move dirs to trunk for git migration
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[General]
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lastDocument=facharbeit.tex
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masterDocument=facharbeit.tex
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name=Facharbeit
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pkg_extIsRegExp=false
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pkg_extensions=.cls .sty
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[Tools]
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MakeIndex=
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QuickBuild=PDFLaTeX+ViewPDF
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[item:facharbeit.kilepr]
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[item:facharbeit.tex]
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\documentclass[fontsize=12pt,a4paper,BCOR=18mm,DIV=13]{scrartcl}
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%neue dt. rechtschreibung
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\usepackage[ngerman]{babel}
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%utf8 benutzen
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\usepackage[utf8]{inputenc}
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\usepackage{lmodern}
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\usepackage[T1]{fontenc}
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% zum Anzeigen von Internetlinks
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\usepackage{url}
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%zeilenabstand
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\usepackage{setspace}
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\onehalfspacing
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% besseres Schriftbild
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% Doku: http://www.ctan.org/tex-archive/macros/latex/contrib/microtype/microtype.pdf
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%\usepackage{microtype}
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%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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%+++ Kopf- und Fusszeilen +++++++++++++++++++++++++++++++++++++++++
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%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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% scrpage2 kommt von KOMA-Script her, kann jedoch auch mit anderen Klassen verwendet werden
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% mit diesem Paket ist es auch möglich, dass rechte und linke Seiten andere Kopfzeilen haben.
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% Dies und vieles mehr findet man in der Dokumentation: scrguide.pdf
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\usepackage{scrpage2}
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%Mehrspaltiges layout
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\usepackage{multicol}
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\usepackage{amsmath}
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%oben nummerieren
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\clearscrheadfoot
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\pagestyle{scrheadings}
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\usepackage[final]{graphicx}
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\KOMAoptions{DIV=current}
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%\setcounter{tocdepth}{4} % \paragraph{} Aufnahme in das Inhaltsverzeichnis *
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%\setcounter{secnumdepth}{4} % Nummerierung vertiefen *
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\title{Simulation der Bewegung geladener Teilchen im E- und B-Feld mit dem Computer}
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\author{Peter Dahlberg}
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\usepackage{ntabbing}
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\begin{document}
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\begin{titlepage}
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%\KOMAoptions{BCOR=0mm}
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\singlespacing
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\setlength{\parindent}{0pt} %absatzeinrückung verhindern
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\begin{center}
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\begin{large}
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\underline{\textbf{Hertzhaimer-Gymnasium Trostberg}}
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\end{large}\\[1em]
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Kollegiatenjahrgang 2007/2009\\[5em]
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\begin{large}
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\textbf{Facharbeit}
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\end{large}\\[2em]
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aus der Physik\\[4em]
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Simulation der Bewegung geladener Teilchen im E- und B-Feld mit dem Computer\\[5em]
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\end{center}
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Verfasser: Peter Dahlberg\\[1em]
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Kursleiter: Helmut Perzl\\[1em]
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Abgabetermin: 30.01.2009\\[2em]
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\begin{multicols}{2}
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Facharbeit:\\[1em]
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Erzielte Punkte: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
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Gesamtergebnis:\\[4em]
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Eintragung des Gesamtergebnisses am:\columnbreak
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Mündliche Prüfung:\\[1em]
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Erzielte Punkte: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
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Einfache Wertung: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[1em]
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Doppelte Wertung: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
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\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[5em]
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\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
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Unterschrift des Kursleiters
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\end{multicols}
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\end{titlepage}
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\titlepage
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\tableofcontents
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\clearpage
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\chead{\pagemark} %ab jetzt seitenzahl
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\section{Einleitung}
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Die Beeinflussung der Bewegung geladener Teilchen durch elektrische und magnetische Felder ist in der Physik von entscheidender Bedeutung beispielsweise in Form von Teilchenbeschleunigern oder Massenspektrographen. Aber auch in unserem Alltag war die Steuerung der Flugbahn geladener Teilchen in Form der Braunschen Röhre, die über ein halbes Jahrhundert lang die Basis für Fernsehgeräte bildete und erst seit einigen Jahren von anderen Techniken abgelöst wird allgegenwärtig.
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Im folgenden wird nun ein Computerprogramm, das den Flug eines geladenen Teilchens durch homogene magnetische und elektrische Felder aus zweidimensionaler Perspektive simuliert beschrieben. Zuvor wird Grundlegendes zu den beiden Feldtypen und der Vorgehensweise bei der Bahnberchnung dargelegt.
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\section{Das elektrische und das magnetische Feld}
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\subsection{Das elektrische Feld}
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\subsubsection{Das elektrische Feld im Allgemeinen} \label{kap:epower}
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\singlespacing
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\begin{quote}
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\begin{small}
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Das elektrische Feld ordnet jedem Raumpunkt die richtungsabhängige Größe der \emph{elektrischen Feldstärke} $\vec {E}$ zu. Diese ist definiert durch die Kraft $\vec {F}$, die auf eine in dem Punkt befindliche Ladung Q wirkt:
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$$ \vec F = Q \vec E $$
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Die Feldstärke ist also, anders gesagt, die Kraft pro Ladungseinheit. Das elektrische Feld ist ein Vektorfeld.\footnote{\cite{wpefeld} (Definition)}
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\end{small}
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\end{quote}
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\onehalfspacing
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Die elektrische Feldstärke hat die Einheit:
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$$ \left[E\right] = 1 \frac{N}{As} = 1 \frac{V}{m} $$
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\subsubsection{Das homogene elektrische Feld}
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\cite{wpefeld} Eine spezielle Art des elektrischen Feldes ist das homogene elektrische Feld. Die Feldstärke $\vec{E}$ hat in jedem Raumpunkt innerhalb dieses Feldes den gleichen Betrag und die gleiche Richtung. Dies vereinfacht das Rechnen, da bei konstantem $Q$ das selbe für die Kraft $\vec{F}$ gilt.
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Zur Erzeugung eines homogenen elektrischen Feldes benötigt man zwei planparallele Kondensatorplatten\footnote{Randeffekte werden vernachlässigt} zwischen denen man eine Spannung $U$ anlegt. Für den Betrag der Feldstärke gilt:
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$$ E = \frac{U}{d}$$
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dabei ist $d$ der Abstand der beiden Platten.
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\begin{figure}[h!]
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\centering
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\includegraphics{grafik/efeld.pdf}
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\caption{Schematische Darstellung eines Plattenkondensators}
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\end{figure}
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Die Feldstärke $\vec{E}$ zeigt in Richtung der negativ geladenen Platte, was der Richtung der Kraft $\vec{F}$ auf eine positive Probeladung entspricht. Bei einer negativen Probeladung ist die Kraft entgegen dem Feld gerichtet.
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\subsubsection{Bewegung geladener Teilchen im homogenen elektrischen Feld}
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\cite{mlmlk1} Wählt man die y-Achse parallel zu den Feldlinien und die x-Achse senkrecht dazu, lässt sich die Bewegung eines geladenen Teilchens analog zum schrägen Wurf bzw. dem freien Fall beschreiben. In x-Richtung bewegt sich das Teilchen mit konstanter Geschwindigkeit \hbox{$v_x\left(t\right) = v_{0x}$} und in y-Richtung beschleunigt durch die vom Feld verursachte Kraft mit \hbox{$v_y\left(t\right) = v_{0y} + a_y t$}. Dabei ist \hbox{$a_y = \frac{F}{m}$}.
