From a0779990cfc410cb760fea27211c8f80baa9f39a Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Thu, 5 Nov 2020 11:27:34 +0100 Subject: [PATCH] added figures and some more text --- lecture.cls | 3 +- theo01.tex | 195 ++++++++++++++++++++++++++++++++++++++++++++++++++++ theo1.tex | 147 --------------------------------------- theoIII.pdf | Bin 175056 -> 252181 bytes theoIII.tex | 2 +- 5 files changed, 198 insertions(+), 149 deletions(-) create mode 100644 theo01.tex delete mode 100644 theo1.tex diff --git a/lecture.cls b/lecture.cls index 202c083..3ab1f19 100644 --- a/lecture.cls +++ b/lecture.cls @@ -76,7 +76,8 @@ \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \renewcommand{\C}{\mathbb{C}} - +\newcommand{\rot}{\operatorname{rot}} +\renewcommand{\div}{\operatorname{div}} % HEADERS %\newpagestyle{main}[\small]{ diff --git a/theo01.tex b/theo01.tex new file mode 100644 index 0000000..9d1956a --- /dev/null +++ b/theo01.tex @@ -0,0 +1,195 @@ +\documentclass{lecture} + +\begin{document} + +\chapter{Phänomenologie der Maxwell-Gleichungen} + +\section{Elektrisches Feld} + +\begin{satz}[Coulomb Gesetz] + Kraft zwischen zwei Ladungen + \[ + F = k \frac{q_1 q_2}{| \vec{r}_1 - \vec{r}_2 |^2} + .\] + Im SI-System ist $k = \frac{1}{4 \pi \epsilon_0}$ mit + $\epsilon_0 = 8.854 \cdot 10^{-12} \frac{\text{As}}{\text{Vm}} = \frac{1}{4 \pi \cdot 9 \cdot 10^{9}} + \frac{\text{As}}{\text{Vm}}$ mit A für Ampère und As $=$ C für Coulomb. + + Im Gauß-System ist $q = \frac{q^{*}}{\sqrt{4 \pi \epsilon_0} }$, so dass $k = 1$. +\end{satz} + +\begin{figure}[h] + \centering +\begin{tikzpicture} + \draw[fill] (-1, 0) circle (0.1cm) node[below left] {$q_1$}; + \draw[fill] (1, 0) circle (0.1cm) node[below left] {$q_2$}; +\end{tikzpicture} +\caption{Zwei Ladungen} +\end{figure} + +\section{Elektrische Feldstärke} + +\begin{satz}[Coulomb Kraft] + \[ + \vec{F} = q \vec{E} + .\] +\end{satz} + +\begin{satz}[Lorentz Kraft] + \[ + \vec{F} = q \vec{v} \times \vec{B} + .\] +\end{satz} + +\section{Maxwell-Gleichungen} + +Die Maxwell Gleichungen bilden die axiomatische Verbindung zwischen Feldern +$\vec{E}, \vec{B}, \vec{D}, \vec{H}$ und den Ladungen $\rho, \vec{\jmath}$ +in Inertialsystemen (oder lokal in frei fallenden Systemen). + +\begin{enumerate}[(1)] + \item $\div \vec{E} = \nabla \cdot \vec{E} = \partial_i E^i = 4\pi \rho$. Mithilfe der 1. Maxwell-Gleichung kann aus dem elektrischen Feld die Ladung bestimmt werden oder aus der Ladungsverteilung auf ein elektrisches Feld geschlossen werden. + \begin{figure}[h] + \centering + \begin{tikzpicture} + \draw (0,0) circle (1.5cm); + \node[fill = black, shape=circle, label=$\rho > 0$] (q) at (0,0) {}; + \foreach \a in {0,30,...,330} {\draw[->] (\a:1.5) -- (\a:2.5);} + \end{tikzpicture} + \caption{Die 1. Maxwell-Gleichung} + \end{figure} + \[ + \int_V \d[3]{r} \div \vec{E} \overset{\text{Gauß}}{=} \int_{\partial V} \d{\vec{S}} \cdot \vec{E} = \underset{\text{el. Fluss}}{\Psi} = \int_V \d[3]{r} 4\pi \rho = 4\pi q + \] + Setzen wir eine sphärische Symmetrie voraus, so erhalten wir + \[ + \int_{\partial V} \d{\vec S} \cdot \vec E = 4\pi r^2 \cdot |\vec E(r)| = 4\pi q \implies |E| \sim \frac{q}{r^2}. + \] + Damit haben wir also das Coulomb-Gesetz in Zusammenhang zu den Maxwell-Gleichungen gesetzt. + \item $\div \vec{B} = \nabla \times \vec{B} = \partial_i B^i = 0$ + \begin{figure}[ht] + \centering + \begin{tikzpicture} + \draw (0,0) circle (1.5cm); + \node[fill=black,label=$\rho_{\mathrm{mag}}\text{?