\documentclass[a7paper]{kartei}

\usepackage[utf8]{inputenc} %UTF8
\usepackage[OT1]{fontenc}
\usepackage[scaled]{helvet}
\usepackage[ngerman]{babel} % Neue Rechtschreibung
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{units}
\usepackage{nicefrac}

% Deutsche Absatzformatierung
\setlength{\parindent}{0pt} 
\setlength{\parskip}{1em}	

% Oft verwendete spacer
\def \bk {\hspace*{2mm}}

% tikz
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.pathmorphing}

% wrapping
\usepackage{wrapfig}

\begin{document}

\fach{SBP Physik}
\kommentar{by Clifford Wolf}
\include{hinweis}

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\begin{karte}{Beschleunigung, Geschwindigkeit, Weg}

\centering

\begin{tabular}{lll}
\\
Beschleunigung & $=\, a(t)$ & \\
Geschwindigkeit & $=\, v(t)$ & $=\, \int a(t)\,dt$ \\
zur\"uckgelegter Weg & $=\, s(t)$ & $=\, \int v(t)\,dt$ \\
\end{tabular}

\begin{tabular}{ccc}

\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$t$};
	\draw[->] (0,0) -- (0,+2) node[above] {$a(t)$};
	\draw[domain=0:+1.8,color=red] plot function{0};
	\draw[domain=0:+1.8,color=green] plot function{0.8};
\end{tikzpicture} &

\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$t$};
	\draw[->] (0,0) -- (0,+2) node[above] {$v(t)$};
	\draw[domain=0:+1.8,color=red] plot function{0.8};
	\draw[domain=0:+1.8,color=green] plot function{x*0.8};
\end{tikzpicture} &

\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$t$};
	\draw[->] (0,0) -- (0,+2) node[above] {$s(t)$};
	\draw[domain=0:+1.8,color=red] plot function{x*0.8};
	\draw[domain=0:+1.8,color=green] plot function{x*x*1.2/2};
\end{tikzpicture}

\end{tabular}

\vskip-1mm
\vbox{ \scriptsize
\null\qquad\textcolor{red}{rot = gleichf\"ormige Bewegung},\hfill\null \\ \hfill
\textcolor{green}{gr\"un = gleichf\"ormig beschleunigte Bewegung}\qquad\null }

\end{karte}

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\begin{karte}{Gleichf\"ormig beschleunigte Bewegung}

\centering

\begin{align*}
a &= \mbox{konstant} \\
v(t) &= a \cdot t \\
s(t) &= \frac{a \cdot t}2 \cdot t = \frac{a \cdot t^2}2 = \frac{v \cdot t}2 \\
\\
t &= \frac{v}{a} \bk \Rightarrow \\
\Rightarrow \bk s &= \frac{a}{2} \cdot \left( \frac{v}{a} \right)^2 = \frac{v^2}{2a} \\
\Rightarrow \bk v &= \sqrt{2as}
\end{align*}

\end{karte}

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\begin{karte}{Kraft}

\centering
\vspace*{1mm}

Kraft beschleunigt eine Masse in eine Richtung.

Kraft bewirkt: Bewegungs\"anderung und/oder Verformung.

\begin{align*}
\overrightarrow{F} &= \mbox{Kraft} \\
\overrightarrow{F} &= m \cdot \overrightarrow{a} \\
\left[F\right] &= \unit{N} = \unitfrac{kg \cdot m}{s^2} = \unit[1]{Newton} \\
\end{align*}

\end{karte}

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\begin{karte}{Newtonsche Axiome}

\begin{itemize}
\item Tr\"agheitsprinzip (Beharrungskonzept) \\
... Ein K\"orper ist bestrebt seine Geschwindigkeit und Richtung beizubehalten.

\item Dynamisches Grundgesetz (Aktionsprinzip) \\
... Eine Kraft, die auf einen K\"orper wirkt, setzt diesen in Bewegung.

\item Gegenwirkungsgesetz (Reaktionsprinzip, actio et reactio) \\
... Eine Aktion bewirkt eine gleich grosse gegengerichtete Reaktion. Kr\"afte
treten immer paarweise auf.

\item Zusatz: Superpositionsprinzip \\
... Wirken auf einen Punkt mehrere Kr\"afte, so addieren sich diese vektoriell
zu einer Kraft auf.
\end{itemize}

\end{karte}

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\begin{karte}{Energie, Arbeit}

\begin{equation*}
E = \overrightarrow{F} \cdot \overrightarrow{s}, \bk \bk
\left[ E \right] = \unit{J} = \unit{N \cdot m} = \unitfrac{kg \cdot m^2}{s^2} = \unit[1]{Joule}
\end{equation*}

\begin{itemize}
\item Kinetische Energie = Bewegungsenergie
\item Potentielle Energie = Energie der Lage
\item Energie = Arbeit = Kraft mal Weg
\end{itemize}

\mbox{Kin. Energie beim freien Fall / Pot. Energie bei hoher Lage:} \\
\begin{equation*}
E = \overrightarrow{F_g} \cdot \overrightarrow{h} = m \overrightarrow{g} \overrightarrow{h} =
m \overrightarrow{g} \frac{\overrightarrow{g} t^2}{2} =
\frac{m g^2  t^2}{2} = \frac{m \left( g t \right)^2}{2} = \frac{m \cdot v^2}{2}
\end{equation*}

\end{karte}

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\begin{karte}{Der freie Fall}

\centering
\vspace*{7mm}

\begin{tikzpicture}
	\draw[-] (-1,0) -- (+2,0);
	\draw[very thin,draw=gray,->] (0,1.8) -- (0,0.5);
	\draw[<->] (-0.5,0.03) -- node[left] {h} (-0.5,2);

	\path (0.0, 2.15) node [shape=rectangle,draw,minimum size=0.5] {}
	      (0.3, 2.25) node [right] {$v_0 = 0$}
	      (2.2, 2.50) node [right] {$E_p = mgh$}
	      (2.2, 2.00) node [right] {$E_k = 0$}
	      (0.0, 0.15) node [shape=rectangle,draw,minimum size=0.5] {}
	      (0.3, 0.25) node [right] {$v = \sqrt{2gh}$}
	      (2.2, 0.50) node [right] {$E_p = 0$}
	      (2.2, 0.00) node [right] {$E_k = \frac{mv^2}{2}$}
	      (3.0, 1.25) node [right] {$mgh = \frac{mv^2}{2}$};
\end{tikzpicture}

\end{karte}

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\begin{karte}{Federwaage}

\centering
\vspace*{1mm}

\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$s$ (Auslenkung)};
	\draw[->] (0,0) -- (0,+2) node[above] {$F$ (Kraft)};
	\draw[domain=0:+1.8,color=blue] plot function{x} node[right] {$F = k \cdot s$};
\end{tikzpicture}

$k \dots$ Federkonstante, \bk $\left[k\right] = \unitfrac{N}{m}$

In der Feder gespeicherte Energie: \bk $E_F = \frac{k \cdot s^2}{2}$

\end{karte}

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\begin{karte}{Leistung}

\centering
\vspace*{3mm}

Leistung ($P$) = Energie (Arbeit) pro Zeiteinheit

$$P = \frac{E}{t}$$
$$\left[P\right] = \unit{W} = \unitfrac{J}{s} = \unitfrac{kg \cdot m^2}{s^3} = \unit[1]{Watt}$$

\end{karte}

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\begin{karte}{Impuls}

\centering
\vspace*{15mm}

Impuls ($\overrightarrow{p}$) = Masse mal Geschwindigkeit

$$\overrightarrow{p} = m \cdot \overrightarrow{v}, \bk\bk
\left[\overrightarrow{p}\right] = \unitfrac{kg \cdot m}{s}$$

\end{karte}

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\begin{karte}{Erhaltungssaetze \\ der Mechanik}

\vspace*{-3mm}

\begin{itemize}
\item Massenerhaltungssatz \\
... In einem abgeschlossenen System ist \\ die Summe aller Massen konstant.

\item Energieerhaltungssatz \\
... In einem abgeschlossenen System ist \\ die Summe aller Energien konstant.

\item Impulserhaltungssatz \\
... In einem abgeschlossenen System ist \\ die Summe aller Impulse konstant.
\end{itemize}

\end{karte}

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\begin{karte}{Elastischer Stoss}

Nach dem elastischen Stoss: \\
{\bf 2 K\"orper, 2 Geschwindigkeiten, keine Deformationsenergie}

\centering

Wegen Impulserhaltungssatz und Energieerhaltungssatz:
\begin{align*}
\mbox{IES:} \bk & m_1 \overrightarrow{v_1} + m_2 \overrightarrow{v_2} = m_1 \overrightarrow{v_1}' + m_2 \overrightarrow{v_2}' \\
\mbox{EES:} \bk & \frac{m_1 \overrightarrow{v_1}^2}2 + \frac{m_2 \overrightarrow{v_2}^2}2 = \frac{m_1 \overrightarrow{v_1}'^2}2 + \frac{m_2 \overrightarrow{v_2}'^2}2
\end{align*}
\begin{align*}
\Longrightarrow \bk \overrightarrow{v_1}' &= \frac{2 m_2 \overrightarrow{v_2} + \overrightarrow{v_1} \left( m_1 - m_2 \right)}{m_1 + m_2} \\
                    \overrightarrow{v_2}' &= \frac{2 m_1 \overrightarrow{v_1} + \overrightarrow{v_2} \left( m_2 - m_1 \right)}{m_1 + m_2}
\end{align*}

\end{karte}

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\begin{karte}{Unelastischer Stoss}

Nach dem unelastischen Stoss: \\
{\bf 1 K\"orper, 1 Geschwindigkeit + Deformationsenergie}

\centering
\vspace*{5mm}

Wegen Impulserhaltungssatz und Energieerhaltungssatz:
\begin{align*}
\mbox{IES:} \bk & m_1 \overrightarrow{v_1} + m_2 \overrightarrow{v_2} = \left( m_1 + m_2 \right) \overrightarrow{v}' \\
\mbox{EES:} \bk & \frac{m_1 \overrightarrow{v_1}^2}2 + \frac{m_2 \overrightarrow{v_2}^2}2 =
		\frac{\left( m_1 + m_2 \right) \overrightarrow{v}'^2}2 + E_{\scriptsize \mbox{Def}}
\end{align*}
$$\Longrightarrow \bk \overrightarrow{v}' = \frac{\overrightarrow{v_1} m_1 + \overrightarrow{v_2} m_2}{m_1 + m_2}$$