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Da die Bewegung analog zur Bewegung einer Masse im Schwerkraftfeld abläuft folgt das Teilchen für alle $v_x \not= 0$ einer Parabelbahn.
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\subsection{Das Magnetfeld}
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\subsubsection{Das homogene Magnetfeld}
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\cite{mlmlk1} Im homogenen Magnetfeld ist die magnetische Flussdichte\footnote{$\left[B\right] = 1 T$ (Tesla)} $\vec{B}$ überall gleich gerichtet und gleich groß.
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Ein annähernd homogenes Magnetfeld findet man beispielsweise zwischen den Schenkeln eines Hufeisenmagneten. Eine andere Möglichkeit ein solches Feld zu erzeugen ist ein Helmholtz-Spulenpaar. Dieses ermöglicht es die Stärke des Feldes beliebig zu verändern.
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\subsubsection{Kraft auf geladene Teilchen im Magnetfeld} \label{kap:magpower}
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\cite{ross} Die Kraft auf einen Ladungsträger im Magnetfeld nennt man Lorentzkraft. Es gilt:
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$$ \vec{F_L} = Q\left(\vec{v} \times \vec{B}\right)$$
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Die Lorentzkraft wirkt also immer senkrecht zur von $\vec{B}$ und $\vec{v}$ aufgespannten Ebene. Es wirkt keine Kraft, wenn sich der Ladungsträger in Feldrichtung oder entgegen der Feldrichtung bewegt \hbox{$\left(\vec{v} \parallel \vec{B}\right)$}. Desweitern wirkt auf eine ruhende Ladung \hbox{$\left(v = 0\right)$} ebenfalls keine Kraft.
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Zur Bestimmung der Richtung der Lorentzkraft kann man die Die Drei-Finger-Regel (Abb. \ref{fig:rhl}) benutzen. Für positiv geladene Teilchen benutzt man die rechte Hand und für negativ geladene die linke Hand.
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.3\textwidth]{grafik/RHR.pdf}
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\caption{\cite{wprhr} Die Drei-Finger-Regel \hbox{$\left(\vec{a} \mathop{\widehat{=}} \vec{v}, \vec{b} \mathop{\widehat{=}} \vec{B}, \vec{a} \times \vec{b} \mathop{\widehat{=}} \vec{F_L} \right)$} }
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\label{fig:rhl}
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\end{figure}
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Für den Betrag der Lorentzkraft gilt: $F_L = QvB \cdot \sin \alpha$ \\
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Dabei ist $\alpha$ der Winkel zwischen $\vec{v}$ und $\vec{B}$.
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\subsubsection{Bewegung geladener Teilchen im homogenen Magnetfeld}
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\cite{ross} Betrachtet man nur den Fall, dass sich ein Teilchen mit der Geschwindigkeit $\vec{v}$ senkrecht zur Feldrichtung bewegt, dann beschreibt es eine Kreisbahn. Dies lässt sich dadurch erklären, dass die Lorentzkraft immer senkrecht zur Flugrichtung wirkt und im laufe der Bewegung durch das Feld immer in Richtung eines bestimmten Punktes wirkt.
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Zur Bestimmung des Radius $r$ der Kreisbahn kann man also den Betrag Lorentzkraft mit dem Betrag Zentripetalkraft $F_Z$ gleichsetzen. Mit \hbox{$\alpha = 90^\circ$} erhält man:
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\begin{eqnarray}
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F_Z &=& \left|F_L\right| \\
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\frac{mv^2}{r} &=& \left|QvB\right|\\
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r &=& \left|\frac{mv}{QB}\right|
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\end{eqnarray}
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\section{Berechnung der Flugbahn einer Probeladung mit dem Computer}
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\subsection{Berechnung der Flugbahn mithilfe kleiner Zeitschritte}
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\label{kap:smallsteps}
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Bei einer beliebigen Anordnung von Feldern, die sich auch beliebig überlagern können ist die exakte Flugbahn eines geladenen Teilchens durch diese Anordnung nicht oder nur schwer rechnerisch zu bestimmen. Es ist aber möglich die Flugbahn anzunähern.
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Die Kraft auf ein Teilchen mit einer bestimmten Ladung, Masse und Geschwindigkeit in einem einzelnen Feld kann berechnet werden\footnote{siehe \ref{kap:epower} und \ref{kap:magpower}}. Überlagern sich mehrere Felder in einem bestimmten Bereich der Anordnung lassen sich die Kräfte der einzelnen Felder zu einer Gesamtkraft Addieren. Da $\vec{F} = m \mathop{\cdot} \vec{a}$ \cite{pfut} gilt, ist auch die Beschleunigung des Teilchens in einem bestimmten Punkt berechenbar.
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Zu Beginn der Simulation befindet sich das Teilchen an einem bestimmten Punkt \hbox{$P_0\left(x_0\mathop{/}y_0\right)$} und hat die Geschwindigkeit $\vec{v_0}$. Näherungsweise nimmt man nun an, dass sich das Teilchen für die Dauer einer bestimmten Zeitspanne $\Delta t$ geradlinig mit konstanter Beschleunigung bis zu einem Punkt \hbox{$P_1\left(x_1\mathop{/}y_1\right)$} bewegt. Mit den Bewegungsgleichungen \cite{pfut} kann $P_1$ und die Geschwindigkeit $\vec{v_1}$ im Punkt $P_1$ berechnet werden:
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\begin{eqnarray}
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\vec{v_1} & = & \vec{v_0} + \frac{\vec{F_0}}{m} \mathop{\cdot} \Delta t \\
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x_1 & = & x_0 + v_{0x} \mathop{\cdot} \Delta t + \frac{F_{0x}}{2m} \mathop{\cdot} {\Delta t}^2 \\
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y_1 & = & y_0 + v_{0y} \mathop{\cdot} \Delta t + \frac{F_{0y}}{2m} \mathop{\cdot} {\Delta t}^2
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\end{eqnarray}
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Nun nimmt man den Punkt $P_1$ als Ausgangspunkt und wiederholt diese Prozedur, dabei erhält man einen Punkt $P_2$. Dies lässt sich über eine beliebige Zahl an Schritten fortsetzen und es ergibt sich eine Näherung der Flugbahn.
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Je kleiner man $\Delta t$ wählt, umso kürzer werden die Geradenstücke und damit wird die Näherung genauer. Allerdings steigt damit natürlich der benötigte Rechenaufwand, um eine bestimmte Gesamtzeitspanne zu berechnen.
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\subsection{Automatische Anpassung der Schrittlänge}
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\label{kap:autostep}
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Es erweist sich in vielen Fällen als unpraktisch für die Zeitspanne $\Delta t$ über die gesamte Simulationsdauer einen konstanten Wert zu verwenden.
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Probleme ergeben sich hier vor allem, wenn das Teilchen durch ein elektrisches Feld einer starken Beschleunigung ausgesetzt ist und sich die Geschwindigkeiten in den einzelnen Bereichen der Simulation stark unterscheiden. Dies führt unter Umständen dazu, dass die Flugbahn in Bereichen mit vergleichsweise hoher Teilchengeschwindigkeit zu ungenau berechnet wird.