}$, shape=circle] (q) at (0,0) {}; + \draw[->] (-2.5, 0) -- (-1.5, 0); + \draw[->] (210:2.5) -- (210:1.5); + \draw[->] (1.5,0) -- (2.5,0); + \draw[->] (30:1.5) -- (30:2.5); + \end{tikzpicture} + \caption{Die 2. Maxwell-Gleichung} + \end{figure} + \[ + \d[3]{r} \div \vec{B} = \int_{ \partial V} \d{\vec{S}} \cdot \vec{B} = \underset{\text{mag. Fluß}}{\Phi} = 0 + \] + Es gibt keine magnetische Ladung (elektromagnetische Dualität) + \item $\rot \vec{E} = \nabla \times \vec E = -\partial_{ct} \vec B$. Die 3. Maxwell-Gleichung ist auch bekannt als Induktionsgesetz von Faraday und stellt eine Verbindung her zwischen magnetischem Fluss durch eine Fläche und induzierter Spannung auf dem Rand der Fläche. Das $-$ auf der rechten Seite führt zur Lenz-Regel. + \begin{figure}[ht] + \centering + \begin{tikzpicture} + \foreach \x in {0,1,2} {\draw[->] (0,\x) to[bend left] (6,\x+2);} + \node (B) at (7,2) {$\vec B, \dot{ \vec{B}} \neq 0$}; + \draw[dashed] (2,2.5) ellipse (1 and 1.5); + \draw[thick] (2,4) arc (90:320:1 and 1.5); + \draw[->] (1.9,4) -- (3,4.2); + \draw[->] (1.1,3.2) -- (1.3,3.9); + \draw[->] (1.3, 1.4) -- (0.8, 2.3); + \draw[->] (1.9,1) -- (1, 1.2); + \node (E) at (2.8,4.5) {$\vec E$}; + \node (S) at (2,1.3) {$S$}; + \node (dS) at (2,.8) {$\d{S}$}; + \end{tikzpicture} + \caption{Die 3. Maxwell-Gleichung} + \end{figure} + \[ + \int_S \d{S} \cdot \rot \vec E \overset{\text{Stokes}}{=} \int_{\partial S} \d{\vec r} \cdot \vec E = \underset{\text{Spannung}}{U} = -\underbrace{\frac{\d}{\d{(ct)}}}_{\frac{1}{c}\frac{\d}{\d t}} \int \d{\vec S} \cdot \vec B = -\frac{\d}{\d{(ct)}} \Phi + \] + \item $\rot \vec B = \nabla \times \vec B = \partial_{ct} \vec E + \frac{4\pi}{c} \vec \jmath$. Die 4. Maxwell-Gleichung ist auch bekannt als Ampere-Gesetz und stellt eine Verbindung her zwischen elektrischen Fluss oder sich änderndem $\vec E$-Feld und einem magnetischen Feld auf dem Rand der Fläche. + \begin{figure}[ht] + \centering + \begin{tikzpicture} + \foreach \x in {0,1,2} {\draw[->] (0,\x) to[bend left] (6,\x+2);} + \node (E) at (7,2) {$\vec E, \dot{\vec{E}} \neq 0$}; + \node (j) at (7, 3) {$\vec \jmath$}; + \draw[dashed] (2,2.5) ellipse (1 and 1.5); + \draw[thick] (2,4) arc (90:320:1 and 1.5); + \draw[->] (1.9,4) -- (.8,3.8); + \draw[->] (1.1,3.2) -- (.9,2.5); + \draw[->] (1.1, 1.8) -- (1.6, .9); + \draw[->] (1.9,1) -- (2.8, .9); + \node (B) at (1,4.1) {$\vec B$}; + \node (S) at (2,3.8) {$S$}; + \node (dS) at (2,4.2) {$\d{S}$}; + \end{tikzpicture} + \caption{Die 4. Maxwell-Gleichung} + \end{figure} + In integraler Form gilt also + \[ + \int_S \d{\vec S}\cdot \rot \vec B + \overset{\text{Stokes}}{=} + \int_{\partial S} \d{\vec