\end{karte}

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\begin{karte}{Reibung}

\centering

{\bf Gleitreibung:} \hspace*{3cm} \\
$F_A > F_R$, \bk \bk $F_R = F_N \cdot \mu$

{\bf Haftreibung:} \hspace*{3cm} \\
$F_A = F_R$, \bk \bk $F_R < F_N \cdot \mu'$

\begin{tikzpicture}
	\draw (-2,0) -- (+2,0);
	\draw (-1,0) -- ++(0,2) -- ++(2,0) -- ++(0,-2);
	\draw (0.0,1) circle (2pt);
	\draw[->] (0,1) -- (0,0.05) node[above right] {$\overrightarrow{F_N}$};
	\draw[->] (0,1) -- (-0.95,1) node[above right] {$\overrightarrow{F_R}$};
	\draw[->] (-2,1) -- (-1.05,1) node[above left] {$\overrightarrow{F_A}$};
\end{tikzpicture}

$\mu$, $\mu'$ ... Materialabh\"angiger Reibungskoeffizient

\end{karte}

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\begin{karte}{Rotation: \\ Drehgr\"ossen und Bahngr\"ossen}

\centering
\vspace*{-1mm}

\begin{tabular}{|l|l|l|}
\hline
 & Bahngr\"osse & Drehgr\"osse \\
\hline \hline
Strecke / Winkel & $b$ (bzw. $s$) & $\varphi$ \\
\hline
Geschwindigkeit & $v$ & $\omega$ \\
\hline
Beschleunigung & $a$ & $\alpha$ \\
\hline
\end{tabular}

Einheiten fuer Drehgr\"ossen: \\ \vspace{1mm}
$\unit[2\pi]{rad} \mathrel{\hat{=}} \unit[360]{^{\circ}} \mathrel{\hat{=}} \unit[1]{Vollkreis}$

Winkel in Radianten (rad) = Bogenl\"ange am Einheitskreis

Drehwinkel in Radianten = Bogenl\"ange durch Radius:
$\varphi = \frac{b}{r}$

\end{karte}

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\begin{karte}{Rotation: Periode, Frequenz, \\ Drehzahl, Kreisfrequenz}

Periodendauer $T$ = Dauer einer Periode, $\left[T\right] = \unit{s}$ \\
(z.B. Dauer einer vollen Umdrehung in Sekunden)

Frequenz $f$ = Anzahl der Perioden pro Sekunde, $\left[f\right] = \unitfrac{1}{s} = \unit{Hz}$

Drehzahl $N$ = Anzahl der Perioden pro Minute, $\left[N\right] = \unitfrac{1}{60s} = \unitfrac{U}{min}$

Winkelgeschwindigkeit $\omega$ = Kreisfrequenz = Bahngeschwindigkeit
am Einheitskreis = Bahngeschwindigkeit durch Radius

$$f = \frac{1}{T}, \bk\bk f = 60 \cdot N, \bk\bk \omega = 2 \pi \cdot f$$

\end{karte}

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\begin{karte}{Rotation: Drehmoment}
Das Drehmoment $\overrightarrow{M}$ bei Rotation $\hat=$ Kraft bei Translation.

\centering

Drehmoment bei tangentialer Kraft $\overrightarrow{F}$ am Radius $\overrightarrow{r}$:
$$\overrightarrow{M} = \overrightarrow{F} \times \overrightarrow{r}$$

Die Vektoren $\overrightarrow{M}$, $\overrightarrow{F}$ und $\overrightarrow{r}$ stehen
rechtwinklig aufeinander.

\begin{tikzpicture}
	\draw[very thin,draw=gray] (0,0) circle (1);
	\draw (0,0) circle (0.1);
	\draw (-0.07,-0.07) -- (+0.07,+0.07);
	\draw (-0.07,+0.07) -- (+0.07,-0.07);
	\draw[->] (0,0.1) -- (0,0.99);
	\draw[->] (-1,1.01) -- (-0.05,1.01);
	\path (0.0, 0.0) ++(0.2,-0.2) node {\tiny $\overrightarrow{M}$};
	\path (0.0, 0.5) ++(0.15,0) node {\tiny $\overrightarrow{r}$};
	\path (-0.5, 1) ++(0,0.15) node {\tiny $\overrightarrow{F}$};
\end{tikzpicture}

\end{karte}

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\begin{karte}{Rotation: Massentr\"agheitsmoment}
Das Massentr\"agheitsmoment $I$ bei Rotation entspricht der tr\"agen Masse bei Translation.

\centering
\vspace*{7mm}

\begin{tabular}{ll}
Bei Translation: & $\overrightarrow{F} = m \cdot \overrightarrow{a}$ \\
Bei Rotation: & $\overrightarrow{M} = I \cdot \overrightarrow{\alpha}$
\end{tabular}

$$I = m \cdot r^2$$

\end{karte}

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\begin{karte}{Energie bei Translation und Rotation}

\centering
\vspace*{2mm}

\begin{eqnarray*}
 & \mbox{Potentielle Energie} & \mbox{Kinetische Energie} \\
\mbox{Bei Translation:} & E = \overrightarrow{F} \cdot \overrightarrow{s} &
	E = \frac{m \cdot \overrightarrow{v}^2}{2} \\ \\
\mbox{Bei Rotation:} & E = \overrightarrow{M} \cdot \overrightarrow{\varphi} &
	E = \frac{I \cdot \overrightarrow{\omega}^2}{2}
\end{eqnarray*}

\end{karte}

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\begin{karte}{Impuls bei Translation und Rotation}
\vspace*{2mm}

Impuls:
$$ \overrightarrow{p} = m \cdot \overrightarrow{v} $$

\vspace*{5mm}
Drehimpuls (Drall):
$$ \overrightarrow{L} = I \cdot \overrightarrow{\omega} $$

\vspace*{8mm}
Der Drehimpuls ist eine Erhaltungsgr\"osse: \bk
In einem abgeschlossenen System ist die Summe aller Drehimpulse konstant.
\end{karte}

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\begin{karte}{Fliehkr\"afte}
Die Fliehkraft ist eine Scheinkraft, die sich aus der Tr\"agheit der Masse ergibt. Wenn
ein K\"orper in Bewegung von einer Zentripetalkraft $\overrightarrow{F_Z}$ in
eine Kreisbahn gezwungen wird, dann wirkt dieser eine gleich grosse Zentrifugalkraft, auch
Fliehkraft genannt, entgegen.

\vspace*{5mm}
Berechnung der Fliehkraft mit der Kreisfrequenz $\omega$ bzw. der Bahngeschwindigkeit $v$:
$$F = m \cdot r \cdot \omega^2 = m \cdot r \cdot \left(\frac{v}{r}\right)^2 = \frac{m \cdot v^2}{r}$$

\end{karte}

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\begin{karte}{Heliozentrisches Weltbild nach Kopernikus}
\vspace*{-3mm}
\begin{itemize}

\item Sonne im Mittelpunkt

\item Fixsterne in Hohlkugel (Fixsternsph\"are)

\item Erde dreht sich um die Sonne (Kreisbahn, 1x pro Jahr)

\item Erde dreht sich um die eigene Achse (1x pro Tag)

\item Planeten drehen sich um die Sonne (Kreisbahnen)

\item Mond dreht sich um die Erde (Kreisbahn)

\end{itemize}
\end{karte}

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\begin{karte}{Heliozentrisches Weltbild nach Kepler}
\begin{itemize}
\item {\bf Elliptische Bahnen} (Sonne im Brennpunkt)

\item {\bf Fl\"achensatz} \\
Die Verbindungslinie zwischen Sonne und Planet \"uberstreift in gleicher Zeit
die gleiche Fl\"ache. \\ (Drehimpuls $L = I \cdot \omega = m \cdot r^2 \cdot \omega = $ konstant)

\item {\bf Halbachsensatz} \\
Das Verh\"altnis vom Quadrat der Periodendauer eines Umlaufes ($T^2$) und von
der 3. Potenz der L\"ange der grossen Halbachse der Umlaufbahn ($a^3$) ist bei
jedem Planeten gleich:

$$ \frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3} \bk \Longleftrightarrow \bk \frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3} $$
\end{itemize}
\end{karte}

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\begin{karte}{Gravitation}
Die Gravitation ist eine Kraft die zwischen allen Massen im Universum wirkt.
Jede zwei Massen ziehen sich mit der Gravitationskraft $F_G$ gegenseitig an:

$$F_G = G \cdot \frac{m_1 \cdot m_2}{r^2}$$
$$G \approx \unitfrac[6.67 \cdot 10^{-11}]{m^3}{kg \cdot s^2}$$

\vspace*{5mm}
Auf der Erdoberfl\"ache k\"onnen die Erdmasse und der Erdradius als konstant
angenommen werden. Hier wird jeder K\"orper durch das Erdschwerefeld mit etwa
$\unitfrac[9,81]{m}{s^2}$ zum Erdmittelpunkt hin beschleunigt.

\end{karte}

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\begin{karte}{Satellitenbedingung}
Bei einem Satelliten erzeugt die Gravitationswirkung der Erde eine
Zentripetalkraft $F_G$ und die Bewegung des Satelliten eine gleich grosse
entgegengerichtete Zentrifugalkraft $F_Z$.

\vspace*{2mm}
$$\overbrace{m_S \cdot r \cdot \omega^2}^{F_Z(\omega)} =
		\overbrace{m_S \cdot \frac{v^2}{r}}^{F_Z(v)} =
		\overbrace{G \cdot \frac{m_S \cdot m_E}{r^2}}^{F_G} $$

\vspace*{-2mm}
$$ \Longrightarrow \bk r^3 \cdot \omega^2 = r \cdot v^2 = G \cdot m_E$$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{1. und 2. kosmische Geschwindigkeit}
Bei einem Satelliten der auf der Erdoberflaeche gestartet wird (dessen Umlaufbahn die
Erdoberflaeche tangential streift), ist..

\begin{itemize}
\item {\bf die 1. kosmische Geschwindigkeit} \bk ($\approx \unitfrac[7,9]{km}{s}$) \\
die kleinste Geschwindigkeit bei der der Satellit ein Satellit ist (also nicht
auf die Erde faellt) und

\item {\bf die 2. kosmische Geschwindigkeit} \bk ($\approx \unitfrac[11,2]{km}{s}$) \\
die gr\"osste Geschwindigkeit bei der der Satellit ein Satellit ist (also
nicht in einer Parabel aus dem Gravitationsfeld der Erde flieht).
\end{itemize}

Die 2. kosmische Geschwindigkeit wird auch Fluchtgeschwindigkeit genannt.

