From 19ccfe21b08fcdb4b1cd46e248f50472883f919c Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Thu, 5 Nov 2020 21:15:31 +0100 Subject: [PATCH] second lecture added --- theo02.tex | 195 ++++++++++++++++++++++++++++++++++++++++++++++++++++ theoIII.pdf | Bin 252177 -> 297139 bytes theoIII.tex | 1 + 3 files changed, 196 insertions(+) create mode 100644 theo02.tex diff --git a/theo02.tex b/theo02.tex new file mode 100644 index 0000000..f79c105 --- /dev/null +++ b/theo02.tex @@ -0,0 +1,195 @@ +\documentclass{lecture} + +\begin{document} + +\section{Elektrodynamik in Materie} +Elektrodynamik in Materie ist im Allgemeinen sehr kompliziert. +Im einfachsten Fall ist die Wirkung +\begin{enumerate} + \item linear + \item isotrop + \item instantan +\end{enumerate} +Dann existiert eine effektive Beschreibung mithilfe von nur zwei Konstanten. Als erstes benötigen wir die Dieelektrizitätskonstante $e$ und die Permeabilitätskonstante $\mu$. Es gilt +\begin{align*} + \vec D &= e\cdot \vec E\\ + \vec B &= \mu \cdot H +\end{align*} +\begin{minipage}[t]{0.49\textwidth} + \centering + im Vakuum (in diesem Fall $e = \mu = 1$) + \begin{align*} + \div \vec E &= 4\pi \rho\\ + \div \vec B &= 0\\ + \rot \vec E &= - \partial_{ct} \vec B\\ + \rot \vec B &= + \partial_{ct} \frac{4\pi}{c} \vec \jmath + \end{align*} +\end{minipage} +\begin{minipage}[t]{0.5\textwidth} + \centering + in Materie + \begin{align*} + \div \vec D &= 4\pi \rho\\ + \div \vec B &= 0\\ + \rot \vec E &= -\partial_{ct} \vec B\\ + \rot \vec H &= +\partial_{ct} \vec D + \frac{4\pi}{c}\vec \jmath + \end{align*} +\end{minipage} +\section{elektrostatisches Potenzial} +Dabei bedeutet elektrostatisch, dass alle Zeitableitungen 0 sind. +\begin{figure}[h] + \centering + \begin{tikzpicture} + \node[fill = black, shape = circle, inner sep = 3pt, label = $q_1$] (q1) at (0,2) {}; + \draw[->] (0,0) -- node[pos=.5, left] {$\vec r_1$} (0,2); + \draw[->] (0,0) -- node[pos=.5, below right] {$\vec r$} (2,1); + \draw[dashed, ->] (0,2) -- node[pos=.5, above right]{$\vec r - \vec r_1$} (2,1); + \node[text width = 2cm] (beobachter) at (3.5,1) {Beobachter mit positiver Probeladung}; + \end{tikzpicture} + \caption{Elektrisches Feld} + \label{efeld} +\end{figure} +Für das elektrische Feld aus Abbildung~\ref{efeld} gilt daher +\[ + \vec E(\vec r) = \frac{q_1}{|\vec r - \vec r_1|} \cdot \vec e_{\vec r - \vec r_1} = q_1 \cdot \frac{\vec r - \vec r_1}{\left|\vec r - \vec r_1\right|^3}. +\] +Bei vielen felderzeugenden Ladungen berechnen wir einfach die Superposition +\[ + \vec E (\vec r) = \sum_{i}^{n} q_i \frac{\vec r - \vec r_i}{\left|\vec r - \vec r_1\right|^3}. +\] +Da es sich bei der Elektrodynamik um eine Kontinuumstheorie handelt, gehen wir von $q$ zu $\rho$ über. Im Kontinuumslimes erhalten wir also +\[ + \vec E(\vec r) = \int \d[3]{r'} \rho(\vec r') \cdot \frac{\vec r - \vec r'}{\left|\vec r - \vec r'\right|^3}. +\] +Im folgenden wirke $\nabla$ auf $r$ und $\nabla'$ auf $r'$. Es gilt +\[ + \frac{\vec r - \vec r'}{\left|\vec r - \vec r'\right|^3} = -\nabla \frac{1}{\left|\vec r - \vec r'\right|} = \nabla' \frac{1}{\left|\vec r - \vec r'\right|}. +\] +Eingesetzt in die Gleichung von oben erhalten wir +\begin{align*} + \vec E(\vec r) &= \int \d[3]{r'} \rho(r') \cdot \nabla'\frac{1}{\left|\vec r - \vec r'\right|}\\ + &= - \int \d[3]{r'} \rho(\vec r') \nabla \frac{1}{\left|\vec r - \vec