From d47f24f04afddd09acdb888b52f48db27281b9d3 Mon Sep 17 00:00:00 2001 From: Luc Date: Fri, 19 Jan 2024 19:55:03 +0100 Subject: [PATCH] Added first two sections of optics. --- config/en/mkdocs.yaml | 12 ++- .../electromagnetism/maxwell-equations.md | 1 + .../optics/electromagnetic-waves.md | 96 +++++++++++++++++++ .../physics/electromagnetism/optics/waves.md | 73 ++++++++++++++ docs/en/physics/index.md | 2 +- .../signal-analysis/fourier-series.md | 1 + .../signal-analysis/fourier-transform.md | 0 7 files changed, 183 insertions(+), 2 deletions(-) create mode 100644 docs/en/physics/electromagnetism/maxwell-equations.md create mode 100644 docs/en/physics/electromagnetism/optics/electromagnetic-waves.md create mode 100644 docs/en/physics/electromagnetism/optics/waves.md create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 20faedb..17c7157 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -117,7 +117,17 @@ nav: - physics/index.md - 'Mathematical physics': - 'Signal analysis': - - 'Signals': mathematics/mathematical-physics/signal-analysis/signals.md + - 'Signals': physics/mathematical-physics/signal-analysis/signals.md + - 'Fourier series': + - 'Fourier transform': + - 'Discrete Fourier transform': + - 'Electromagnetism': + - 'Electrostatics': + - 'Magnetostatics': + - 'Maxwell-equations': physics/electromagnetism/maxwell-equations.md + - 'Optics': + - 'Waves': physics/electromagnetism/optics/waves.md + - 'Electromagnetic waves': physics/electromagnetism/optics/electromagnetic-waves.md - 'Chemistry': - chemistry/index.md diff --git a/docs/en/physics/electromagnetism/maxwell-equations.md b/docs/en/physics/electromagnetism/maxwell-equations.md new file mode 100644 index 0000000..33fb93e --- /dev/null +++ b/docs/en/physics/electromagnetism/maxwell-equations.md @@ -0,0 +1 @@ +# Maxwell equations \ No newline at end of file diff --git a/docs/en/physics/electromagnetism/optics/electromagnetic-waves.md b/docs/en/physics/electromagnetism/optics/electromagnetic-waves.md new file mode 100644 index 0000000..ab73f12 --- /dev/null +++ b/docs/en/physics/electromagnetism/optics/electromagnetic-waves.md @@ -0,0 +1,96 @@ +# Electromagnetic waves + +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 + +$$ +\begin{align*} + &\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\\\ + &\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t) +\end{align*} +$$ + +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](waves.md). 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. diff --git a/docs/en/physics/electromagnetism/optics/waves.md b/docs/en/physics/electromagnetism/optics/waves.md new file mode 100644 index 0000000..cdd6577 --- /dev/null +++ b/docs/en/physics/electromagnetism/optics/waves.md @@ -0,0 +1,73 @@ +# Waves + +> *Definition*: a wave is a propagating disturbance transporting energy and momentum. A $1 + 1$ dimensional wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ travelling can be defined by a linear combination of a right and left travelling function $f,g: \mathbb{R} \to \mathbb{R}$ obtaining +> +> $$ +> \Psi(x,t) = f(x - vt) + g(x + vt), +> $$ +> +> for all $(x,t) \in \mathbb{R}^2$ and $v \in \mathbb{R}$ the speed of the wave. Satisfies the $1 + 1$ dimensional wave equation +> +> $$ +> \partial_x^2 \Psi(x,t) = \frac{1}{v^2} \partial_t^2 \Psi(x,t). +> $$ + +The derivation of the wave equation can be obtained in section... + +> *Theorem*: a right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ with $\lambda, T, A, \varphi \in \mathbb{R}$ the wavelength, period, amplitude and phase of the wave is given by +> +> $$ +> \begin{align*} +> \Psi(x,t) &= A \sin \big(k(x-vt) + \varphi\big), \\ +> &= A \sin(kx-\omega t + \varphi), \\ +> &= A \sin \Big(2\pi \Big(\frac{x}{\lambda} - \frac{t}{T} \Big) + \varphi \Big), +> \end{align*} +> $$ +> +> for all $(x,t) \in \mathbb{R}^2$. With $k = \frac{2\pi}{\lambda}$ the wavenumber, $\omega = \frac{2\pi}{T}$ the angular frequency and $v = \frac{\lambda}{T}$ the wave speed. + +A right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ can also be represented in the complex plane given by + +$$ + \Psi(x,t) = \text{Im} \big(A \exp i(kx - \omega t + \varphi )\big), +$$ + +for all $(x,t) \in \mathbb{R}^2$. + +> *Theorem*: let $\Psi: \mathbb{R}^4 \to \mathbb{R}$ be a $3 + 1$ dimensional wave then it satisfies the $3 + 1$ dimensional wave equation given by +> +> $$ +> \nabla^2 \Psi(\mathbf{x},t) = \frac{1}{v^2} \partial_t^2 \Psi(\mathbf{x},t), +> $$ +> +> for all $(\mathbf{x},t) \in \mathbb{R}^4$. + +We may formulate various solutions $\Psi: \mathbb{R}^4 \to \mathbb{R}$ for this wave equation. + +The first solution may be the plane wave that follows cartesian symmetry and can therefore best be described in a cartesian coordinate system $\mathbf{v}(x,y,z)$. The solution is given by + +$$ + \Psi(\mathbf{v}, t) = \text{Im}\big(A \exp i(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t + \varphi) \big), +$$ + +for all $(\mathbf{v}, t) \in \mathbb{R}^4$ with $\mathbf{k} \in \mathbb{R}^3$ the wavevector. + +The second solution may be the cylindrical wave that follows cylindrical symmetry and can therefore best be described in a cylindrical coordinate system $\mathbf{v}(r,\theta,z)$. The solution is given by + +$$ + \Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\sqrt{\|\mathbf{v}\|}} \exp i(k \|\mathbf{v} \| - \omega t + \varphi) \Bigg), +$$ + +for all $(\mathbf{v}, t) \in \mathbb{R}^4$. + +The third solution may be the spherical wave that follows spherical symmetry and can therefore best be described in a spherical coordinate system $\mathbf{v}(r, \theta, \varphi)$. The solution is given by + +$$ + \Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\|\mathbf{v}\|} \exp i(k\|\mathbf{v}\| - \omega t + \varphi) \Bigg) +$$ + +for all $(\mathbf{v}, t) \in \mathbb{R}^4$. + +> *Principle*: the principle of superposition is valid for waves, since the solution space of the wave equation is linear. + +From this principle we obtain the property of constructive and destructive interference of waves. \ No newline at end of file diff --git a/docs/en/physics/index.md b/docs/en/physics/index.md index f6a882e..98d1e61 100644 --- a/docs/en/physics/index.md +++ b/docs/en/physics/index.md @@ -1,6 +1,6 @@ # Physics -Welcome to the physics page. Some special physical environments that will be used in this seection are listed and explained below. +Welcome to the physics page. Some special physical environments that will be used in this section are listed and explained below. * *Principles*: a fundamental rule or concept in physics serving as a basis for reasoning. * *Assumptions*: a less fundamental rule or concept in physics that is taken to be true such that certain phenoma can be simplified. diff --git a/docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md b/docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md new file mode 100644 index 0000000..ff5ddae --- /dev/null +++ b/docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md @@ -0,0 +1 @@ +# Fourier series \ No newline at end of file diff --git a/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md b/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md new file mode 100644 index 0000000..e69de29