From 338b077345d9ac20cf190878ceee5016d6383c75 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 4 Nov 2020 14:57:50 +0100 Subject: [PATCH] add first theo lecture without graphics --- theo1.tex | 147 ++++++++++++++++++++++++++++++++++++++++++++++++++++ theoIII.pdf | Bin 27471 -> 175056 bytes theoIII.tex | 2 + 3 files changed, 149 insertions(+) create mode 100644 theo1.tex diff --git a/theo1.tex b/theo1.tex new file mode 100644 index 0000000..3228f34 --- /dev/null +++ b/theo1.tex @@ -0,0 +1,147 @@ +\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 93bae6e8bccfa8ae52edb1685fbfc1c64cb4f01b..8465bf85e86399630d827d0d39c10c1070af0b9a 100644 GIT binary patch delta 140369 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z75RVD(Hy>0q_H4Bahvr~{T_!b6dJQ5rI+5$O>41LmUDTdchFCm2UMklg;UB!jmOHh z;@)T$EoZ$=8tbntjpKBsH@Dj|e*$vp+{B99oUv(7jmoyUz-G)#-AOvsDty57>75)9 z^YOg*?q+At&U5C+hsuQ=qf@h&YK%=X2Wt%L;=kaHeok3Fwy-#f3JlvF6&)Gq7b-&G e{(DhT*l54V=!21gK`0uXX-Y$BYuj$IL;VlCB1?$? diff --git a/theoIII.tex b/theoIII.tex index f60f5c2..88f2818 100644 --- a/theoIII.tex +++ b/theoIII.tex @@ -22,4 +22,6 @@ Rui Yang (\href{mailto:rui.yang@stud.uni-heidelberg.de}{rui.yang@stud.uni-heidel \tableofcontents +\input{theo1.tex} + \end{document}