r'\right|}\\ + &= -\nabla \underbrace{\int \d[3]{r'} \frac{\rho(\vec r')}{\left|\vec r - \vec r'\right|}}_{\eqqcolon \phi(\vec r)}\\ + \to \vec E(\vec r) &= -\nabla \phi(\vec r) +\end{align*} +\begin{definition} + Das elektrostatische Potenzial $\phi$ ist gegeben durch + \[\phi(\vec r) = \int \d[3]{r'} \rho(\vec r') \cdot \frac{1}{\left|\vec r - \vec r'\right|}. + \] +\end{definition} +Für die Elektrostatik gilt also $\rot \vec E = - \rot \vec \phi = 0$. Aus den Maxwell-Gleichungen erhalten wir konsistenterweise $\rot \vec E = -\partial_{ct} \vec B = 0$. +\begin{align*} + W_{AB} &= q \int_A^B \d{\vec S} \cdot \vec E\;\sim \text{ Verschiebearbeit}\\ + &= q\cdot \left| \phi(\vec r_B) - \phi(\vec r_A)\right|\; \sim \text{Interpretation von $\phi$ als potentielle Energie} +\end{align*} +Aus +\begin{align*} + \vec E(\vec r) &= - \nabla \phi(\vec r)\\ + \intertext{und dem Gauß-Gesetz} + \div \vec E &= 4 \pi \rho + \intertext{erhalten wir} + - \div \nabla \phi &= -\Delta \phi = 4\pi \rho +\end{align*} +\begin{definition}[Laplace-Operator] + \[ + \Delta = \div \nabla = \delta^{ij}\partial_i\partial_j + \] +\end{definition} +\begin{satz}[Poisson-Gleichung] + Es gilt + \[ + \Delta \phi = - 4\pi \rho + \] beziehungsweise im Vakuum + \[ + \Delta \phi = 0 + \] +\end{satz} +%figure äquipotentiallinien, kästchen mit fluss rein oder raus +Wir berechnen nun $\Delta \phi$ an einem Ort $\neq 0$ im Feld einer Punktladung. +\begin{align*} + \phi(\vec r) &= \frac{1}{|\vec r - \vec r'|} &&\text{Potenzial einer Punktladung }\vec r'\\ + \Delta \phi &= \Delta \frac{1}{r}\\ + &= \frac{1}{r}\frac{\partial^2}{\partial r^2}\underbrace{\left(r \cdot \frac{1}{r}\right)}_{= \text{const}}\\ + &= 0 +\end{align*} +Nun möchten wir $\Delta \phi$ bei der Punktladung berechnen. Wir erhalten +\begin{salign*} + \int_V \d[3]{r} \Delta \phi &\stackrel{\Delta = \div \nabla}{=} \int_{V} \d[3]{r}\div \nabla \phi\\ + &\stackrel{\text{Gauss}}{=} \int_{\partial V} \d{\vec S} \cdot \nabla \phi\\ + &\stackrel{\text{sph. Symmetrie}}{=} \int_{\partial V} r^2\d{\Omega} \cdot \underbrace{\frac{\partial \phi}{\partial r}}_{-\frac{1}{r^2}}\\ + &= -\int_{\partial V}\d{\Omega}\\ + &= -4\pi +\end{salign*} + +Die Gleichung +\[ + \Delta \frac{1}{\left|\vec r - \vec r'\right|} = -4\pi \delta_D \left|\vec r - \vec r'\right| +\] +kombiniert beide Fälle. Dabei ist $\delta_D$ die Dirac-Funktion. +\section{Dirac $\delta_D$-Funktion} +Möchte man von der Ladungsdichte $\rho(\vec r)$ auf die Ladung im Volumen $V$ schließen, berechnet man einfach +\[q = \int_V \d[3]{r}\rho(\vec r).\] +In die andere Richtung ist es nicht offensichtlich, hier gilt +\[ + \rho(\vec r) = q\delta_D (\vec r - \vec r') +\] +wenn $q$ bei $\vec r'$ liegt. +\begin{enumerate} + \item Für das Potenzial gilt + \begin{salign*} + \phi(\vec r) &= \int_V \d[3]{R'} \frac{\rho(\vec r')}{|\vec r - \vec r'},\quad \rho \to 0. + \intertext{Dann erhalten wir} + \Delta \phi &= \int_V \d[3]{r'} \Delta \frac{1}{|\vec r - \vec r'|}\\ + &= -4\pi \int \d[3]{r'} \rho(\vec r') \cdot \delta_D(\vec r - \vec r') + \intertext{Es gilt $\int_V \d[3]{r'}\varphi(\vec r') \delta_D(\vec r - \vec r') = \varphi(\vec r)$ und $\int \d[3]{r'} \delta_D(\vec r - \vec r') = 1$} + &= - 4\pi \rho(\vec r) + \end{salign*} + Man kann sich die Dirac $\delta$-Funktion vorstellen als + \[ + \lim\limits_{\sigma^2 \to 0} \frac{1}{\sqrt{2\pi \sigma^2}}\cdot \exp\left(-\frac{x^2}{2\sigma^2}\right). + \] + \item Für diskrete Ladungen gilt + \begin{salign*} + \phi(\vec r) &= \sum_{i} \frac{q_i}{\left|\vec r - \vec r'\right|}\\ + \Delta \phi(\vec r) &= \sum_{i} q_i \Delta \frac{1}{\left|\vec r - \vec r'\right|}\\ + &= \sum_{i} q_i(-4\pi)\delta_D (\vec r - \vec r_i)\\ + &= -4\pi \rho(\vec r)\\ + &= \sum_{i}q_i \delta_D(\vec r - \vec r_i) + \end{salign*} +\end{enumerate} +\section{Eigenschaften der Dirac $\delta_D$-Funktion} +\begin{enumerate} + \item Normierung $\int_-\infty^\infty\d{x} \delta_D(x) = 1$ + \item Verschiebung $\int_{-\infty}^\infty\d{x}\varphi(x) \delta_D(x-a) = \varphi(a)$ + \item Skalierung $\int_{-\infty}^\infty\d{x} \delta_D(ax) = \int_{-\infty}^\infty \d{y}\delta_D(y)$ + \item Ableitung $\int_{-\infty}^\infty \d{x} \varphi(x) \delta_D'(x-a) = \underbrace{\varphi(x) \delta_D(x-a)\bigg|_{-\infty}^{+\infty}}_{\to 0} - \int_{-\infty}^\infty\d{x} \varphi'(x)\delta_D(x-a) = \varphi'(a)$. +\end{enumerate} +\section{Potenzielle Energie einer Ladungsverteilung} +\begin{enumerate}[(1)] + \item Bei einer Ladung $q_1$ an der Stelle $\vec r_1$ erhalten wir das Potenzial $\phi_1 = \frac{q_1}{\left|\vec r - \vec r_1\right|}$. + \item Nun schieben wir eine Ladung $q_2$ aus dem Unendlichen an die Stelle $\vec r_2$. Dabei verrichten wir eine Arbeit $W_2 = q_2\phi_1(\vec r_2)$. + \item Wir schieben eine dritte Ladung $q_3$ an die Stelle $\vec r_3$ und verrichten die Arbeit $W_3 = q_3(\phi_1(\vec r_3) + \phi_2(\vec r_3))$.\\ + $\vdots$ + \item[(n)] Schließlich schieben wir die Ladung $q_n$ an die Stelle $\vec r_n$ und verrichten die Arbeit $W_n = q_n\sum_{i = 1}^{n-1} \phi_i(\vec r_n)$ +\end{enumerate} +Als Gesamtenergie ergibt sich daher +\begin{salign*} + W_\text{ges} &= \sum_{n = 1}^{N} W_n\\ + &= \sum_{n = 1}^{N} q_n \sum_{i = 1}^{n-1} \frac{q_i}{\left|\vec r - \vec r_1\right|}\\ + &\stackrel{\text{Doppelzählung}}{=} \frac{1}{2} \sum_{n = 1}^{N} \sum_{i = 1}^{N} \frac{q_iq_n}{\left|\vec r - \vec r_1\right|} +\end{salign*} +Gehen wir nun zum Kontinuum über, so erhalten wir +\begin{salign*} + W &= \frac{1}{2}\int \d[3]{r} \int \d[3]{r'} \frac{\rho(\vec r)\rho(\vec r')}{\left|\vec r - \vec r'\right|}\\ + &= \frac{1}{2} \int \d[3]{r} \rho(\vec r) \cdot \underbrace{\int \d[3]{4'} \frac{\rho(\vec r)}{\left|\vec r - \vec r'\right|}}_{= \phi(\vec r)}\\ + &= \frac{1}{2} \int \d[3]{r} \rho(\vec r) \cdot\phi(\vec r)\\ + &\stackrel{\Delta \phi = -4\pi\rho}{=} - \frac{1}{8\pi}\int \d[3]{r} \Delta \phi \cdot \phi + \intertext{Es gilt die Produktregel $\phi \nabla \phi = \phi \div \nabla \phi = \div (\phi \nabla \phi) - \nabla\phi \cdot \nabla\phi$} + &= -\frac{1}{8\pi} \int \d[3]{r} \div (\phi\nabla \phi) + \frac{1}{8\pi}\int \d[3]{r} (\nabla \phi)^2\\ + &= \frac{1}{8\pi} \int \d[3]{r} \left|\vec E\right|^2\\ + &= \int \d[3]{r} W_{\mathrm{el}} +\end{salign*} +\begin{definition}[Energiedichte] + Wir definieren die Energiedichte $W_{\mathrm{el}} = \frac{1}{8\pi} |\vec E|^2$. +\end{definition} +\end{document} \ No newline at end of file diff --git a/theoIII.pdf b/theoIII.pdf index 3aebbb33962cbc732365498e6d143c52dc87f308..24cca9ea3f3b449a94369251fb11dd27e184d160 100644 GIT binary patch delta 224107 zcmZs?Q*b6g_pcj!V%z2$+qP|IV*8D4+cqY)ZQGjInQ(u5|LdGOb@o@)7k$|mtNQn> 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\input{theo01.tex} +\input{theo02.tex} \end{document}