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Desweiteren ergibt sich dadurch auch ein Nachteil für den Benutzer, da dieser für jede Simulationsanordnung einen passenden Wert für $\Delta t$ finden muss.
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Diese Probleme lassen sich entschärfen, indem man versucht $\Delta t$ für jeden Schritt abhängig von der Beschleunigung $a$ und der Geschwindigkeit $v$ des Teilchens so zu wählen, dass es in jedem Schritt näherungsweise einen vorher festgelegten Weg $\Delta s$ zurücklegt.
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Zur Bestimmung von $\Delta t$ wird die Beziehung \hbox{$2 \cdot a \cdot \Delta s = v_1^2 - v_0^2$} \cite{pfut} mit der Annahme $v_1 \approx \frac{\Delta s}{\Delta t}$ herangezogen. Eingesetzt und aufgelößt erhält man:
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$$ \Delta t \approx \frac{\Delta s}{\sqrt{2 \cdot a \cdot \Delta s \cdot v_0^2}} $$
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Dies stellt sich als eine in diesem Anwendungsfall brauchbare Näherung heraus, die auf jeden Fall besser ist, als nur die Näherung $\Delta s \approx \frac{v_1}{\Delta t}$ zu verwenden, die wie oben beschrieben zu Problemen bei starker Beschleunigung führt.
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\subsection{Relativistische Massenzunahme bei hohen Geschwindigkeiten}
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\cite{bsmasse} Da die Masse $m$ eines Körpers mit zunehmender Geschwindigkeit $v$ zunimmt muss dies natürlich auch in der Simulation berücksichtigt werden. Es gilt:
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$$m = \frac{m_0}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$
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Dabei ist $m_0$ die Ruhemasse des Körpers und $c$ ist die Lichtgeschwindigkeit im Vakuum\footnote{$c = 2,99792458 \cdot 10^8 \frac{m}{s}$}.
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Für \hbox{$v \mathop{\le} 0,1 \cdot c$} gilt in guter Näherung \hbox{$m = m_0$}. Diese Näherung wird auch im Programm verwendet.
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Bei großen Geschwindigkeiten in der Nähe von $c$ kann es in der Simulation vorkommen, dass das Teilchen innerhalb eines Zeitschrittes $\Delta t$ die Grenzgeschwindigkeit $c$ geringfügig überschreitet. Dies geschieht, da die Massenkorrektur nur mit der vor der Ausführung des Schrittes bekannten Geschwindigkeit $v_0$ erfolgen kann. Sollte dieser Fall eintreten wird im darauffolgenden Schritt mit einer Geschwindigkeit, die geringfügig kleiner\footnote{\hbox{$v_{etwas\ kleiner} = c - 3 \cdot 10^{-8} \frac{m}{s}$} } ist, als $c$ weitergerechnet.
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\section{Das Programm "`lsim"}
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\subsection{Aufbau und Bedienung des Programms}
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\subsubsection{Der grundlegende Aufbau des Programmfensters}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.98\textwidth]{grafik/screen1.pdf}
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\caption{Die Benutzeroberfläche von lsim nach dem Start mit farblich markierten Hauptbereichen}
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\label{fig:scrfull1}
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\end{figure}
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\noindent Im folgenden wird nun der Aufbau der Benutzeroberfläche und die Bedienung des Programms beschrieben.
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||||||
|
Wie in Abb. \ref{fig:scrfull1} zu erkennen ist gliedert sich das Fenster in vier Hauptbereiche.
|
||||||
|
Im oberen Bereich befinden sich eine Menüleiste und eine Werkzeugleiste (Rot). Die Symbole in der Werkzeugleiste und die Menüeinträge dienen dazu bestimmte Aktionen auszulösen, wie das Berechnen der Flugbahn, öffnen und speichern von Dateien oder das Einfügen von Feldern.
|
||||||
|
Im mittleren Teil des Fensters befinden sich auf der linken Seite mehrere Reiter (Orange), die dazu dienen Einstellungen und Zahlenwerte zu verändern oder anzuzeigen. Auf der rechten Seite befindet sich eine Zeichenebene, in der die Felder und das Teilchen angeordnet werden und das Ergebnis der Simulation sichtbar wird (Blau).
|
||||||
|
Am unteren Rand befindet sich eine Statusleiste (Grün) in der situationsabhängig verschiedene Informationen angezeigt werden.
|
||||||
|
|
||||||
|
\subsubsection{Bearbeitungs- und Simulationsmodus}
|
||||||
|
Das Programm kennt zwei verschiedene Zustände. Zum einen gibt es den Bearbeitungsmodus, der dazu dient Feldanordnungen zu erstellen und die Simulationsparameter zu verändern, die vor der Berechnung bekannt sein müssen. Nach dem Start des Programms befindet man sich in diesem Modus.
|
||||||
|
|
||||||
|
Zum anderen gibt es den Simulationsmodus, in den man mit folgendem Knopf in der Werkzeugleiste gelangt: \includegraphics[width=1.5em]{grafik/window-simulmode.pdf}\\
|
||||||
|
Beim Wechsel in den Simulationsmodus wird automatisch die Flugbahn berechnet, was je nach eingestellter bzw. benötigter Anzahl an Rechenschritten einige Sekunden in Anspruch nehmen kann. Zurück in den Bearbeitungsmodus gelangt man mit folgendem Knopf: \includegraphics[width=1.5em]{grafik/window-editmode.pdf}
|
||||||
|
|
||||||
|
\subsubsection{Erstellen und Bearbeiten einer Anordnung}
|
||||||
|
\label{kap:editmode}
|
||||||
|
Wenn man das Programm startet oder wenn man im Menü "`Datei"' den Punkt "`Alles Zurücksetzen"' auswählt sieht man eine leere Zeichenfläche mit einem Koordinatensystem in dessen Mittelpunkt sich die Probleadung befindet. Das Koordinatensystem enthält bis auf die x- und y-Richtung keine Beschriftung aber es lassen sich in der Statusleiste die Koordinaten ablesen, an denen sich der Mauszeiger befindet.
|
||||||
|
|
||||||
|
\paragraph{Einfügen eines elektrischen Feldes} $~~$\\
|
||||||
|
Um ein elektrisches Feld einzufügen, muss man in der Werkzeugleiste auf folgenden Knopf klicken: \includegraphics[width=1.5em]{grafik/insert-efield.pdf}\\
|
||||||
|
Dieser erscheint daraufhin versenkt. Jetzt klickt man mit der linken Maustaste an eine beliebige Stelle inerhalb der Zeichenfläche, hält die Maustate gedrückt und zieht das Feld nach rechts unten auf. Ein elektrisches Feld erscheint in blauer Farbe und die Linien im inneren geben die Feldrichtung an.
|
||||||
|
|
||||||
|
\paragraph{Einfügen eines Magnetfeldes} $~~$\\
|
||||||
|
Analog zum elektrischen Feld läuft das Einfügen beim Magentfeld ab. \includegraphics[width=1.5em]{grafik/insert-bfield.pdf} ist der dafür zuständige Knopf. Ein Magnetfeld erscheint in grauer Farbe.
|
||||||
|
|
||||||
|
\paragraph{Einfügen einer Metallplatte} $~~$\\
|
||||||
|
Eine Metallplatte dient dazu die Probeladung zu absorbieren. Dies wirkt sich im Programm dadurch aus, dass die Simulation an dem Punkt endet, an dem die Flugbahn die Metallplatte kreuzt oder berührt. Eine Metallplatte wird nach einem Klick auf den Knopf \includegraphics[width=1.5em]{grafik/insert-stopper.pdf} von links nach rechts "`aufgezogen"'. Sie erscheint in Hellblau und hat eine Dicke von $10px$.
