102 lines
3.5 KiB
Markdown
102 lines
3.5 KiB
Markdown
# Electromagnetic waves
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This section is a direct follow up on the section [Maxwell equations](../maxwell-equations.md). Where 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, given by
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$$
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\begin{align*}
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&\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\\\
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&\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t)
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\end{align*}
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$$
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for all $(\mathbf{v}, t) \in U$.
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It may be observed that the eletric and magnetic field comply with the $3 + 1$ dimensional wave equation posed in the section [waves](waves.md). Obtaining the speed $v \in \mathbb{R}$ given by
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$$
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v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = c,
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$$
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defined by $c$ the speed of light, or more generally the speed of information in the universe. Outside vacuum we have
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$$
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v = \frac{1}{\sqrt{\varepsilon \mu}} = \frac{c}{n},
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$$
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with $n = \sqrt{K_E K_B}$ the index of refraction.
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> *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
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>
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> $$
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> \begin{align*}
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> \mathbf{E}(\mathbf{v}, t) &= \text{Im}\Big(\mathbf{E}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \\ \\ \mathbf{B}(\mathbf{v}, t) &= \text{Im} \Big(\mathbf{B}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big)
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> \end{align*}
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> $$
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>
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> for all $(\mathbf{v}, t) \in U$ with $\mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3$.
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??? note "*Proof*:"
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Will be added later.
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The above proposition gives an example of a light wave, but note that there are much more solutions that comply to Maxwell's equations.
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> *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.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ in a electromagnetic wave are orthogonal to each other; $\langle \mathbf{E}, \mathbf{B} \rangle = 0$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> E(\mathbf{v}, t) = v B(\mathbf{v}, t)
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> $$
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>
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> for all $(\mathbf{v}, t) \in U$ with $v = \frac{c}{n}$ the wave speed.
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??? note "*Proof*:"
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Will be added later.
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## Energy flow
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> *Law*: the energy flux density $\mathbf{S}: U \to \mathbb{R}^3$ of an electromagnetic wave is given by
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>
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> $$
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> \mathbf{S}(\mathbf{v}, t) = \frac{1}{\mu_0} \mathbf{E}(\mathbf{v}, t) \times \mathbf{B}(\mathbf{v}, t),
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> $$
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>
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> for all $(\mathbf{v}, t) \in U$. $\mathbf{S}$ is also called the Poynting vector.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Definition*: the time average of the magnitude of $\mathbf{S}$ is called the irradiance.
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<br>
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> *Proposition*: the irradiance $I \in \mathbb{R}$ for harmonic linearly polarized plane electromagnetic waves is given by
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>
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> $$
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> I = \frac{\varepsilon_0 c}{2} E_0^2,
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> $$
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>
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> with $E_0$ the magnitude of $\mathbf{E}_0$.
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??? note "*Proof*:"
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Will be added later.
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