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 \(\mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3\).

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.

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; \(\langle \mathbf{E}, \mathbf{B} \rangle = 0\).

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.

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.

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 \(\mathbf{E}_0\).

Proof:

Will be added later.