}=\underline{\underline{\varepsilon}}\underline{e}, \underline{b}=\underline{\underline{\mu}}\underline{h}, \underline{j}=\sigma\underline{e}$$
The shape of the constitutive relation is not unique since it's deduced in each case by the microscopic mechanisms that occur in the material.

Conduction
Some materials have a large number of electron free to move, without a field the motion of electrons is random, but under the effect of an electromagnetic filed a new velocity is added to the random one and cause a shift of the electron in the opposite direction of the electromagnetic field. Thanks to some mechanism of friction, created by the clash of the electron, they don't accelerate but migrate with average drift velocity proportional to the intensity of the field. This effect is called conduction.

Polarization
In the dielectric material the electrons aren't free to move and under the effect of an electric field, the electrons that move around the nucleus undergo a shift such that the center of gravity of the electronic cloud changes compared to the core. The set of the electronic cloud of the core create an electric dipole and so the atom is polarized.
This is the electric polarization but in reality there are more kinds of polarization.

Magnetization
The motion of revolution of the electrons around the nucleus and the motion of
spin of individual electrons create elementary coils of  electric current.
Summing the effects produced by the individual coils is obtained a field
said magnetization

Poynting's theorem

In the future articles I'll add other details and explain better the poynting's theorem, for now I'll just write a superficial description. Let's first define the pointing vector :

$\underline{s}=\underline{e}\times\underline{h}$
Now let's make the divergence

$\nabla \cdot \underline{s}=\nabla\cdot(\underline{e}\times\underline{h})=\underline{h}\cdot \nabla\times \underline{e} -\underline{e}\cdot \nabla \times \underline{h}$

Combining the equations and integrating in a volume we obtain:
$$\int_{\partial V} \underline{s} \cdot \underline{i}_n dS+\iiint_V\bigg(\underline{h}\cdot \frac{\partial \underline{b}}{\partial t}+\underline{e}\cdot\frac{\partial \underline{d}}{\partial t}\bigg)+\iiint_V \sigma\cdot e^2 dV=-\iiint_V \underline{e} \cdot \underline{j_0} dV$$
This represent an energetic balance, and establish that the instant power generated by the sources in V is equal to:
1. the power dissipated by the Joule effect
2. the variation of electromagnetic energy
3. the power that flows through the surface S
The poynting theorem basically represents a formalization of the transport of the electromagnetic energy.