notes/docs/physics/electromagnetism/optics/electromagnetic-waves.md

Electromagnetic waves

Priorly the Laplacian of the electric field \mathbf{E}: U \to \mathbb{R}^3 and magnetic field \mathbf{B}: U \to \mathbb{R}^3 in vacuum (\varepsilon = \varepsilon_0, \mu = \mu_0) have been determined, and are given by

\begin{align*} &\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\ &\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t) \end{align*}

for all (\mathbf{v}, t) \in U.

It may be observed that the eletric and magnetic field comply with the 3 + 1 dimensional wave equation posed in the section waves. Obtaining the speed v \in \mathbb{R} given by

v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = c,

defined by c the speed of light, or more generally the speed of information in the universe. Outside vacuum we have

v = \frac{1}{\sqrt{\varepsilon \mu}} = \frac{c}{n},

with n = \sqrt{K_E K_B} the index of refraction.

Proposition: let \mathbf{E},\mathbf{B}: U \to \mathbb{R}^3, a solution for the wave equations of the electric and magnetic field may be harmonic linearly polarized plane waves satisfying Maxwell's equations given by

\begin{align*} \mathbf{E}(\mathbf{v}, t) &= \text{Im}\Big(\mathbf{E}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \ \ \mathbf{B}(\mathbf{v}, t) &= \text{Im} \Big(\mathbf{B}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \end{align*}

for all (\mathbf{v}, t) \in U with \mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3.

??? note "Proof:"

Will be added later.

The above proposition gives an example of a light wave, but note that there are much more solutions that comply to Maxwell's equations.

Law: the electric field \mathbf{E} and the magnetic field \mathbf{B} for all solutions of the posed wave equations are orthogonal to the direction of propagation \mathbf{k}. Therefore electromagnetic waves are transverse.

??? note "Proof:"

Will be added later.

Law: the electric field \mathbf{E} and the magnetic field \mathbf{B} in a electromagnetic wave are orthogonal to each other; \langle \mathbf{E}, \mathbf{B} \rangle = 0.

??? note "Proof:"

Will be added later.

Corollary: it follows from the above law that the magnitude of the electric and magnetic fields E, B: U \to \mathbb{R} in a electromagnetic wave are related by

E(\mathbf{v}, t) = v B(\mathbf{v}, t)

for all (\mathbf{v}, t) \in U with v = \frac{c}{n} the wave speed.

??? note "Proof:"

Will be added later.

Energy flow

Law: the energy flux density \mathbf{S}: U \to \mathbb{R}^3 of an electromagnetic wave is given by

\mathbf{S}(\mathbf{v}, t) = \frac{1}{\mu_0} \mathbf{E}(\mathbf{v}, t) \times \mathbf{B}(\mathbf{v}, t),

for all (\mathbf{v}, t) \in U. \mathbf{S} is also called the Poynting vector.

??? note "Proof:"

Will be added later.

Definition: the time average of the magnitude of \mathbf{S} is called the irradiance.

Proposition: the irradiance I \in \mathbb{R} for harmonic linearly polarized plane electromagnetic waves is given by

I = \frac{\varepsilon_0 c}{2} E_0^2,

with E_0 the magnitude of \mathbf{E}_0.

??? note "Proof:"

Will be added later.