\end{karte}

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\begin{karte}{Ekliptik und Jahreszeiten}
Die Ekliptik ist die Ebene in der die Umlaufbahn der Erde liegt.
Die Erde rotiert um die eigene Achse. Diese Achse steht in einem Winkel von
$\unit[67]{^{\circ}}$ zur Ekliptik.

\centering
\begin{tikzpicture}
	\draw (-5,0) circle (1.2) node [above] {Sonne};
	\draw[very thin,draw=gray] (-6.7,0) -- node [above] {Ekliptik} (+1.5,0);

	\draw (0,0) circle (1) node [below,rotate=67] {Erde};

	\draw[very thin,draw=gray] (canvas polar cs:angle=67-90,radius=1cm) -- (canvas polar cs:angle=67+90,radius=1cm);
	\draw (canvas polar cs:angle=67+180,radius=1cm) -- (canvas polar cs:angle=67,radius=1cm) node [above,rotate=-23] {Achse};

	\draw[->] (canvas polar cs:angle=67+45,radius=1.5cm) node [left] {Winter} -- (canvas polar cs:angle=67+45,radius=1.1cm);
	\draw[->] (canvas polar cs:angle=67+135,radius=1.5cm) node [left] {Sommer} -- (canvas polar cs:angle=67+135,radius=1.1cm);

	% \draw (-0.07,+0.07) -- (+0.07,-0.07);
	% \draw[->] (0,0.1) -- (0,0.99);
	% \draw[->] (-1,1.01) -- (-0.05,1.01);
	% \path (0.0, 0.0) ++(0.2,-0.2) node {\tiny $\overrightarrow{M}$};
	% \path (0.0, 0.5) ++(0.15,0) node {\tiny $\overrightarrow{r}$};
	% \path (-0.5, 1) ++(0,0.15) node {\tiny $\overrightarrow{F}$};
\end{tikzpicture}

\small
An den Polen scheint ein halbes Jahr lang die Sonne bzw. scheint ein halbes
Jahr lang die Sonne nicht. $\Rightarrow$ Polartag bzw. Polarnacht
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Voll- und Neumond}

\centering
\vspace*{-3mm}
\begin{tikzpicture}
	\foreach \x in {0,45,90,135,180,225,270,315} {
		\filldraw [fill=gray] (canvas polar cs:angle=\x,radius=2cm) circle (0.3);
		\draw[fill=white] (canvas polar cs:angle=\x,radius=2cm) ++(0,-.3) arc (-90:90:.3);
	}
	\draw (0,0) circle (0.5) node {Erde};
	\foreach \x in {-2.5,-2,-1.5,-1,-.5,0,.5,1,1.5,2,2.5} {
		\draw[->] (5,\x) -- (4,\x);
	}
	\draw (5.5,0) node [rotate=90] {Sonne};
\end{tikzpicture}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Sonnen- und Mondfinsternis}

\small
Bei einer Sonnenfinsternis schiebt sich der Mond zwischen Erde und Sonne.
Eine Sonnenfinsternis kann daher nur bei Neumond auftreten.

\vspace*{2mm}
Bei einer Mondfinsternis schiebt sich die Erde zwischen Sonne und Mond.
Eine Mondfinsternis kann daher nur bei Vollmond auftreten.

\vspace*{8mm}
\centering
\begin{tikzpicture}
	\draw (-2,0) circle (0.3) node {\tiny Mond};
	\draw (-2,0.5) node {\small Mondfinsternis};

	\draw (0,0) circle (0.5) node {Erde};

	\draw (2,0) circle (0.3) node {\tiny Mond};
	\draw (2,0.5) node {\small Sonnenfinsternis};
	\draw (5,0) circle (1) node {Sonne};
\end{tikzpicture}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Ebbe und Flut}
Wegen $F_G = G \cdot \frac{m_1 \cdot m_2}{r^2}$ wird das Wasser auf der
mondabgewandten Seite der Erde, die Erde selbst und das Wasser auf der
mondzugewandten Seite der Erde unterschiedlich stark zum Mond hin beschleunigt.

\begin{center}
\begin{tikzpicture}
	\draw[fill=blue!50] (1,0) arc (0:360:1 and 0.6);
	\draw[fill=white] (0,0) circle (0.7) node {Erde};
	\foreach \x in {-1,+1} {
		\draw[->] (-0.8,\x) -- ++(0.2,0);
		\draw[->] (0,\x) -- ++(0.4,0);
		\draw[->] (0.7,\x) -- ++(0.6,0);
	}
	\draw[fill=white] (3,0) circle (0.4) node {\tiny Mond};
\end{tikzpicture}
\end{center}

So kommt es zu den Gezeiten (Tide): Entlang der Mond-Erde-Achse entstehen Flutberge (Flut)
und normal dazu Ebbe.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Dichte und Druck}
{\bf Dichte}:
$$ \rho = \frac{m}{V}, \bk\bk \left[\rho\right] = \unitfrac{kg}{m^3} $$

\vspace{10mm}
{\bf Druck}:
$$ p = \frac{F}{A}, \bk\bk \left[p\right] = \unitfrac{N}{m^2} = \unit{Pa} = \unit[1]{Pascal} $$

$$ \unit[1]{bar} = \unit[100\,000]{Pa}, \bk\bk \unit[1]{Torr} = \unit[1]{mm\,Hg} \approx \unit[133,3]{Pa} $$

Die Einheit $\unit{\text{at\"u}}$ bezeichnet den Druck in $\unit{bar}$ \"uber dem normalen Umgebunsgdruck.
$\bk\bk \Longrightarrow \bk\bk \unit[x]{\text{at\"u}} \bk\approx\bk \unit[x+1]{bar}$
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Ideale Fl\"ussigkeit}
\vspace*{-5mm}
\begin{itemize}
\item inkompressibel
\item daher konstante Dichte $\rho$
\item keine innere Reibung
\item keine Oberfl\"achenspannung
\end{itemize}

\vspace*{5mm}
\"Uber die L\"ange eines Rohres sind \\
\hspace*{10mm} der Massendurchsatz $\nicefrac{m}{t}$ und \\
\hspace*{10mm} der Volumensdurchsatz $\nicefrac{V}{t}$ \\
fuer jeden Schnitt durch das Rohr konstant.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Ideales Gas}
\vspace*{-2mm}
\begin{itemize}
\item kompressibel
\item daher variable Dichte $\rho$
\item keine innere Reibung
\end{itemize}

\vspace*{8mm}
\"Uber die L\"ange eines Rohres ist \\
\hspace*{10mm} der Massendurchsatz $\nicefrac{m}{t}$ \\
fuer jeden Schnitt durch das Rohr konstant.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Hydrostatischer Druck}
Der hydrostatische Druck ist der Druck der durch die Gewichtskraft einer
Fl\"ussigkeitss\"aule ausge\"ubt wird.

$$F = m g = \rho V g = \rho A h g$$

$$p = \frac{F}{A} = \frac{\rho A h g}{A} = \rho h g$$

$$\unit[10]{Meter\,Wasser} \mathrel{\hat=} \unit[1]{bar} = \unit[10^5]{Pa}$$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Auftrieb}
Ein K\"orper in einer Fl\"ussigkeit (mit der Dichte $\rho_{F}$) wird bedingt durch
die Druckdifferenz zwischen seiner Ober- und Unterseite nach oben gedr\"uckt. Diese
Kraft nennt man Auftrieb.

\vspace*{-5mm}
\begingroup
\begin{wrapfigure}{l}{2.2cm}
\begin{tikzpicture}
	\fill [fill=blue!10,draw=none] (-1,1.8) -- (+1,1.8) -- (1,-0.5) -- (-1,-0.5);

	\draw (-1,2) -- (-1,-0.5) -- (1,-0.5) -- (1,2);
	% \draw [decorate,decoration=snake] (-1,1.8) -- (+1,1.8);
	\draw[fill=white!10] (-0.3,0.5) rectangle (0.3,1.1);
	\draw[<->] (0.5,0.5) -- node [right] {$h$} (0.5,1.1);
	\draw[<->] (-0.3,0.9) -- node [below] {$A$} (0.3,0.9);

	\draw[->] (-0.2,1.5) -- (-0.2,1.15);
	\draw (0,1.35) node {$p_0$};
	\draw[->] (+0.2,1.5) -- (+0.2,1.15);

	\draw[->] (-0.2,-0.3) -- (-0.2,0.45);
	\draw (0,0.1) node {$p_1$};
	\draw[->] (+0.2,-0.3) -- (+0.2,0.45);
\end{tikzpicture}
\end{wrapfigure}

\begin{gather*}
p_1 = p_0 + \rho_F h g, \bk\bk p = p1 - p0 = \rho_F h g \\
F_G = p A = \rho_F A h g = \rho_F V g
\end{gather*}

\hspace*{2.2cm} Der Auftrieb ist also so gross wie die Gewichts- \\
\hspace*{2.2cm} kraft der verdr\"angten Fl\"ussigkeitsmenge.
\endgroup

Dem Auftrieb wirkt die Schwerkraft die auf den K\"orper wirkt entgegen. Daher
schwimmt ein K\"orper wenn $\rho_K < \rho_F$, schwebt wenn $\rho_K = \rho_F$
und sinkt wenn $\rho_K > \rho_F$.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Kompressionsdruck \\ und statische Druckenergie}
Kompressionsdruck (statischer Druck) ist jener Druck der durch eine Kraft {$F$} ausge\"ubt wird.
\vspace*{-3mm} $$ p = \frac{F}{A} $$
Wird gegen einen Druck eine Fl\"ussigkeitsmenge verdr\"angt (z.B. in einem Kolben)
so wird damit Arbeit verrichtet:
\vspace*{-4mm} \begin{center}
\begin{tikzpicture}
	\draw[fill=blue!10,draw=none] (0,-0.05) rectangle (3,0.65);

	\draw[fill=white] (0,0) rectangle (0.1,0.6);
	\draw[color=gray,fill=white] (1,0) rectangle (1.1,0.6);
	\draw (0,-0.05) -- (3,-0.05);
	\draw (0,+0.65) -- (3,+0.65);

	\draw[->] (0,+0.8) -- node [above] {$s$} (1,+0.8);
	\draw[<->] (2.95,0) -- node [right] {$A$} (2.95,0.6);
	\draw[->] (-0.7,0.3) -- node [above] {$F$} (-0.1,0.3);

	\draw[color=blue] (0.55,0.3) node {$V$};

	\draw[->] (1.6,0.1) -- (1.2,0.1);
	\draw[->] (1.6,0.5) -- (1.2,0.5);
	\draw (1.4,0.3) node {$p$};
\end{tikzpicture}
\end{center}
\vspace*{-2mm}
$$ E = F s = p A s = p V = \mbox{statische Druckenergie} $$

Hydraulische Presse: hydraulische Verbindung zweier Kolben mit unterschiedlicher
Querschnittsfl\"ache.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Hydrodynamischer Druck}
Der hydrodynamische Druck ist jener Druck, den eine str\"omende Fl\"ussigkeit gegen
einen K\"orper in dieser Str\"omung aus\"ubt.