|
||||||
|
|
||||||
|
\paragraph{Platzieren der Probeladung} $~~$\\
|
||||||
|
Nach einem Klick den Knopf \includegraphics[width=1.5em]{grafik/place-charge.pdf} lässt sich die Probeladung durch einfaches klicken innerhalb der Zeichenfläche an eine beliebige Stelle setzen.
|
||||||
|
|
||||||
|
\paragraph{Verändern der Feldanordnung} $~~$\\
|
||||||
|
Ein einzelnes Feld oder eine Metallplatte innerhalb der Zeichenfläche kann durch einen Klick ausgewählt werden und mit gedrückter linker Maustaste verschoben werden. Neben einem ausgewählten Objekt erscheinen schwarze Kästchen, mithilfe derer man die Größe des Objektes verändern kann. Durch drücken der Taste "`Entf"' wird ein ausgewähltes Objekt wieder entfernt.
|
||||||
|
|
||||||
|
Da sich die einzelnen Objekte überlappen können, ist es manchmal notwendig einzelne davon nach hinten bzw. nach vorne zu schieben. Der Knopf \includegraphics[width=1.5em]{grafik/go-up.pdf} verschiebt das gewählte Objekt um eine Ebene nach vorne und der Knopf \includegraphics[width=1.5em]{grafik/go-down.pdf} ensprechend um eine Ebene nach hinten. Um ein Objekt nach ganz vorne bzw. nach ganz hinten zu schieben verwendet man folgende Knöpfe: \includegraphics[width=1.5em]{grafik/go-top.pdf} \includegraphics[width=1.5em]{grafik/go-bottom.pdf}
|
||||||
|
|
||||||
|
Zur Durchführung dieser Aktionen muss der Knopf \includegraphics[width=1.5em]{grafik/edit-scene.pdf} hervorgehoben sein.
|
||||||
|
|
||||||
|
\paragraph{Eingabe und Darstellung numerischer Werte} $~~$\\
|
||||||
|
Es ist zu beachten, dass bei der Eingabe und Darstellung von Zahlenwerten das auf dem Computer eingestellte Dezimaltrennzeichen verwendet wird. Zahlen mit Zehnerpotenzen haben die Form \hbox{"`2,3e+21"'}, was \hbox{$2,3 \cdot 10^{21}$} entspricht. Bei negativem Exponenten entspricht \hbox{"`2,3e-21"'} der Zahl \hbox{$2,3 \cdot 10^{-21}$}. Die Eingabefelder, bei denen es zweckmäsig ist unterstützen die Eingabe von Zehnerpotenzen in der angegebenen Form.
|
||||||
|
|
||||||
|
\paragraph{Der Reiter "`Feldeinstellungen"'} $~~$\\
|
||||||
|
Der Inhalt dieses Reiters ist abhängig vom gewählen Feld bzw. der gewählten Metallplatte. Für alle Fälle befinden sich dort Eingabefelder mit denen sich Größe und Position des Objektes ändern lassen. Metallplatten und elektrische Felder können um einen beliebigen ganzzahligen Winkel gedreht werden. Die Drehung erfolgt dabei rechtsherum. Bei elektrischen Feldern und Magnetfeldern lässt sich die Feld\-stärke bzw. die Flussdichte einstellen. Zusätzlich kann bei Magnetfeldern ausgewählt werden, ob das Feld in die Ebene hinein oder aus der Ebene heraus zeigen soll.
|
||||||
|
|
||||||
|
\paragraph{Der Reiter "`Probeladung"'} $~~$\\
|
||||||
|
Im Reiter "`Probeladung"' lässt sich die Position der Probleladung festlegen. Zusätzlich kann eine Startgeschwindigkeit für das Teilchen, aufgeteilt in x- und y-Richtung festgelegt werden. Ebenfalls eingestellt werden können Ruhemasse und Ladung des Teilchens. Hierbei enstsprechen die voreingestellten Werte denen des Elektrons.
|
||||||
|
|
||||||
|
|
||||||
|
\paragraph{Der Reiter "`Allgemeines"'} $~~$\\
|
||||||
|
Hier kann die maximale Anzahl an Rechenschritten angegeben werden. Wenn das Teilchen während des Fluges an eine Metallplatte stößt sind mitunter weniger Schritte nötig, als angegeben. Es kann außerdem die Länge $\Delta t$ eines Zeitschrittes angegeben werden\footnote{vgl. \ref{kap:smallsteps}}.
|
||||||
|
|
||||||
|
Dieser Wert wird ignoriert, wenn die automatische Bestimmung der Zeitschrittlänge aktiviert wird. Ist diese Option gewählt, was in der Voreinstellung der Fall ist, kann auch der anzunähernde Wegunterschied $\Delta s$ in Pixel eingestellt werden\footnote{vgl. \ref{kap:autostep}}.
|
||||||
|
|
||||||
|
Das letzte Eingabefeld ermöglicht die Einstellung des Maßstabes innerhalb der Simulation. Die Voreinstellung ist: \hbox{$1 px \ \widehat{=} \ 1 cm$}.
|
||||||
|
|
||||||
|
\subsubsection{Betrachten des Berechnungsergebnises}
|
||||||
|
Befindet man sich im Simulationsmodus wird im Zeichenbereich die errechnete Flugbahn des Teilchens angezeigt. Diese lässt sich im Reiter "`Allgemeines"' auch ausblenden.
|
||||||
|
|
||||||
|
Durch einen Klick auf \includegraphics[width=1.5em]{grafik/media-playback-start.pdf} startet das Teilchen seinen simulierten Flug entlang der Bahn. Der Knopf verändert sich dabei zu \includegraphics[width=1.5em]{grafik/media-playback-pause.pdf}. Durch einen erneuten Klick darauf hält das Teilchen an seiner aktuellen Position an. Ein Klick auf \includegraphics[width=1.5em]{grafik/media-playback-stop.pdf} hält die Animation ebenfalls an, versetzt das Teilchen aber wieder an seinen Anfangspunkt. Im Reiter "`Allgemeines"' lässt sich festlegen, wie schnell die Animation sein soll. Man gibt entweder eine durchschnittliche Geschwindigkeit in $\frac{px}{s}$ an oder einen Wert in Sekunden, wie lange die Animation über die ganze Flugbahn dauern soll.
|
||||||
|
|
||||||
|
In "`Allgemeines"' wird auch die "`reale Simulationsdauer"' angezeigt. Gemeint ist damit die Zeit, die ein Teilchen mit den eingestellten Eigenschaften in der Realität benötigen würde, um die Flugbahn zu durchlaufen.
|
||||||
|
|
||||||
|
Im Reiter "`Daten"' wird während der Animation die aktuelle Position des Teilchens angezeigt. Desweiteren kann hier, während die Animation läuft die reale Geschwindigkeit des Teilchens in diesem Punkt abgelesen werden. Die Anzeige dieser Werte kann in machen Situationen ein starkes Ruckeln der Animation hervorrufen und muss deshalb im selben Reiter extra aktiviert werden.