r} \cdot \vec B + = + \frac{\d{}}{\d{(ct)}} \int_S \underbrace{\d{\vec S} \cdot \vec E}_{=\Psi} + \frac{4\pi}{c}\underbrace{\int \d{\vec S} \cdot \vec \jmath}_{= I \text{ Strom}}. + \] + Setzen wir zylindrische Symmetrie voraus, so erhalten wir + \[ + \int_{\partial S} \d{\vec r} \cdot \vec B = 4\pi r \left|\vec B(r)\right| = \frac{4\pi}{c} I \implies \left|\vec B\right| \sim \frac{1}{r}. + \] +\end{enumerate} + +\section{Eigenschaften der Maxwell-Gleichungen} + +\begin{itemize} + \item Die Maxwell-Gleichungen sind lineare ($\to$ Superposition), + \item partielle (Ableitungen in $x^i$ und $ct$ $\to \nabla, \partial_{ct}$), + \item hyperbolische Differenzialgleichungen (wird später noch erklärt). + \begin{tikzpicture} + \node (q) at (0,0) {$\rho, \vec \jmath$}; + \node (E) at (3,0) {$\vec E, \vec B$}; + \draw[->] (q) to[out = 30, in=150] node[above] {Quelle} (E); + \draw[->] (E) to[out = -150, in=-30] node[below] {Lokalisierung} (q); + \end{tikzpicture} + \item Die Maxwell-Gleichungen ergeben nur Sinn in einem Bezugssystem, wir legen also ein Inertialsystem fest. + \item Wir haben 2 $\div$-Gleichungen und 2 $\rot$-Gleichungen, also effektiv $1 + 1 + 3 + 3 = 8$ Gleichungen, wir untersuchen aber nur die Dynamik von $\vec E$- und $\vec B$-Feld ($3 + 3 = 6$ Komponenten). Sind die Maxwell-Gleichungen daher überbestimmt? +\end{itemize} + +\section{Erhaltung der elektrischen Ladung} + +Betrachte Maxwell (1) und (4): + +\begin{salign*} + \text{div } \vec{E} &= 4 \pi \rho \mid \partial_{ct}\\ + \text{rot } \vec{B} &= \partial_{ct} \vec{E} + \frac{4 \pi}{c} \vec{\jmath} \mid \text{div } + \intertext{Damit folgt} + \text{div } \text{rot } \vec{B} = 0 &= \partial_{ct} \text{div } \vec{E} + \frac{4 \pi}{c} \text{div } + \vec{\jmath} \\ + &= 4 \pi \partial_{ct} \rho + \frac{4\pi}{c} \text{div } \vec{\jmath} + \intertext{Also insgesamt die sog. Kontinuitätsgleichung} + \partial_t \rho + \text{div } \vec{\jmath} &= 0 \tageq\label{eq:cont} +.\end{salign*} +In Integralform ergibt sich daraus: +\begin{salign*} + \frac{\d}{\d t} \int_{V}^{} \d[3]{r} \rho = \frac{\d }{\d t} q = - \int_{V}^{} \d[3]{r} \text{div } + \vec{\jmath} &\stackrel{\text{Gauß}}{=} - \int_{\partial V}^{} \d{\vec{S}} \cdot \vec{\jmath} +.\end{salign*} +Die Elektrodynamik ist eine Kontinuumstheorie. Ladung ist dabei eine Art ,,Fluid``. + +\section{Elektromagnetische Dualität} + +Betrachte die Maxwell-Gleichungen im Vakuum, d.h. $\rho = 0$, $\vec{\jmath} = 0$. Dann gilt +\begin{enumerate}[(1)] + \item $\text{div } \vec{E} = 0$ + \item $\text{div } \vec{B} = 0$ + \item $\text{rot } \vec{E} = - \partial_{ct} \vec{B}$ + \item $\text{rot } \vec{B} = \partial_{ct} \vec{E}$ +\end{enumerate} +Vertauschen wir $\vec E \to \vec B,\quad \vec B \to -\vec E$, so ändert sich an den Maxwell-Gleichungen im Vakuum nichts (Dualität). +Warum existieren eigentlich keine magnetischen Ladungen? +\begin{enumerate}[(1)] + \item $\text{div } \vec{E} = 4 \pi \rho$ + \item $\text{div } \vec{B} = 4 \pi \rho_{m}$ + \item $\text{rot } \vec{E} = - \partial_{ct} \vec{B} + \frac{4 \pi}{c} \vec{\jmath}_m$ + \item $\text{rot } \vec{B} = \partial_{ct} \vec{E} + \frac{4 \pi}{c} \vec{\jmath}$ +\end{enumerate} +Es folgt also, analog zu \ref{eq:cont}, eine Kontinuitätsgleichung für die magnetische Ladung \[\partial_t \rho_m + \text{div } \vec{\jmath} = 0.