\vspace{10mm}
$$ p = \frac{\rho v^2}{2} $$

\vspace{10mm}
Der hydrodynamische Druck wird auch ``Geschwindigkeitsdruck'' oder ``Staudruck'' genannt.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Bernoulligleichung}
Die Bernoulligleichung ist eine Energieerhaltungsgleichung. Aus ihr geht hervor, dass
bei einer Erh\"ohung der Geschwindigkeit eines Fluids der statische Druck sinken muss.

\centering
\begin{tikzpicture}[scale=0.8]
	\def\a{-3.5}
	\def\b{-2.5}
	\def\c{-2.0}
	\def\d{-1.5}
	\def\e{-1.0}
	\def\f{+1.0}
	\def\g{+1.5}
	\def\h{+2.0}
	\def\i{+2.5}
	\def\j{+4.0}

	\def\x{-2.0}
	\def\y{-1.0}
	\def\z{+1.0}
	\def\zbc{+0.5}
	\def\zde{+0.8}
	\def\zfg{+0.8}
	\def\zhi{+0.2}

	\fill[fill=blue!10] (\a,0) -- (\a,\x) --
		(\e,\x) -- (0,\y) -- (\j,\y) -- (\j,0) --
		(\i,0) -- (\i,\zhi) -- (\h,\zhi) -- (\h,0) --
		(\g,0) -- (\g,\zfg) -- (\f,\zfg) -- (\f,0) --
		(\e,0) -- (\e,\zde) -- (\d,\zde) -- (\d,0) --
		(\c,0) -- (\c,\zbc) -- (\b,\zbc) -- (\b,0);

	\draw (\a,\x) -- (\e,\x) -- (0,\y) -- (\j,\y);
	\draw (\j,0) -- (\i,0) -- (\i,\z);
	\draw (\h,\z) -- (\h,0) -- (\g,0) -- (\g,\z);
	\draw (\f,\z) -- (\f,0) -- (\e,0) -- (\e,\z);
	\draw (\d,\z) -- (\d,0) -- (\c,0) -- (\c,\z);
	\draw (\b,\z) -- (\b,0) -- (\a,0);

	\draw (\e,0) arc (0:-90:0.5);
	\draw (\g,0) arc (0:-90:0.5);

	\draw (\b/2+\c/2,\z) node {$p_1$};
	\draw (\d/2+\e/2,\z) node {$p_2$};
	\draw (\f/2+\g/2,\z) node {$p_3$};
	\draw (\h/2+\i/2,\z) node {$p_4$};

	\draw (0,-1) node [below right] { \vbox{
		\setlength{\baselineskip}{2mm}
		\hbox{\small $p_1$ ... stat. Druck bei $v_1$}
		\hbox{\small $p_2$ ... stat. + dyn. Druck bei $v_1$}
		\hbox{\small $p_3$ ... stat. + dyn. Druck bei $v_2$}
		\hbox{\small $p_4$ ... stat. Druck bei $v_2$}
	}};

	\draw (-3.3,-2.3) node [right] {\tiny ($v_2 > v_1$ wg. Verkleinerung};
	\draw (-3.3,-2.6) node [right] {\tiny des Rohrquerschnitts)};

	\draw[->] (-3.2,-1) -- node [above] {$v_1$} (-2.8,-1);
	\draw[->] (3.0,-0.6) -- node [above] {$v_2$} (3.8,-0.6);
\end{tikzpicture}

$E = E_\text{Kin} + E_\text{Druck} = \frac{\rho V v^2}{2} + V p = \mbox{konstant}$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Aerostatik}

\small
Da Gase kompressibel sind ist die Dichte in einer Gass\"aule nicht konstant.
Daher ist der Druck am Boden einer Gass\"aule nicht linear, sondern quadratisch
von ihrer H\"ohe abh\"angig.

\centering
\vspace{0mm}
\begin{tabular}{p{3.5cm}p{2mm}p{3.5cm}}

\hbox{Hydrostatischer Druck:} & &
Aerostatischer Druck: \\

\begin{tikzpicture}
	\draw[->] (0,0) -- (+3,0) node[right] {$h$};
	\draw[->] (0,0) -- (0,+2) node[above] {$p$};
	\draw[domain=0:+2.8] plot function{x*0.8};
\end{tikzpicture} & &
\begin{tikzpicture}
	\draw[->] (0,0) -- (+3,0) node[right] {$h$};
	\draw[->] (0,0) -- (0,+2) node[above] {$p$};
	\draw[domain=0:+2.5] plot function{x*x*0.8/2};
\end{tikzpicture} \\

\small $h$ ... Tiefe in der Fl\"ussigkeit & &
\small $h$ ... Strecke zum Ende der Atmosph\"are \\

\end{tabular}
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Schwingung}
Eine Schwingung ist eine periodische Hin-Her-Bewegung.

\begin{center}
\begin{tabular}{ccc}
Federpendel & & Fadenpendel \\ \\
\begin{tikzpicture}
	\draw (-1,0) -- (+1,0);
	\draw [decorate,decoration={coil,segment length=0.25cm,aspect=1.5}] (0,0) -- (0,-2);
	\draw [fill=black] (0,-2) circle (0.2cm);
	\draw [<->] (0.5,-0.5) -- (0.5,-2.5);
\end{tikzpicture} &
\hspace*{1cm} &
\begin{tikzpicture}
	\draw (-1,0) -- (+1,0);
	\draw (0,0) -- (-1,-1.5);
	\draw [fill=black] (-1,-1.5) circle (0.2cm);
	\draw [->] (0.0,-2.5) arc (-90:-45:1.8);
	\draw [->] (0.0,-2.5) arc (-90:-135:1.8);
\end{tikzpicture} \\ \\
geradlinig & & kreisf\"ormig \\
\end{tabular}
\end{center}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Harmonische Schwingung}
Harmonische Schwingung = Eine Gr\"osse \"andert sich sinusf\"ormig \"uber die Zeit.

\vspace*{3mm}
\begin{center}
\begin{tikzpicture}
	\draw[domain=-0.1:+6.1,color=red,samples=100] plot function{sin(pi*x)/pi};
	\draw[->] (0,0) -- (+7,0) node[above] {$t$};
	\draw[->] (0,-1) -- (0,+1);
	\draw (0,0.5) node[above,rotate=90] {Auslenkung};
\end{tikzpicture}
\end{center}

\vspace*{3mm}
Jede nicht harmonische Schwingung kann als \"Uberlagerung \\ harmonischer Schwingungen
betrachtet werden.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Frequenz bei Feder- \\ und Fadenpendel}

\centering
\vspace*{3mm}

{\bf Federpendel: \hspace*{2cm}}
$$ \omega = \sqrt{\frac{k}{m}} $$

\vspace*{3mm}
{\bf Fadenpendel: \hspace*{2cm}}
$$ \omega = \sqrt{\frac{g}{l}} $$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Schwingungsarten}

\vspace*{3mm}
{\bf frei:} selbstt\"atige Schwingung nach einmaliger Erregung

{\bf erzwungen:} Schwingung aufgrund kontinuierlicher Erregung

\vspace*{5mm}
{\bf ged\"ampft:} schwingendes System gibt Energie ab \\
\hspace*{1cm} $\longrightarrow$ Amplitude wird kleiner

{\bf unged\"ampft:} schwingendes System gibt keine Energie ab \\
\hspace*{1cm} $\longrightarrow$ Amplitude ist gleichbleibend

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Resonanz}
Resonanz ist, wenn ein schwingungsf\"ahiges System mit seiner Eigenfrequenz angeregt wird.

\vspace*{5mm}
Die Resonanzkurve eines solchen Systems gibt seine Schwingungsamplitude in
Abh\"angigkeit von der Erregerfrequenz an.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Welle, Wellenarten}
Welle = Ausbreitung von Schwingungen \"uber die Aneinanderreihung und Kopplung
von schwingungsf\"ahigen Systemen.

\vspace*{5mm}
{\bf Transversalwelle:} \\
Schwingung normal zur Ausbreitungsrichtung

\vspace*{5mm}
{\bf Longitudinalwelle:} \\
Schwingung parallel zur Ausbreitungsrichtung

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Akustik}
Schall = Schwingung der Luftdichte \\
Ausbreitungsgeschwindigeit der Schallwellen: $c = \unitfrac[330]{m}{s}$

\centering

\vspace*{3mm}
\begin{tabular}{lc}
Infraschall: & $< \unit[16]{Hz}$ \\
Schall: & $\unit[16]{Hz} \mathrel{-} \unit[20]{kHz}$ \\
Ultraschall: & $> \unit[20]{kHz}$ \\
\end{tabular}

\vspace*{3mm}
\begin{tabular}{rcl}
Tonh\"ohe & $\hat=$ & Frequenz \\
Lautst\"arke & $\hat=$ & Amplitude \\
\end{tabular}
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Dopplereffekt bei Schall}
Dopplereffekt bei Schall = Ver\"anderung der Frequenz durch Bewegung von Sender S und/oder
Beobachterin B:

{\bf Fall 1: B bewegt sich zu S hin:}
$$ f_\text{B} = f_\text{S} \cdot \left( 1 + \frac{v}{c} \right) $$

{\bf Fall 2: S bewegt sich zu B hin:}
$$ f_\text{B} = f_\text{S} \cdot \frac{1}{1 - \frac{v}{c}} $$

\vspace*{-2mm}
Das Bezugssystem ist immer das ruhende Medium (Luft).
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Dopplereffekt beim Licht}
Im Gegensatz zum Schall gibt es beim Licht kein Medium.