|
||||||
|
|
||||||
|
\subsection{Anwendungsbeispiel: Simulation eines Wienfilters}
|
||||||
|
\subsubsection{Funktionsweise des Wienfilters}
|
||||||
|
|
||||||
|
\begin{figure}[h!]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.7\textwidth]{grafik/wien.pdf}
|
||||||
|
\caption{Schematische Darstellung eines Wien Filters}
|
||||||
|
\label{fig:wien}
|
||||||
|
\end{figure}
|
||||||
|
\noindent \cite{wpwien} Der Wien Geschwindigkeitsfilter lässt nur Teilchen einer bestimmten Geschwindigkeit $v$, wie in Abb. \ref{fig:wien} zu sehen ist passieren. Weicht die Geschwindigkeit von diesem Wert ab, passiert das Teilchen den Filter nicht geradlinig und kann die Blende nicht durchqueren. Dabei ist die Größe und das Vorzeichen der Ladung $Q$, die das Teilchen trägt egal, solange \hbox{$Q \not = 0$} ist. Wenn \hbox{$Q = 0$} ist wirken die Felder gar nicht auf das Teilchen und jedes Teilchen würde den Filter geradlinig durchqueren.
|
||||||
|
|
||||||
|
Die Felder sind senkrecht zueinander angeordnet und die Flugbahn des Teilchens steht jeweils senkrecht auf beiden Feldern. Außerdem sind die Felder so angeordnet, dass die Lorentzkraft $F_L$ genau entgegen der Coulombkraft $F_C$ des elektrischen Feldes wirkt. Setzt man jetzt die Beträge der beiden Kräfte gleich, erhält man eine Bedingung für $v$.
|
||||||
|
\begin{eqnarray}
|
||||||
|
\left|F_L\right| & = & \left|F_C\right|\\
|
||||||
|
\left|QvB\right| & = & \left|QE\right|\\
|
||||||
|
v & = & \left|\frac{E}{B}\right| \label{eqn:vWien}
|
||||||
|
\end{eqnarray}
|
||||||
|
$~~$
|
||||||
|
\subsubsection{Aufbau eines Wienfilters im Programm}
|
||||||
|
\begin{figure}[h!]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.9\textwidth]{grafik/lsim_wien.pdf}
|
||||||
|
\caption{Beispielaufbau eines Wienfilters in lsim mit der Flugbahn eines Teilchens, das die Geschwindigkeit \hbox{$v=5,0 \cdot 10^5 \frac{m}{s}$} besitzt}
|
||||||
|
\label{fig:lsim_wien}
|
||||||
|
\end{figure}
|
||||||
|
Wie in Abb. \ref{fig:lsim_wien} zu erkennen ist muss man ein elektrisches Feld und ein Magnetfeld erzeugen\footnote{Zur Erzeugung und Veränderung von Objekten siehe auch \ref{kap:editmode}} und Beide genau übereinander legen. Die Ausrichtung der Felder muss nicht verändert werden, da die Voreinstellungen passen. Die elektrische Feldstärke $E$ berägt in diesem Beispiel \hbox{$10,0 \frac{V}{m}$} und die magnetische Flussdichte $B$ beträgt \hbox{$2,0 \cdot 10^{-5} T$}.
|
||||||
|
|
||||||
|
Rechts vom Feld ordnet man zwei Metallplatten zu einer Blende an. Die obere Platte ist um $270^\circ$ gedreht, die untere um $90^\circ$. Im abgebildeten Beispiel haben beide jeweils den Abstand $0,1px$ von der x-Achse, so dass ein schmaler Spalt offen bleibt. Dahinter kann noch eine Metallplatte platziert werden, um das Teilchen aufzufangen.
|
||||||
|
|
||||||
|
Die Probleadung platziert man auf der linken Seite vor den Feldern auf der x-Achse und gibt ihr eine positive Anfangesgeschwindigkeit in x-Richtung. Man stellt fest, dass das Teilchen nur bei der aus (\ref{eqn:vWien}) berechneten Geschwindigkeit \hbox{$v=5,0 \cdot 10^5 \frac{m}{s}$} die Anordnung geradlinig durchfliegt und die Blende durchqueren kann.
|
||||||
|
|
||||||
|
Dieses Beispiel ist auch in der beigefügten Datei "`wien\_klein.lsm"' zu finden, die mit dem Programm geöffnet werden kann.
|
||||||
|
|
||||||
|
\begin{figure}[h!]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=0.9\textwidth]{grafik/lsim_wien_slower.pdf}
|
||||||
|
\caption{Hier hat das Teilchen keine passende Geschwindigkeit und durchfliegt die Blende somit nicht; \hbox{$v=4,5 \cdot 10^5 \frac{m}{s}$} }
|
||||||
|
\label{fig:lsim_wien_slower}
|
||||||
|
\end{figure}
|
||||||
|
\newpage
|
||||||
|
|
||||||
|
\section*{Nachwort}
|
||||||
|
\addcontentsline{toc}{section}{Nachwort}
|
||||||
|
Das Programm "`lsim"' ist in der Programmiersprache C++ geschrieben und verwendet die Klassenbibliothek Qt\footnote{für weitere Informationen siehe \url{http://www.qtsoftware.com/}}. Für die Verwendung von Qt habe ich mich entschieden, da es damit möglich ist relativ einfach grafische Benutzeroberflächen zu erstellen. Außerdem sind damit erstellte Programme unter allen gängigen Betriebsystemen übersetzbar und lauffähig. Dies war mir besonders wichtig, da ich Linux verwende und das Programm trotzdem auf dem weit verbreiteten Windows lauffähig sein sollte.
|
||||||
|
|
||||||
|
Das Programm "`lsim"' ist freie Software und steht unter der GPL\footnote{Kopien der Softwarelizenzen finden sich im Quellcode Verzeichnis von "`lsim"'} Version 2 oder höher. Qt ist in der verwendeten "`Open Source Edition"' unter GPL Version 2 und GPL Version 3 erhältlich. Die verwendeten Symbolgrafiken entstammen dem "`Oxygen Icon Theme"'\footnote{Das "`Oxygen Icon Theme"' ist Teil des KDE Projekts, siehe \url{http://www.kde.org/} }, das unter der LGPL Version 3 oder höher erhältlich ist und wurden zum Teil von mir verändert.
|
||||||
|
|
||||||
|
Das schreiben dieses Programms bedeutete für mich eine gewisse Herausforderung, da ich mich erst in C++ und Qt einarbeiten musste. Allerdings konnte ich dadurch meine Programmierkenntnise um einiges erweitern. Ich hoffe, dass das Programm auch zu etwas anderem nützlich ist, als Gegenstand einer Facharbeit zu sein wünsche dem Benutzer viel Freude bei der Benutzung.
|
||||||
|
|
||||||
|
Dieses Dokument wurde mit dem Textsatzsystem \LaTeX \ erstellt.