\] +\end{document} diff --git a/theo1.tex b/theo1.tex deleted file mode 100644 index 3228f34..0000000 --- a/theo1.tex +++ /dev/null @@ -1,147 +0,0 @@ -\documentclass{lecture} - -\begin{document} - -\chapter{Phänomenologie der Maxwell-Gleichungen} - -\section{Elektrisches Feld} - -\begin{satz}[Colomb Gesetz] - Kraft zwischen zwei Ladungen - \[ - F = k \frac{q_1 q_2}{| \vec{r}_1 - \vec{r}_2 |^2} - .\] - Im SI-System ist $k = \frac{1}{4 \pi \epsilon_0}$ mit - $\epsilon_0 = 8.854 \cdot 10^{-12} \frac{\text{As}}{\text{Vm}} = \frac{1}{4 \pi \cdot 9 \cdot 10^{9}} - \frac{\text{As}}{\text{Vm}}$ mit A für Ampère und As $=$ C für Coulomb. - - Im Gauß-System ist $q = \frac{q^{*}}{\sqrt{4 \pi \epsilon_0} }$, so dass $k = 1$. -\end{satz} - -\begin{figure}[h] - \centering -\begin{tikzpicture} - \draw[fill] (-1, 0) circle (0.1cm) node[below left] {$q_1$}; - \draw[fill] (1, 0) circle (0.1cm) node[below left] {$q_2$}; -\end{tikzpicture} -\caption{Zwei Ladungen} -\end{figure} - -\section{Elektrische Feldstärke} - -\begin{satz}[Coulomb Kraft] - \[ - \vec{F} = q \vec{E} - .\] -\end{satz} - -\begin{satz}[Lorentz Kraft] - \[ - \vec{F} = q \vec{v} \times \vec{B} - .\] -\end{satz} - -\section{Maxwell-Gleichungen} - -Die Maxwell Gleichungen bilden die axiomatische Verbindung zwischen Feldern -$\vec{E}, \vec{B}, \vec{D}, \vec{H}$ und den Ladungen $\rho, \vec{\jmath}$ -in Inertialsystemen (oder lokal in frei fallenden Systemen). - -\begin{enumerate}[(1)] - \item - \begin{salign*} - \text{div } \vec{E} &= \nabla \cdot \vec{E} = \partial_i E^{i} = 4 \pi \rho - \intertext{Integralform} - \int_{V}^{} \text{div } \vec{E} \d[3]{x} &\stackrel{\text{Gauß}}{=} - \int_{\partial V}^{} \d{\vec{S}} \cdot \vec{E} = \underbrace{\Psi}_{\text{elektrischer Fluss}} - = \int_{V}^{} 4 \pi\rho \d[3]{r} = \underbrace{4 \pi q}_{\text{Ladung}} - \intertext{Im Fall von sphärischer Symmetrie gilt} - \int_{\partial V}^{} \d{\vec{S}} \cdot \vec{E} &= 4 \pi r^2 | \vec{E}(r) | = 4 \pi q - \implies |E| \sim \frac{q}{r^2} - .\end{salign*} - \item - \begin{salign*} - \text{div } \vec{B} &= \nabla \times \vec{B} = \partial_i B^{i} = 0 - \intertext{Integralform} - \int_{V}^{} \d[3]{r} \text{div } \vec{B} &= \int_{\partial V}^{} \d{\vec{S}} \cdot \vec{B} - = \underbrace{\Phi}_{\text{magnetischer Fluss}} = 0 - .\end{salign*} - Es gibt keine magnetische Ladung, es gilt also immer $\rho_m = 0$. Elektromagnetische Dualität. - \item Faraday-Induktionsgesetz - \begin{salign*} - \text{rot } \vec{E} &= \nabla \times \vec{E} = \underbrace{-}_{\text{Lenz-Regel}} \partial_{ct} \vec{B} - \intertext{Integralform} - \int_{S}^{} \d{\vec{S}} \cdot \text{rot } \vec{E} - &\stackrel{\text{Stokes}}{=} \int_{\partial S}^{} \d{\vec{r}} \cdot \vec{E} - = \underbrace{U}_{\text{Spannung}} - = - \frac{\d}{\d (ct)} \int_{}^{} \d{\vec{S}} \cdot \vec{B} = - \frac{\d}{\d (ct)} \Phi - .