Daher ist das Bezugssystem frei w\"ahlbar und das Ph\"anomen ist unabh\"angig
davon ob sich Sender S oder Beobachterin B bewegt:

\vspace*{-6mm}
$$ f_\text{B} = f_\text{S} \cdot \frac{\sqrt{1 - \left(\frac{v}{c}\right)^2}}{1 - \frac{v}{c}} = f_\text{S} \cdot \sqrt{\frac{c+v}{c-v}} $$

Ein positives $v$ entspricht einer Ann\"aherung, ein negatives $v$ einer
Entfernung von Sender und Beobachterin.

Beim optischen Dopplereffekt spricht man bei einer Entfernung von Sender und
Beobachterin auch von der Rotverschiebung.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Reflexion von Licht}

{\bf Diffuse Reflexion:} \hspace*{0.5cm} (z.B. Wand) \\
Reflexion an einer nicht spiegelnden Oberfl\"ache. Das einfallende Licht wird
in alle Richtungen gleichermassen gestreut. Die wahrgenommene Helligkeit
der Oberfl\"ache ist unabh\"angig von der Position des Beobachters.

{\bf Regul\"are Reflexion:} \hspace*{0.5cm} (z.B. Spiegel) \\
Reflexion an einer spiegelnden Oberfl\"ache. Das einfallende Licht wird gem\"ass
Einfallswinkel $=$ Ausfallswinkel reflektiert. Daher ist die wahrgenommene Helligkeit
der Oberfl\"ache abh\"angig von der Position des Beobachters.

{\bf Gemischte Reflexion:} \hspace*{0.5cm} (z.B. Lack) \\
Reflexion mit diffusen und regul\"aren Anteilen.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{karte}{Kategorisierung eines \\ optischen Abbildes}

\vspace*{3mm}
{\bf Gr\"osse:} \\
\hspace*{5mm} kleiner, gleich oder gr\"osser

\vspace*{3mm}
{\bf Richtung:} \\
\hspace*{5mm} aufrecht oder verkehrt

\vspace*{3mm}
{\bf Position:} \\
\hspace*{5mm} virtuell oder reell

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Strahlengang beim Hohlspiegel}

\centering

Sph\"arischer Spiegel = Konstruktion aus Kugeloberfl\"ache \\
Parabolspiegel = Konstruktion aus Parabel

\vspace*{-2mm}
\begin{center}
\begin{tikzpicture}
	\draw (-0.5,0) -- (+3,0);

	\draw[fill=black] (2.5,0) circle (0.05) node [above left] {\scriptsize M};
	\draw[fill=black] (0.87868,0) circle (0.05) node [above right] {\scriptsize F};
	\draw[fill=black] (-0.05,0) circle (0.05) node [above left] {\scriptsize S};

	\draw[<-,red] (3,1) -- (0.17157,1);
	\draw[->,green] (0.17157,1) -- (1.5858,-1);
	\draw[<->,magenta] (1.5,+1.5) -- (-0.03,0) -- (1.5,-1.5);
	\draw[<->,blue!80] (+3,+0.24) -- (0.21457,-1.1142);

	\draw (-0.05,0) arc (180:140:2.5);
	\draw (-0.05,0) arc (180:220:2.5);

	\draw (3.5,1.5) node [right] {F = Brennpunkt (Fokus)}
	    ++(0,-0.5) node [right] {M = Mittelpunkt}
	    ++(0,-0.5) node [right] {S = Scheitelpunkt};

	\draw[red] (5.5,-0.5) node [left] {Parallelstrahl};
	\draw (5.75,-0.5) node {$\Leftrightarrow$};
	\draw[green] (6.0,-0.5) node [right] {Brennstrahl};

	\draw[blue!80] (5.5,-1.0) node [left] {Mittelstrahl};
	\draw (5.75,-1.0) node {$\Leftrightarrow$};
	\draw[blue!80] (6.0,-1.0) node [right] {Mittelstrahl};

	\draw[magenta] (5.5,-1.5) node [left] {Scheitelstrahl};
	\draw (5.75,-1.5) node {$\Leftrightarrow$};
	\draw[magenta] (6.0,-1.5) node [right] {Scheitelstrahl};
\end{tikzpicture}
\end{center}

Mittellinie = ``Hauptstrahl'' = ``optische Achse''

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Objekt im W\"olbspiegel}
{\small Konstruktion eines Abbildes im Spiegel bzw. durch eine Linse: Urbild
als aufrechter Pfeil auf der optischen Achse. Strahlengang von zwei Strahlen
durch die Pfeilspitze einzeichnen. Kreuzungspunkt der projizierten Strahlen ist
die Pfeilspitze des Abbilde-Pfeils. Im folgenden Beispiel kreuzen sich die
Strahlen ``hinter dem Spiegel''.}

\vspace*{-5mm}
\begin{center}
\begin{tikzpicture}
	\draw (-2,0) -- (+2,0);

	\draw[red] (2,1) -- (-0.1,1);
	\draw[green] (-1.4645,0) -- (0.6,1.5);
	\draw[magenta] (1.5,+1.5) -- (0,0);
	\draw[magenta] (-1.5,1.5) -- (1,-1);

	\draw[fill=black] (-1.4645,0) circle (0.05) node [above left] {\scriptsize F};
	\draw[fill=black] (0,0) circle (0.05) node [above left] {\scriptsize S};

	\draw[->,very thick] (1,0) -- (1,1);
	\draw[->,very thick] (-0.62,0) -- (-0.62,0.62);

	\draw (0,0) arc (0:20:5);
	\draw (0,0) arc (0:-15:5);

	\draw[<-,gray] (-0.55,-0.1) arc(225:315:1);
\end{tikzpicture} \\ \vspace*{1mm}
Linker Pfeil: Abbild, \bk
Rechter Pfeil: Urbild
\end{center}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Abbilder im Hohl- \\ und W\"olbspiegel}

\vspace*{3mm}
\centering

\begin{tabular}{|l|c|c|}
\hline
 & \bf Hohlspiegel & \bf W\"olbspiegel \\
\hline
\bf Objekt zwischen & verg\"ossert & \\
\bf S und F & aufrecht & \\
 & virtuell & verkleinert \\
\cline{1-2}
\bf Objekt zwischen & verg\"ossert & \\
\bf F und M & verkehrt & aufrecht \\
 & reell & \\
\cline{1-2}
\bf Objekt jenseits & verkleinert & virtuell \\
\bf von M & verkehrt & \\
 & reell & \\
\hline
\end{tabular}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Strahlengang bei Linsen}

\centering
\vspace*{-3mm}

\begin{tabular}{ccc}
\begin{tikzpicture}
	\draw (-0.28,0) arc (180:155:3);
	\draw (-0.28,0) arc (180:205:3);
	\draw (+0.28,0) arc (0:+25:3);
	\draw (+0.28,0) arc (0:-25:3);

	\draw (-1.5,0) -- (+1.5,0);

	\draw[fill=black] (0,0) circle (0.05) node [above] {\scriptsize M};
	\draw[fill=black] (1,0) circle (0.05) node [above] {\scriptsize F};

	\draw[<-,red] (-1.5,1) -- (0,1);
	\draw[->,green] (0,1) -- (1.5,-0.5);

	\draw[<->,blue!80] (-1.3,-0.8) -- (+1.3,+0.8);

	\draw[|<-] (0,-0.5) -- (0.4,-0.5);
	\draw[->|] (0.6,-0.5) -- (1,-0.5);
	\draw (0.5,-0.5) node {\small $f$};
\end{tikzpicture}
& \hspace*{10mm} &
\begin{tikzpicture}
	\draw (+0.1,0) arc (180:155:3);
	\draw (+0.1,0) arc (180:205:3);
	\draw (-0.1,0) arc (0:+25:3);
	\draw (-0.1,0) arc (0:-25:3);

	\draw (-0.39,+1.26) -- (+0.39,+1.26);
	\draw (-0.39,-1.26) -- (+0.39,-1.26);

	\draw (-1.5,0) -- (+1.5,0);

	\draw[<-,red] (-1.5,0.5) -- (0,0.5);
	\draw[green!20,thin] (0,0.5) -- (-1.5,-0.25);
	\draw[->,green] (0,0.5) -- (+1.5,1.25);

	\draw[fill=black] (0,0) circle (0.05) node [above] {\scriptsize M};
	\draw[fill=black] (-1,0) circle (0.05) node [above] {\scriptsize F};

	\draw[<->,blue!80] (-1.3,-0.8) -- (+1.3,+0.8);

	\draw[|<-] (0,-0.5) -- (-0.4,-0.5);
	\draw[->|] (-0.6,-0.5) -- (-1,-0.5);
	\draw (-0.5,-0.5) node {\small $f$};
\end{tikzpicture} \\
konvexe Linse & & konkave Linse \\
(Sammellinse) & & (Zerstreuungslinse) \\
\end{tabular}

\vspace*{0mm}
\begin{tikzpicture}
	\draw (3.5,0.2) node {$f$ = Brennweite};

	\draw (-0.5,-0.5) node [right] {F = Brennpunkt (Fokus)}
	    ++(0,-0.5) node [right] {M = Mittelpunkt};

	\draw[red] (5.5,-0.5) node [left] {Parallelstrahl};
	\draw (5.75,-0.5) node {$\Leftrightarrow$};
	\draw[green] (6.0,-0.5) node [right] {Brennstrahl};

	\draw[blue!80] (5.5,-1.0) node [left] {Mittelstrahl};
	\draw (5.75,-1.0) node {$\Leftrightarrow$};
	\draw[blue!80] (6.0,-1.0) node [right] {Mittelstrahl};
\end{tikzpicture}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Abbilder bei Linsen}