|
||||||
|
|
||||||
|
\section*{Literaturverzeichnis}
|
||||||
|
\addcontentsline{toc}{section}{Literaturverzeichnis}
|
||||||
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|
||||||
|
%\renewcommand*{\refname}{}
|
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|
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|
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|
||||||
|
\begin{thebibliography}{1234567}
|
||||||
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\bibitem[mlmlk1]{mlmlk1}
|
||||||
|
Müller, A., Leitner, E., Mráz, F., Physik, Leistungskurs 1.Semester, Elektrische und magnetische Felder, München, Ehrenwirth, $1991^{11}$
|
||||||
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|
||||||
|
\bibitem[pfut]{pfut}
|
||||||
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Hammer, A., Hammer, H., Hammer, K., Physikalische Formeln und Tabellen, München, J. Lindauer Verlag, $2002^{8}$
|
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|
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|
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|
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\renewcommand{\refname}{Internetquellen}
|
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|
||||||
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\bibitem[ross]{ross}
|
||||||
|
Rossbach, J., Notizen zur Vorlesung Physik II – WS 2006/07, Das magnetische Feld, \url{http://www.desy.de/~schleper/lehre/physik2/WS2006_07/V_Magnetismus.pdf} (26.01.2009)
|
||||||
|
|
||||||
|
\bibitem[bsmasse]{bsmasse}
|
||||||
|
Szallies, Bernhard, Die relativistische Masse, \url{http://www.szallies.de/RelativistischeMasse.pdf} (25.01.2009)
|
||||||
|
|
||||||
|
\bibitem[wpefeld]{wpefeld}
|
||||||
|
Wikipedia, Seite "`Elektrisches Feld“, \url{http://de.wikipedia.org/w/index.php?title=Elektrisches_Feld&oldid=54383375&printable=yes} (29.12.2008)
|
||||||
|
|
||||||
|
\bibitem[wpwien]{wpwien}
|
||||||
|
Wikipedia, Seite "`Geschwindigkeitsfilter“, \url{http://de.wikipedia.org/w/index.php?title=Geschwindigkeitsfilter&oldid=55777339&printable=yes} (25.01.2009)
|
||||||
|
|
||||||
|
\bibitem[wprhr]{wprhr}
|
||||||
|
Wikipedia, Datei:RHR.svg, \url{http://de.wikipedia.org/w/index.php?title=Datei:RHR.svg&printable=yes} (06.01.2009)
|
||||||
|
\end{thebibliography}
|
||||||
|
|
||||||
|
\end{flushleft}
|
||||||
|
|
||||||
|
\clearpage
|
||||||
|
\singlespacing
|
||||||
|
\noindent\textbf{"`Ich erkläre hiermit, dass ich die Facharbeit ohne fremde Hilfe angefertigt und nur die im Literaturverzeichnis angeführten Quellen und Hilfsmittel benützt habe."'} \\[5em]
|
||||||
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|
||||||
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|
||||||
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\noindent\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }, den \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
|
||||||
|
$\mathop{\ }$\ \ \ \ \ \ \ \ \ \ Ort \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Datum
|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
\noindent\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
|
||||||
|
Unterschrift des Schülers \ \
|
||||||
|
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|
||||||
|
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|
||||||
|
|
||||||
|
|
||||||
|
\end{multicols}
|
||||||
|
|
||||||
|
\end{document}
|
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<text
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xml:space="preserve"
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style="font-size:8.49097347px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans"
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|
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\documentclass[fontsize=12pt,a4paper,BCOR=18mm,DIV=13]{scrartcl}
|
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|
|
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|
%neue dt. rechtschreibung
|
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|
\usepackage[ngerman]{babel}
|
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|
%utf8 benutzen
|
||||||
|
\usepackage[utf8]{inputenc}
|
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|
% zum Anzeigen von Internetlinks
|
||||||
|
\usepackage{url}
|
||||||
|
|
||||||
|
%zeilenabstand
|
||||||
|
\usepackage{setspace}
|
||||||
|
\onehalfspacing
|
||||||
|
|
||||||
|
% besseres Schriftbild
|
||||||
|
% Doku: http://www.ctan.org/tex-archive/macros/latex/contrib/microtype/microtype.pdf
|
||||||
|
\usepackage{microtype}
|
||||||
|
|
||||||
|
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
|
||||||
|
%+++ Kopf- und Fusszeilen +++++++++++++++++++++++++++++++++++++++++
|
||||||
|
%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
|
||||||
|
% scrpage2 kommt von KOMA-Script her, kann jedoch auch mit anderen Klassen verwendet werden
|
||||||
|
% mit diesem Paket ist es auch möglich, dass rechte und linke Seiten andere Kopfzeilen haben.
|
||||||
|
% Dies und vieles mehr findet man in der Dokumentation: scrguide.pdf
|
||||||
|
\usepackage{scrpage2}
|
||||||
|
|
||||||
|
%Mehrspaltiges layout
|
||||||
|
\usepackage{multicol}
|
||||||
|
|
||||||
|
\usepackage{amsmath}
|
||||||
|
|
||||||
|
%oben nummerieren
|
||||||
|
\clearscrheadfoot
|
||||||
|
\pagestyle{scrheadings}
|
||||||
|
\usepackage{graphicx}
|
||||||
|
|
||||||
|
\setcounter{tocdepth}{4} % \paragraph{} Aufnahme in das Inhaltsverzeichnis *
|
||||||
|
\setcounter{secnumdepth}{4} % Nummerierung vertiefen *
|
||||||
|
|
||||||
|
\KOMAoptions{DIV=current}
|
||||||
|
|
||||||
|
\title{Ein Testdokument}
|
||||||
|
\author{Peter Dahlberg}
|
||||||
|
\usepackage{ntabbing}
|
||||||
|
\begin{document}
|
||||||
|
|
||||||
|
\begin{titlepage}
|
||||||
|
\KOMAoptions{BCOR=0mm}
|
||||||
|
\singlespacing
|
||||||
|
\setlength{\parindent}{0pt} %absatzeinrückung verhindern
|
||||||
|
\begin{center}
|
||||||
|
\begin{large}
|
||||||
|
\underline{\textbf{Hertzhaimer-Gymnasium Trostberg}}
|
||||||
|
\end{large}\\[1em]
|
||||||
|
Kollegiatenjahrgang 2007/2009\\[5em]
|
||||||
|
\begin{large}
|
||||||
|
\textbf{Facharbeit}
|
||||||
|
\end{large}\\[2em]
|
||||||
|
aus der Physik\\[4em]
|
||||||
|
Simulation der Bewegung geladener Teilchen im elektrischen- und magnetischen Feld mit dem Computer\\[5em]
|
||||||
|
\end{center}
|
||||||
|
Verfasser: Peter Dahlberg\\[1em]
|
||||||
|
Kursleiter: Helmut Perzl\\[1em]
|
||||||
|
Abgabetermin: 30.01.2009\\[2em]
|
||||||
|
\begin{multicols}{2}
|
||||||
|
Facharbeit:\\[1em]
|
||||||
|
Erzielte Punkte: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
|
||||||
|
Gesamtergebnis:\\[4em]
|
||||||
|
Eintragung des Gesamtergebnisses am:\columnbreak
|
||||||
|
|
||||||
|
Mündliche Prüfung:\\[1em]
|
||||||
|
Erziehlte Punkte: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
|
||||||
|
Einfache Wertung: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[1em]
|
||||||
|
Doppelte Wertung: \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[2em]
|
||||||
|
\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\[5em]
|
||||||
|
\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
|
||||||
|
Unterschrift des Kursleiters
|
||||||
|
\end{multicols}
|
||||||
|
\end{titlepage}
|
||||||
|
\titlepage
|
||||||
|
\tableofcontents
|
||||||
|
\clearpage
|
||||||
|
\chead{\pagemark} %ab jetzt seitenzahl
|
||||||
|
\section{Übersicht}
|
||||||
|
\subsection{Lorem Ipsum}
|
||||||
|
"`Lorem ipsum"'\footnote{\raggedright Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.} dolor sit amet, consectetur adipiscing elit. Sed mattis. Mauris metus est, ultrices ac, aliquet a, ultricies in, nisi. Mauris risus. Etiam fermentum, nisi id condimentum tincidunt, massa leo viverra leo, vitae eleifend diam ipsum at ante. Praesent massa justo, egestas id, cursus eu, consequat vel, ligula. Nulla rutrum neque vel tellus. Nulla lacinia. Etiam varius felis eget ipsum. Vestibulum dapibus pulvinar justo. Suspendisse enim massa, mollis eget, hendrerit sit amet, varius in, turpis. Aliquam quam nisi, dignissim vitae, condimentum vulputate, vulputate hendrerit, pede. Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Etiam dignissim, mi non interdum pellentesque, est quam congue purus, vitae tincidunt nunc sem sit amet risus. Vivamus massa.