\end{salign*} - \item Ampère-Gesetz - \begin{salign*} - \text{rot } \vec{B} &= \nabla \times \vec{B} = \partial_{ct} \vec{E} + \frac{4 \pi}{c} \vec{\jmath} - \intertext{Integralform} - \int_{S}^{} \d{\vec{S}} \cdot \vec{B} &\stackrel{\text{Stokes}}{=} - \int_{\partial S}^{} \d{\vec{r}} \cdot \vec{B} - = \frac{\d }{\d ct} \underbrace{\int_{S}^{} \d{\vec{S}} \cdot \vec{E} }_{\Psi} - + \frac{4\pi}{c} \underbrace{\int_{}^{} \d{\vec{S}} \cdot \vec{\jmath}}_{I \text{ Strom}} - \intertext{Im Fall von zylindrischer Symmetrie gilt:} - \int_{\partial S}^{} \d{\vec{r}} \cdot \vec{B} &= - 4 \pi r | \vec{B}(r)| = \frac{4 \pi}{c} I \implies | \vec{B} | \sim \frac{1}{r} - .\end{salign*} -\end{enumerate} - -\section{Eigenschaften der Maxwell-Gleichungen} - -\begin{itemize} - \item linear ($\to$ Superposition) - \item partiell - \item hyperbolisch - \item Inertialsystem - \item Überbestimmtheit? -\end{itemize} - -\section{Erhaltung aus elektrischer Ladung} - -Betrachte Maxwell (1) und (4): - -\begin{salign*} - \text{div } \vec{E} &= 4 \pi \rho \mid \partial_{ct}\\ - \text{rot } \vec{B} &= \partial_{ct} \vec{E} + \frac{4 \pi}{c} \vec{\jmath} \mid \text{div } - \intertext{Damit folgt} - \text{div } \text{rot } \vec{B} = 0 &= \partial_{ct} \text{div } \vec{E} + \frac{4 \pi}{c} \text{div } - \vec{\jmath} \\ - &= 4 \pi \partial_{ct} \rho + \frac{4\pi}{c} \text{div } \vec{\jmath} - \intertext{Also insgesamt die sog. Kontinuitätsgleichung} - \partial_t \rho + \text{div } \vec{\jmath} &= 0 \tageq\label{eq:cont} -.\end{salign*} -In Integralform ergibt sich daraus: -\begin{salign*} - \frac{\d}{\d t} \int_{V}^{} \d[3]{r} \rho = \frac{\d }{\d t} q = - \int_{V}^{} \d[3]{r} \text{div } - \vec{\jmath} &\stackrel{\text{Gauß}}{=} - \int_{\partial V}^{} \d{\vec{S}} \cdot \vec{\jmath} -.\end{salign*} -Die Elektrodynamik ist eine Kontinuumstheorie. Ladung ist dabei eine Art ,,Fluid``. - -\section{Elektromagnetische Dualität} - -Betrachte die Maxwell-Gleichungen im Vakuum, d.h. $\rho = 0$, $\vec{\jmath} = 0$. Dann gilt -\begin{enumerate}[(1)] - \item $\text{div } \vec{E} = 0$ - \item $\text{div } \vec{B} = 0$ - \item $\text{rot } \vec{E} = - \partial_{ct} \vec{B}$ - \item $\text{rot } \vec{B} = \partial_{ct} \vec{E}$ -\end{enumerate} - -Vertauschung $\vec{E} \longrightarrow \vec{B}$, $\vec{B} \longrightarrow - \vec{E}$: Dualität - -Warum existieren eigentlich keine magnetischen Ladungen? - -\begin{enumerate}[(1)] - \item $\text{div } \vec{E} = 4 \pi \rho$ - \item $\text{div } \vec{B} = 4 \pi \rho_{m}$ - \item $\text{rot } \vec{E} = - \partial_{ct} \vec{B} + \frac{4 \pi}{c} \vec{\jmath}_m$ - \item $\text{rot } \vec{B} = \partial_{ct} \vec{E} + \frac{4 \pi}{c} \vec{\jmath}$ -\end{enumerate} - -Es folgt also direkt $\partial_t \rho_m + \text{div } \vec{\jmath} = 0$ analog zu \ref{eq:cont}. - -\end{document} diff --git a/theoIII.pdf b/theoIII.pdf index 8465bf85e86399630d827d0d39c10c1070af0b9a..0e7d970cc78ee5ae62a721a0703d6efac0e65d65 100644 GIT binary patch delta 228107 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