\vspace*{3mm}
\centering

\begin{tabular}{|l|c|c|}
\hline
 & \bf konvexe Linse & \bf konkave Linse \\
 & \bf\small (Sammellinse) & \bf\small \kern-0.5em(Zerstreuungslinse)\kern-0.5em \\
\hline
\bf Objekt innerhalb & verg\"ossert & \\
\bf von $f$ & aufrecht & \\
 & virtuell & verkleinert \\
\cline{1-2}
\bf Objekt zwischen & verg\"ossert & \\
\bf $f$ und $2f$ & verkehrt & aufrecht \\
 & reell & \\
\cline{1-2}
\bf Objekt ausserhalb & verkleinert & virtuell \\
\bf von $2f$ & verkehrt & \\
 & reell & \\
\hline
\end{tabular}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Brechung}
Wenn Licht durch die Grenzschicht zweier dispersiver (licht\-durch\-l\"assiger)
Medien mit unterschiedlicher mediumspezifischer Lichtgeschwindigkeit $c_1$,
$c_2$ f\"allt, so \"andert sich der Winkel des Strahls zur Grenzschicht:

\vspace*{-5mm}
\begin{center}
\begin{tikzpicture}
	\draw (-2,0) -- (0.5,0);
	\draw[gray] (0,1.5) -- (0,-1.5);

	\draw[<->] (-1,1) -- (0,0) -- (0.5,-1);

	\draw[gray] (0,0.7) arc (90:90+45:0.7cm);
	\draw (-0.19134,0.46194) node {\tiny $\alpha$};

	\draw[gray] (0,-0.7) arc (-90:-90+25:0.7cm);
	\draw (0.10822,-0.48815) node {\tiny $\beta$};

	\draw (-1.8,0.3) node [right] {\small $n_1 = \frac{c_V}{c_1}$};
	\draw (-1.8,-0.3) node [right] {\small $n_2 = \frac{c_V}{c_2}$};

	\draw[gray] (-0.2,0) arc (180:180+90:0.2cm);
	\draw[gray,fill=gray] (-0.075,-0.075) circle (0.2mm);

	\draw (0.8,0.8) node[right] {\large $ \frac{\sin \alpha}{\sin \beta} = \frac{c_1}{c_2} = \frac{n_2}{n_1} $}
	      ++(0,-0.8) node[right] {\small $n_1$, $n_2$ = ``Brechungsindex''}
	      ++(0,-0.6) node[right] {\small $c_V$ = Vakuumlicht-}
	      ++(+0.8,-0.4) node[right] {\small geschwindigkeit};
\end{tikzpicture}
\end{center}

\centering
\vspace*{-2mm}
$n_\text{Vakuum} = 1$ \bk
$n_\text{Luft} \approx 1.000\dots{}1$ \bk
$n_\text{Wasser} \approx 1,3$ \bk
$n_\text{Diamant} \approx 2,4$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Totalreflektion}

\small
Bei Brechung von einem dichteren Medium (Medium mit kleinerem $c_M$, also
gr\"osserem $n_M$) zu einem weniger dichten Medium (Medium mit gr\"osserem
$c_M$, also kleinerem $n_M$) kommt es ab einem bestimmten Grenzwinkel zur
Totalreflektion:

\vspace*{-3mm}
\begin{center}
\begin{tikzpicture}
	\draw[->,red!80,very thin] (0,0) -- (1.5,-0.1);
	\draw[red!80,very thin] (0,-0.7) arc (-90:-90+87:0.7cm);
	\draw (0.34418,-0.36269) node {\tiny $\beta$};

	\draw (-2,0) -- (+2,0);
	\draw[gray] (0,1.5) -- (0,-1);

	\draw[->] (-1,1) -- (0,0) -- (1,1);

	\draw[gray] (0,0.7) arc (90:90+45:0.7cm);
	\draw (-0.19134,0.46194) node {\tiny $\alpha$};

	\draw[gray] (0,0.7) arc (90:90-45:0.7cm);
	\draw (+0.19134,0.46194) node {\tiny $\alpha$};

	\draw (-1.8,0.3) node [right] {\small $n_1 = \frac{c_V}{c_1}$};
	\draw (-1.8,-0.3) node [right] {\small $n_2 = \frac{c_V}{c_2}$};

	\draw[gray] (-0.2,0) arc (180:180+90:0.2cm);
	\draw[gray,fill=gray] (-0.075,-0.075) circle (0.2mm);

	\draw (1,0.5) node[right] {\large $ \frac{\sin \alpha}{\sin \beta} = \frac{c_1}{c_2} = \frac{n_2}{n_1} $};
\end{tikzpicture}
\end{center}

\vspace*{-5mm}
Bei $n_1 > n_2$ und entsprechend grossem $\sin \alpha$ wird $\sin \beta$
g\"osser als 1. Da es keinen Winkel $\beta$, der diese Bedingung erf\"ullen
k\"onnte, gibt, kommt es zu keiner Brechung und die Grenzschicht wird spiegelnd.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Mischfarbe, Spektralfarbe}
\small

Licht ist eine elektromagnetische Welle. Die Frequenz (Wellenl\"ange) bestimmt
die Farbe. Langwelliges (niederfrequentes) Licht ist rot, kurzwelliges
(hochfrequentes) Licht ist violett:

\vspace*{-3mm}
\begin{center} \scriptsize
\begin{tabular}{lrrr}
\bf Farbton & \bf Wellenl\"ange & \bf Wellenfrequenz & \bf Energie pro Photon \\
\hline
Violett & 380-420 nm & 789,5-714,5 THz & 3,26-2,955 eV \\
Blau & 420-490 nm & 714,5-612,5 THz & 2,95-2,535 eV \\
Gr\"un & 490-575 nm & 612,5-522,5 THz & 2,53-2,165 eV \\
Gelb & 575-585 nm & 522,5-513,5 THz & 2,16-2,125 eV \\
Orange & 585-650 nm & 513,5-462,5 THz & 2,12-1,915 eV \\
Rot & 650-750 nm & 462,5-400,5 THz & 1,91-1,655 eV \\
\end{tabular}
\end{center}

\vspace*{-3mm}
Weisses Licht = Gemisch aus allen Wellenl\"angen \\
Spektralfrabe = Licht mit nur einer Wellenl\"ange (monochromatisch) \\
Mischfarbe =  Weiss - 1 Farbe = erscheint als Kompliment\"arfarbe

F\"ur das freie Auge ist der Unterschied zwischen einer Spektralfarbe und einer
Mischfarbe nicht zu erkennen.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Farben durch Dispersion}
Der Brechungsindex $n$ ist nicht nur vom Medium sondern auch von der
Wellenl\"ange des Lichts ($\lambda$) abh\"angig.

\begin{center}
\begin{tabular}{cc}
\begin{tikzpicture}
	\draw[->] (-0.1,0) -- (1.5,0) node [right] {$\lambda$};
	\draw[->] (0,0.05) -- (0,1.5) node [above] {$n$};
	\draw[domain=0.1:1.4] plot function{x*x*0.4 + x*-1.3 + 1.3};
	\draw[blue!60,fill=blue!60] (0.1,-0.15) circle (1pt);
	\draw[red!60,fill=red!60] (1.4,-0.15) circle (1pt);
\end{tikzpicture} &
\begin{tikzpicture}
	\draw (-1,0) -- (1,0);

	\draw (-1,0.5) -- (0,0);
	\draw[->,blue!60] (0,0) -- (0.2,-1);
	\draw[->,green!60] (0,0) -- (0.4,-1);
	\draw[->,yellow] (0,0) -- (0.6,-1);
	\draw[->,orange] (0,0) -- (0.8,-1);
	\draw[->,red!60] (0,0) -- (1.0,-1);
\end{tikzpicture}
\end{tabular}
\end{center}

\vspace*{5mm}
Dadurch wird weisses Licht bei Brechung in das Spektrum zerlegt.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Farben durch Streuung}
Licht wird beim Durchtritt durch ein Medium an Partikeln im Medium (Staub, Wasserdampf, etc.) 
gestreut (aufgef\"achert). Der Grad der Streuung ist von der Wellenl\"ange
abh\"angig. Blaues Licht wird 4 mal st\"arker gestreut als rotes.

Morgenrot und Himmelblau durch Streuung des weissen Sonnenlichts in der Erdathmosphaere:

\centering
\begin{tikzpicture}
	\draw (-2,0) circle (1cm);
	\draw (-2,0) node {Sonne};

	\draw (+2,0) circle (0.5cm);
	\draw[gray,very thin] (+2,0) circle (0.8cm);
	\draw (+2,0) node {Erde};

	\draw[->] (-2+1.05,0.2) -- (2-0.58,0.6);
	\draw[->,red!60] (2-0.52,0.6) -- (2,0.5);
	\draw[->,blue!60] (2-0.52,0.6) -- (2-0.3,0.4);
\end{tikzpicture}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Farben durch Interferenz \\ an d\"unnen Schichten}
\small

Bei einer Reflexion an einer d\"unnen Schicht (\"Olfilm, etc.) kommt es zu
einer Doppelreflexion: Einmal an der Oberseite und einmal an der Unterseite
der Schicht.

\begin{center}
\vspace*{-10mm}
\begin{tikzpicture}
	\draw[gray] (-1.5,0) -- (+1.5,0);
	\draw[gray] (-1.5,-0.32) -- (+1.5,-0.32);

	\draw[->] (-1,1) -- node [above,rotate=-45] {$\lambda$} (-0.02,0.02);
	\draw[->] (0.02,0.02) -- (1,1);
	\draw[->] (0.02,-0.02) -- (0.3,-0.3) -- (1.3,0.7);

	\draw (1.15,0.85) ++(0.1,0.1) node[rotate=-45] {\tiny Interferenz};

	\draw[|<->|] (-1.7,0) -- node [left] {$d$} (-1.7,-0.32);
\end{tikzpicture}
\end{center}

\vspace*{-5mm}
Der Strahl der an der Unterseite der Schicht reflektiert wurde legt einen
um ca. $2d$ l\"angeren Weg zur\"uck als der Strahl der an der Oberseite reflektiert
wurde. Dadurch kommt es zu einem Phasenversatz dieser beiden Strahlen und zur
Interferenz. Manche spektralen Anteile werden ausgel\"oschte (destruktive
Interferenz) und andere verst\"arkt (konstruktive Interferenz).