|
||||||
|
Das ist $$a = 77 \cdot b^2$$
|
||||||
|
\underline{Das sind die Bewegungsgleichungen:}
|
||||||
|
\begin{eqnarray}
|
||||||
|
\dddot{s} &=& 0 \\
|
||||||
|
\ddot{s} &=& a \\
|
||||||
|
\dot{s} &=& v_0 + a \cdot t \\
|
||||||
|
s &=& s_0 + v_0 \cdot t + \frac{1}{2} \cdot a \cdot t^2
|
||||||
|
\end{eqnarray}
|
||||||
|
Summen gehen wie folgt: $$\sum_{n=0}^{9} 55\cdot x = y$$
|
||||||
|
$$\Delta t_{1,2} = \left|\frac{-\left|v_0\right| \pm \sqrt{v_0^2 + 2\left|a\right|s}}{\left|a\right|}\right|$$
|
||||||
|
\section{Baum}
|
||||||
|
\subsection{Definition und taxonomische Verbreitung}
|
||||||
|
Die Botanik definiert Bäume als mehrjährige, holzige Samenpflanzen, die einen dominierenden Spross aufweisen, der durch sekundäres Dickenwachstum an Umfang zunimmt. Diese Merkmale unterscheiden einen Baum von Sträuchern, Farnen, Palmen und anderen holzigen Pflanzen. Im Gegensatz zu ihren entwicklungsgeschichtlichen Vorläufern verfügen die meisten Bäume zudem über wesentlich differenziertere Blattorgane, die mehrfach verzweigten Seitentrieben (Lang- und Kurztrieben) entspringen. Stamm, Äste und Zweige verlängern sich jedes Jahr durch Austreiben von Endknospen und Seitenknospen, verholzen dabei und nehmen kontinuierlich an Umfang zu. Im Gegensatz zum Strauch ist es das besondere Merkmal der Bäume, dass die Endknospen über die Seitenknospen dominieren (apikale Dominanz) und sich dadurch ein vorherrschender Haupttrieb herausbildet (Akrotonie).
|
||||||
|
|
||||||
|
Baumförmige Lebensformen kommen in verschiedenen Pflanzengruppen vor: „Echte“ Bäume sind die Laubbäume unter den Bedecktsamern und die baumförmigen Nacktsamer, zu denen Nadelholzgewächse wie die bekannten Nadelbäume gehören, aber auch Ginkgo biloba (als einziger noch existierender Vertreter der Ginkgogewächse) sowie zahlreiche Vertreter der fiederblättrigen Nacktsamer (Cycadophytina). Eigentümlichster Baum ist wohl die in Namibia vorkommende Welwitschia mirabilis, deren Stamm im Boden verbleibt. Daneben können auch die Palmen und die Baumfarne eine baumähnliche Form ausbilden. Diese Gruppen besitzen aber kein echtes Holz (sekundäres Xylem) und gelten daher nicht als Bäume. Eine Sonderstellung nimmt der Drachenbaum (Dracaena) ein. Dieser gehört zwar zu den Einkeimblättrigen, hat aber ein sekundäres Dickenwachstum.
|
||||||
|
|
||||||
|
Baumähnliche Formen finden sich hauptsächlich in rund 50 höheren Pflanzenfamilien. Dagegen fehlt die Baumform bei Algen, Moosen, Liliengewächsen, Iridaceae, Hydrocharitaceae, Orchideen, Chenopodiaceae, Primelgewächsen und meist auch bei den Convolvulaceae, Enziangewächsen, Glockenblumengewächsen, Cucurbitaceae, Doldengewächsen, Saxifragaceae, Papaveraceae, Ranunculaceae oder Caryophyllaceae.
|
||||||
|
|
||||||
|
Bäume kommen innerhalb der bedeutenden botanischen Abteilung der Nacktsamer (Gymnospermen) in Form der nadelblättrigen Nacktsamer (Coniferophytina) vor. Hier ist besonders die Unterklasse der Nadelbäume (Coniferae) aus der Klasse Pinatae zu nennen. Dominiert werden die Arten vor allem von der Ordnung Pinales mit den Familien Pinaceae (Fichten, Kiefern, Tannen, Douglasie, Lärchen, Goldlärche), Cupressaceae (Zypressen, Scheinzypressen, Thuja, Wacholder, der ebenfalls zu den Zypressengewächsen gehörenden Unterfamilie Taxodiaceae (Mammutbaum, Küstensequoie, Urweltmammutbaum, Sumpfzypresse) sowie den Familien Podocarpaceae (Podocarpus, Dacrydium), Araucariaceae (Araucarien, Agathis), Taxaceae (Eibe) und Cephalotaxaceae (Kopfeibe).
|
||||||
|
|
||||||
|
Viele Baumarten kommen aber auch innerhalb der Abteilung der Bedecktsamer (Angiospermen) vor. Die verschiedenen Unterklassen haben hier unterschiedliche Laubbaumtypen hervorgebracht. Zu den bedeutendsten zählen die Unterordnung Hamamelididae mit den Buchengewächsen (Fagaceae), zu denen neben den Buchen (Fagus spp.) auch die Eichen (Quercus spp.), und die Edelkastanie (Castanea sativa) gehören. Zur gleichen Unterordnung gehören die Birkengewächse (Betulaceae) mit den Birken und Erlen, sowie die Nussbäume (Juglandaceae), die Ulmen (Ulmaceae) und die Maulbeergewächse (Moraceae). Zur Unterklasse der Rosidae zählen die Obstgehölze aus der Familie der Rosengewächse (Rosaceae) sowie die Leguminosen (Fabales) mit sehr zahlreichen, vor allem tropischen Arten. Neben der Gattung Dalbergia (Palisanderbäume) gehört auch die Gattung Robinia in diese Gruppe. Wirtschaftlich bedeutsam sind die Zedrachgewächse (Meliaceae) mit den Gattungen Enthandophragma (Mahagonibäume) und Cedrela sowie die der Unterklasse Dilleniidae zugehörige Familie der Dipterocarpaceae mit der Gattung Shorea (Meranti, Bangkirai).