\vspace*{-5mm}
$$ \mbox{Konstruktiv:} \bk d = k \cdot \frac{\lambda}{2}, \bk \bk \bk \bk
\mbox{Destruktiv:} \bk d = \frac{\lambda}{4} + k \cdot \frac{\lambda}{2} $$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Atommodell}
\small
Der Kern besteht aus Protonen ($p^+$) und Neutronen.
Der Radius des Kerns ist viel kleiner als der Radius des ganzen Atoms

Elektronen ($e^-$) beschreiben Kreisbahnen um den Kern und
werden durch die Coulombkraft in der Bahn gehalten (Zentripetalkraft).

Neutronen sind etwa so schwer wie Protonen.
Ein Proton bzw. ein Neutron wiegt etwa 1836 mal mehr als ein Elektron.

Es gibt nur einige erlaubte Bahnen fuer die Elektronen (``Schalen'').

Wenn ein Elektron auf eine niedrigere Bahn springt wird Energie abgestrahlt.
Diese Energie kann nur in bestimmten ``Paketen'' abgestrahlt werden. Wegen
$E = h \cdot f$ sind nur gewisse Frequenzen m\"oglich ($h$ = Plancksches
Wirkungsquantum = konstant).
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Elemente und Isotope}
Element = Sammelbezeichnung f\"ur alle Atome mit der selben Anzahl an Protonen
im Kern. Alle Atome vom selben Element haben dieselben chemischen Eigenschaften.

\vspace*{-4mm}
\begin{center}
\begin{tikzpicture}
	\draw (-0.5,0) node[left] {Schreibweise fuer Elemente:};
	\draw (0,0) node {$E$};
	\draw (-0.19,+0.15) node {\scriptsize $M$};
	\draw (-0.19,-0.15) node {\scriptsize $Z$};
\end{tikzpicture}
\end{center}

\vspace*{-4mm}
$M$ ... Massenzahl (Protonen + Neutronen) \\
$Z$ ... Ordnungszahl (Anzahl Protonen) \\
$E$ ... Chemisches K\"urzel f\"ur das Element

Isotope = Atome vom gleichen Element aber unterschiedlicher Neutronenanzahl.
Zum Beispiel die Isotope von Wasserstoff:

\vspace*{-4mm}
\begin{center}
\begin{tikzpicture}
	\draw (0,0) node {H}
		++(-0.19,+0.15) node {\scriptsize 1}
		++(0,-0.3) node {\scriptsize 1}
		++(0.3,+0.15) node [right] {= Wasserstoff};
	\draw (3,0) node {H}
		++(-0.19,+0.15) node {\scriptsize 2}
		++(0,-0.3) node {\scriptsize 1}
		++(0.3,+0.15) node [right] {= Deuterium};
	\draw (6,0) node {H}
		++(-0.19,+0.15) node {\scriptsize 3}
		++(0,-0.3) node {\scriptsize 1}
		++(0.3,+0.15) node [right] {= Tritium};
\end{tikzpicture}
\end{center}
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Radioaktive Strahlung}
\small
Radioaktive Strahlung entsteht bei Zerfallsprozessen von Atomkernen.

\vspace*{3mm}
{\bf $\alpha$-Strahlung:} \bk (He-Kerne: 2 Protonen + 2 Neutronen, $v \approx \unitfrac[10^7]{m}{s}$) \\
\bk\bk $(Z,\bk M) \bk \longrightarrow \bk (Z-2,\bk M-4) \bk + \bk 1\bk\text{He-Kern}$ 

\vspace*{3mm}
{\bf $\beta$-Strahlung:} \bk (1 Elektron, $v \approx \unitfrac[10^8]{m}{s}$) \\
\bk\bk $(Z,\bk M) \bk \longrightarrow \bk (Z+1,\bk M) \bk + \bk 1\bk\text{Elektron} \bk + \bk 1\bk\text{Antineutrino}$ \\
\bk\bk Der Strahler wird beim Zerfall ionisiert

\vspace*{3mm}
{\bf $\gamma$-Strahlung:} \bk (Elektromagnetische Welle, $f \approx \unit[10^{20}]{Hz}$)

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Massendefekt}
\small
Die Einzelteile einen Atoms sind in Summe schwerer als das ``zusammengesetzte''
Atom.  Die Differenzmasse wird beim bilden eines Atoms in Energie umgesetzt
bzw. muss beim Zerlegen des Atoms als Energie zugef\"uhrt werden.

\bk\bk Energieerhaltungssatz + Massenerhaltungssatz \\
\bk\bk\bk\bk\bk\bk\bk\bk\bk\bk $\Rightarrow$ Energiemasseerhaltungssatz ($E = mc^2$)

\bk\bk\bk\bk\bk\bk Bindungsenergie = $\nicefrac{\Delta m}{\text{Nuklid}}$ = immer negativ

Bei Fe (Eisen) ist die Bindungsenergie am kleinsten (negativsten).

\vspace*{-3mm}
Bei leichteren Elementen als Eisen: Energiegewinnung durch Fusion

\vspace*{-3mm}
Bei schwereren Elementen als Eisen: Energiegewinnung durch Fission
(Kernspaltung, Funktionsprinzip AKW)

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{W\"armeenergie, Temperatur \\ und Temperaturskalen}
Temperatur ($T$, $\left[T\right] = \unit{K} = \unit[1]{Kelvin}$) ist die
mittlere kinetische Energie der Teilchen.

\vspace*{-2mm}
W\"arme ($Q$) ist eine Energieform ($\left[Q\right] = \unit[1]{kcal} = \unit[4,18]{kJ}$).
Durch Zuf\"uhren von W\"armeenergie wird die Temperatur erh\"oht.

\vspace*{-5mm}
\begin{center}
\begin{tabular}{cc}
\large $ Q = m \cdot c \cdot \Delta T $ &
\scriptsize
\begin{tabular}{lrrr}
\bf Material & \bf $c$ in $\unitfrac{kcal}{kg \cdot \Delta K}$ & \bf $c$ in $\unitfrac{kJ}{kg \cdot \Delta K}$ \\
\hline
Wasser & 1 & 4,18 \\
Luft & 0,24 & 1,00 \\
Landmassen & $\approx 0,2$ & $\approx 0,8$ \\
Blei & 0,08 & 0,33
\end{tabular}
\end{tabular}
\end{center}

\vspace*{-3mm}
Achtung: $c$ ist nicht wirklich eine Materialkonstante sondern hat selbst eine Temperaturabh\"angigkeit.

\vspace*{-2mm}
Temperaturskalen: Celsius ($\unit{^{\circ}C}$), Kelvin ($\unit{K}$), Fahrenheit ($\unit{^{\circ}F}$)

\vspace*{-2mm}
\centering
$ \unit[273,15]{K} = \unit[0]{^{\circ}C} = \unit[32]{^{\circ}F}, \bk\bk\bk\bk
\unit[\Delta 1]{K} = \unit[\Delta 1]{^{\circ}C} = \unit[\Delta 1,8]{^{\circ}F} $
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Thermodynamische Zustandsgr\"ossen}
\vspace*{2mm}
{\bf Druck} \\
Formelzeichen: $p$, \bk Definition: $p = \frac{F}{A}$, \bk Einheit: $\unit{Pa} = \unitfrac[1]{N}{m^2}$

\vspace*{5mm}
{\bf Temperatur} \\
Formelzeichen: $T$, \bk Definition: $T = \frac{m \cdot c}{Q}$, \bk Einheit: $\unit{K}$

\vspace*{5mm}
{\bf Volumen} \\
Formelzeichen: $V$, \bk Definition: $V = l \cdot b \cdot h$, \bk Einheit: $\unit{m^3}$
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Mechanisches W\"arme\"aquivalent}

\begin{align*}
E &= m \cdot g \cdot h         & \left[E\right] &= \unit{J} \\
Q &= m \cdot c \cdot \Delta T  & \left[Q\right] &= \unit{kcal}
\end{align*}

\begin{align*}
\unit[1]{kcal} &= \unit[4,18]{kJ} = \unit[4\,180]{J} \\
\unit[1]{kcal} &\mathrel{\hat{=}} \unit[1]{kg} \mbox{ Wasser um } \unit[1]{^{\circ}C} \mbox{ erw\"armen } \\
\unit[1]{kcal} &\mathrel{\hat{=}} \unit[1]{kg} \mbox{ um } \unit[427]{m} \mbox{ heben } \\
\unit[1]{kcal} &\mathrel{\hat{=}} \unit[1]{kg} \mbox{ auf } \unitfrac[329]{km}{h} \mbox{ beschleunigen }
\end{align*}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Haupts\"atze der Thermodynamik}
\small
{\bf 0. Hauptsatz:} \\
Die Temperaturen zweier miteinander verbundener Systeme gleichen sich aus.

{\bf 1. Hauptsatz:} \\
W\"arme ist eine Energieform. Der Energieerhaltungssatz ist auch f\"ur
W\"arme\-energie g\"ultig.

{\bf 2. Hauptsatz:} \\
Thermische Energie ist nicht in beliebigem Ma\ss{}e in andere Energiearten umwandelbar.

{\bf 3. Hauptsatz:} \\
Der absolute Nullpunkt der Temperatur ($\unit[0]{K} \approx \unit[-273,15]{^{\circ}C}$) ist unerreichbar.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Gasgleichung}

\centering

\vspace*{-3mm}
{\huge $ p V = n R T $}

\vspace*{-1mm}
\begin{tabular}{lllll}
$p$ & Druck & \hspace*{1cm} & $V$ & Volumen \\
$n$ & Stoffmenge & & $T$ & Temperatur \\
$R$ & \multicolumn{4}{l}{allgemeine Gaskonstante ($R \approx \unitfrac[8\,314]{J}{mol \cdot K}$)}
\end{tabular}

\vspace*{+2mm}
\begin{tabular}{cc}
\bf Boyle-Mariottesches Gesetz & \bf Gay-Lussacsches Gesetz \\
$T,n = \text{const} \Rightarrow p V = \text{const}$ & $V,n = \text{const} \Rightarrow \frac{p}{T} = \text{const}$ \\
\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$p$};
	\draw[->] (0,0) -- (0,+1) node[above] {$V$};
	\draw[domain=0.2:+1.8] plot function{0.2/x};
\end{tikzpicture} &
\begin{tikzpicture}
	\draw[->] (0,0) -- (+2,0) node[right] {$p$};
	\draw[->] (0,0) -- (0,+1) node[above] {$T$};
	\draw[domain=0:+1.8] plot function{x/2};
\end{tikzpicture}
\end{tabular}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{W\"arme\"ubertragung}

\vspace*{-2mm}
{\bf W\"armeleitung:} \\
W\"arme\"ubertragung in Festk\"orpern bzw. ruhenden Fl\"ussigkeiten und Gasen.
Die W\"armeleitf\"ahigkeit $k$ ($\left[k\right] = \unitfrac{W}{m \cdot K}$) ist eine
temperaturabh\"angige Materialkonstante.