|
||||||
|
\subsection{Morphologie baumförmiger Lebensformen}
|
||||||
|
Baumartige Lebensformen zeigen eine große Variationsbreite im Aufbau (Morphologie). Assoziiert wird mit dem Begriff Baum in der Regel der Aufbau aus Baumkrone, Baumstamm und Baumwurzeln. Bei den baumartigen Farnen und den meisten Palmen finden sich einfache Stämme, die keine Äste ausbilden, sondern schopfartig angeordnete, häufig gefiederte Blätter. Vor allem zeigen sie kein sekundäres Dickenwachstum und sind damit keine echten Bäume.
|
||||||
|
|
||||||
|
Bei den echten Bäumen wächst aus dem Spross des Keimpflänzchens durch Längen- und sekundäres Dickenwachstum der künftige Baumstamm heran. Entweder bildet sich der Spross an der Spitze durch die sich ständig erneuernde Gipfelknospe aufrecht weiter und wird zum geraden, bis zur höchsten Kronenspitze durchgehenden Baumstamm (Monopodium). Beispiele dafür sind die meisten Nadelgehölze, aber auch einige Laubgehölze wie Eichen. Bei den Nadelgehölzen bildet sich durch die horizontal ansetzenden Äste eine pyramidenförmige Krone. Diese wird schlanker, wenn sich auch die Äste steil am Stamm aufwärts richten (z. B. Pyramidenpappel). Die monopodiale Wuchsform wird durch die apikale Dominanz der Gipfelknospe verursacht. In der Knospe gebildete Wuchsstoffe (Auxine) unterdrücken die Aktivität der Seitenknospen. Bei der Eiche und anderen Baumarten lässt diese Dominanz des Haupttriebes mit dem Alter nach und es bildet sich eine typische, verzweigte Laubbaumkrone.
|
||||||
|
|
||||||
|
Bei anderen Gehölzen (z. B. Buche, Hainbuche) übernimmt eine subtermale Seitenknospe die Führung (Sympodium). Bei Bäumen entsteht so eine aufrechte „Scheinachse“ (Monochasium). Im späteren Verlauf lässt auch hier die Dominanz der führenden Knospe nach und aus weiteren Seitenknospen entwickeln sich stärkere Äste, die schließlich eine Krone formen. Dies geschieht aber meist früher als bei Bäumen mit monopodialem Wuchs. Sträucher sind durch das Fehlen der apikalen Dominanz gekennzeichnet. Zahlreiche Seitentriebe bilden eine weit verzweigte Wuchsform. Bei Gehölzen, die im Jahreszeitenklima wachsen, bildet sich an den Wuchsachsen während der Vegetationsperiode je ein Triebabschnitt (Jahrestrieb). Kommt es, wie bei der Eiche, während der Vegetationsperiode zu einem weiteren Austrieb, spricht man vom Johannistrieb (Prolepsis). Tropische Arten neigen zu mehrfachem Austrieb. Aus der Zahl der Jahrestriebe und dem Grad der Verzweigung lässt sich das Alter eines Astes ermitteln. Diese Altersbestimmung wird jedoch bei zahlreichen Arten (z. B. Fichte, Tanne) und regelmäßig bei älteren Bäumen durch die Ausbildung von sogenannten Proventivtrieben erschwert, die aus „schlafenden“ Knospen austreiben. Die regelmäßige Bildung von Proventivtrieben wird auch als Reiteration (sprich: Re-Iteration) bezeichnet. Diese Wiederholungstriebe dienen der Erneuerung der Krone und verschaffen Bäumen die Möglichkeit, alternde Äste zu ersetzen sowie auf Stress (Schneebruch, Insektenkalamitäten) zu reagieren.
|
||||||
|
|
||||||
|
Bäume können ein Alter von mehreren hundert Jahren, an bestimmten Standorten sogar von mehreren tausend Jahren erreichen. Als älteste Bäume der Welt gelten gesichert derzeit die Grannenkiefern in den White Mountains in Kalifornien, USA, die dort an der Baumgrenze wachsen und fast 5.000 Jahre alt sind. Die nachweislich ältesten Bäume Mitteleuropas werden auf etwa 600-700 Jahre datiert. Wächst der Baum unter im Jahresrhythmus schwankenden klimatischen Bedingungen, wird während der Vegetationsperiode ein Jahresring angelegt. Mit Hilfe dieser Ringe lassen sich sowohl das Alter eines Baumes als auch dessen Wuchsbedingungen in den einzelnen Jahren ablesen. Die Dendrochronologie nutzt dies, um altes Holz zu datieren und das Klima einer Region bis zu mehreren tausend Jahren zu rekonstruieren.
|
||||||
|
|
||||||
|
Seine Entwicklung bringt für den Baum zahlreiche Probleme und Schädigungen mit sich. Hierunter fallen vor allem Pilz- und Insektenschäden, Windbruch (Baumteile brechen ab), Windwurf (der Baum wird mit den Wurzeln aus dem Boden gehebelt), Schneebruch (Baumteile unter schweren Schneelasten brechen ab), Blitzschaden (Stammteile werden abgesprengt), Frost (Trockenschaden durch Transpiration bei gefrorenem Boden, Stammrisse) und bei Jungbäumen übermäßiger Wildverbiss und Schälung. Die verschiedenen Krankheiten, von denen Bäume befallen werden können, bezeichnet man als Brand, Krebs, Rost, Mehltau, Rotfäule, Weißfäule, Braunfäule, Harzfluss. Zu Missbildungen an Bäumen zählt man die Maserkröpfe, die Hexenbesen oder Wetterbüsche sowie die Gallen.
|
||||||
|
Seine Entwicklung bringt für den Baum zahlreiche Probleme und Schädigungen mit sich. Hierunter fallen vor allem Pilz- und Insektenschäden, Windbruch (Baumteile brechen ab), Windwurf (der Baum wird mit den Wurzeln aus dem Boden gehebelt), Schneebruch (Baumteile unter schweren Schneelasten brechen ab), Blitzschaden (Stammteile werden abgesprengt), Frost (Trockenschaden durch Transpiration bei gefrorenem Boden, Stammrisse) und bei Jungbäumen übermäßiger Wildverbiss und Schälung. Die verschiedenen Krankheiten, von denen Bäume befallen werden können, bezeichnet man als Brand, Krebs, Rost, Mehltau, Rotfäule, Weißfäule, Braunfäule, Harzfluss. Zu Missbildungen an Bäumen zählt man die Maserkröpfe, die Hexenbesen oder Wetterbüsche sowie die Gallen.
|
||||||
|
\end{document}
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fa_material/RelativistischeMasse.pdf
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|||||||
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Format der Datei:
|
||||||
|
Dateiname: url
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
wpefeld: http://de.wikipedia.org/w/index.php?title=Elektrisches_Feld&oldid=54383375&printable=yes
|
||||||
|
|
||||||
|
wprhr.pdf und RHR.svg: http://de.wikipedia.org/w/index.php?title=Datei:RHR.svg&printable=yes
|
||||||
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|
||||||
|
wpwien.pdf: http://de.wikipedia.org/w/index.php?title=Geschwindigkeitsfilter&oldid=55777339&printable=yes
|
||||||
|
|
||||||
|
RelativistischeMasse.pdf: http://www.szallies.de/RelativistischeMasse.pdf
|
||||||
|
|
||||||
|
V_Magnetismus.pdf: http://www.desy.de/~schleper/lehre/physik2/WS2006_07/V_Magnetismus.pdf
|
||||||
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Reference in New Issue
Block a user