\vspace*{+2mm}
{\bf W\"armestr\"omung:} \\
W\"arme\"ubertragung in bewegten Fl\"ussigkeiten und Gasen. \\
Die W\"arme wird mit der Materie mitbewegt.

\vspace*{+2mm}
{\bf W\"armestrahlung:} \\
W\"arme\"ubertragung durch eine elektromagnetische Welle.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Phasen\"uberg\"ange (Aggregatzust\"ande)}

\centering

\begin{tikzpicture}[bb/.style={rectangle,minimum size=6mm,rounded corners=3mm,very thick,
                draw=black!50,top color=white,bottom color=black!20,font=\ttfamily},yscale=0.8]
	\node (f) [bb] {fest};
	\node (l) [bb,right of=f,node distance=40mm] {fl\"ussig};
	\node (g) [bb,right of=l,node distance=40mm] {gasf\"ormig};

	\draw[<-] ($(f.north) + (-2mm,0)$) -- ++(0,10mm) -- node [above] {verfestigen / resublimieren} ($ (g.north) + (+2mm,10mm) $) -- ++(0,-10mm);
	\draw[<-] ($(f.north) + (+2mm,0)$) -- ++(0,3mm) -- node [above] {erstarren} ($ (l.north) + (-2mm,3mm) $) -- ++(0,-3mm);
	\draw[<-] ($(l.north) + (+2mm,0)$) -- ++(0,3mm) -- node [above] {kondensieren} ($ (g.north) + (-2mm,3mm) $) -- ++(0,-3mm);

	\draw[->] ($(f.south) + (-2mm,0)$) -- ++(0,-10mm) -- node [below] {sublimieren} ($ (g.south) + (+2mm,-10mm) $) -- ++(0,+10mm);
	\draw[->] ($(f.south) + (+2mm,0)$) -- ++(0,-3mm) -- node [below] {schmelzen} ($ (l.south) + (-2mm,-3mm) $) -- ++(0,+3mm);
	\draw[->] ($(l.south) + (+2mm,0)$) -- ++(0,-3mm) -- node [below] {sieden} ($ (g.south) + (-2mm,-3mm) $) -- ++(0,+3mm);
\end{tikzpicture}

\begin{tabular}{cp{5cm}}
\begin{tikzpicture}
	\draw[->] (0,0) -- (3,0) node [above] {$Q$};
	\draw[->] (0,0) -- (0,1.5) node [right] {$T$};

	\draw[red] (0,0) -- node [right,black] {\tiny fest} (0.5,0.5) --
		(1.0,0.5) -- node [right,black] {\tiny fl\"ussig} (1.5,1.0) --
		(2.0,1.0) -- node [right,black] {\tiny gasf\"ormig} (2.5,1.5);

	\draw[gray] (1.1,0.5) -- (1.5,0.5) node [right] {\tiny 1};
	\draw[gray] (2.1,1.0) -- (2.5,1.0) node [right] {\tiny 2};
\end{tikzpicture} &
	\small \vspace*{-1.7cm}
	An den Phasen\"uberg\"angen m\"ussen zus\"atzlich zu $Q = m c \Delta T$ noch die
	Schmelzw\"arme (1) und die Verdampfungsw\"arme (2) zum weiteren Erh\"ohen der
	Temperatur zugef\"uhrt werden.
\end{tabular}

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Druck-Temperatur-Phasendiagramm}
Das Druck-Temperatur-Phasendiagramm zeigt f\"ur einen Stoff die Phase in
Abh\"anigkeit von Druck und Temperatur:

\vspace*{-2mm}
\begin{tabular}{lp{3.6cm}}
\begin{tikzpicture}
	\draw (0,0) arc (-60:-45:5) coordinate (tp);
	\draw (tp) arc (-30:-15:8) coordinate (m1);
	\draw (tp) arc (-80:-50:6) coordinate (kp);

	\fill[red!10] let \p1 = (m1), \p2 = (kp) in (tp) -- (kp) -- (\x2,\y1) -- (m1);
	\fill[blue!10] let \p1 = (m1) in (0,0) -- (tp) -- (m1) -- (0,\y1);
	\fill[green!10] let \p1 = (kp), \p2 = (m1) in (0,0) -- (tp) -- (kp) -- (\x1,\y2) -- (4.7,\y2) -- (4.7,0);

	% redraw and fill the arcs
	\draw[fill=blue!10] (0,0) arc (-60:-45:5);
	\draw[fill=blue!10] (tp) arc (-30:-15:8);
	\draw [->] (tp) arc (-30:-14:8);
	\draw[fill=red!10] (tp) arc (-80:-50:6);

	\draw[fill=black] (tp) circle (1pt) node [below right] {\scriptsize Tripelpunkt};
	\draw[fill=black] (kp) circle (1pt) node [above left] {\scriptsize kritischer Punkt};

	\draw (0.7,1.5) node {fest};
	\draw (2.3,1.7) node {fl\"ussig};
	\draw (3.5,0.7) node {gasf\"ormig};

	\draw[->] (0,0) -- (5,0) node [above] {$T$};
	\draw[->] (0,0) -- (0,3) node [right] {$p$};
\end{tikzpicture} &
\vspace*{-2.5cm}
Bei Temperaturen unterhalb des Tripelpunktes gibt es keine fl\"ussige Phase.
Der Stoff sublimiert und resublimiert im Phasen\"ubergang. 
\end{tabular}

\vspace*{-1mm}
Bei Temperaturen jenseits des kritischen Punkts (\"uber der kritischen Temperatur)
kann ein Gas durch Druck nicht mehr verfl\"ussigt werden.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Luftfeuchtigkeit}
\centering

\vspace*{-3mm}
Einheit der absoluten Luftfeuchtigkeit: $\unitfrac{g}{m^3}$

\vspace{1mm}
Die S\"attigungsmenge der Luftfeuchtigkeit ist temperaturabh\"angig: \\
\vspace{2mm}
\begin{tikzpicture}
	\draw[->] (0,0) -- (3,0) node [above] {$T$};
	\draw[->] (0,0) -- (0,2) node [right] {$\unitfrac{m}{V}$};
	\draw[domain=0.1:2.9] (0,0) plot function{0.2 + x*x*0.2};
\end{tikzpicture}

\vspace{1mm}
relative Luftfeuchtigkeit = \% der S\"attigungsmenge
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{W\"armestrahlung}
Jeder Gegenstand strahlt in Abh\"angigkeit von seiner Temperatur
elektromagnetische Wellen ab.

\vspace{0.5mm}
Wird ein Gegenstand heisser so nimmt die ausgesendete Strahlungsenergie mit der
4. Potenz der Temperatur zu ($E \sim T^4$). Mit einer Erh\"ohung der Temperatur
wird auch das Strahlungsmaximum ins kurzwelligere Spektrum verschoben.

\vspace{0.5mm}
Beispiel: Eine Gl\"uhwendel leuchtet bei geringer Temperatur r\"otlich-orange
und bei hoher Temperatur weiss.

\vspace{0.5mm}
Jeder K\"orper absorbiert W\"armestrahlung im gleichem Ma\ss{}e in dem er
W\"armestrahlung aussendet. (0. Hauptsatz der Thermodynamik)
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Elektrische Ladungen}
Es gibt positive ($+$) und negative ($-$) Ladungen.

Elektrische Ladung ist immer an Materie gebunden.

Negative Elementarladung: Elektron ($\text{e}^-$)

Positive Elementarladung: Proton ($\text{p}^+$)

Positive und negative Ladung heben sich gegenseitig auf.

Ladungen k\"onnen nicht erzeugt sondern nur getrennt werden.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Coulombkraft}
Gleichnamige Ladungen ($+$/$+$ bzw. $-$/$-$) stossen sich ab.

Ungleichnamige Ladungen ($+$/$-$) ziehen sich an.

Diese Kraft ist die Coulombkraft $F_C$:

$$ F_C = k_C \cdot \frac{Q_1 \cdot Q_2}{r^2}, \bk \bk k_C = \frac{1}{4 \pi \epsilon_0} \approx \unitfrac[8,99 \cdot 10^9]{V \cdot m}{A \cdot s} $$

Formelzeichen der Ladung: $Q$

Einheit der Ladung: $\left[Q\right] = \unit{C} = \unit{A \cdot s} = \unit[1]{Coulomb}$
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Elektrostatisches Feld}
Jede Ladung $Q$ verursacht ein elektrostatisches Feld
\hbox{$\overrightarrow{E} = k_C \cdot \frac{Q}{r^2}$}.

Das elektrostatische Feld ist ein Quellenfeld mit positiven Ladungen als
Quellen und negativen Ladungen als Senken.

Zur Veranschaulichung dienen Feldlinien:
\vspace*{-2mm}\begin{itemize}
\item Die Feldlinien stehen normal auf die Ladungen
\item Dichte der Feldlinien $\hat=$ St\"arke des Feldes
\end{itemize}

Kraft $F_{e^-}$ auf ein $e^-$ im E-Feld:
$$ \overrightarrow{E} = k_C \cdot \frac{Q}{r^2}, \bk F_C = k_C \cdot \frac{Q_1 \cdot Q_2}{r^2}
\bk\bk \Rightarrow \bk\bk F_{e^-} = \overrightarrow{E} \cdot e^- $$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Strom}

\centering
\vspace*{8mm}

Strom $I$ = bewegte Ladung $\frac{Q}{t}$
$$ I = \frac{Q}{t}, \bk\bk \left[I\right] = \unit{A} = \unitfrac{C}{s} = \unit[1]{Ampere} $$

Gleichstrom = geichf\"ormig bewegte Ladung

Wechselstrom = ungeichf\"ormig bewegte Ladung
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Elektrischer Widerstand und \\ elektrische Leistung}
Der elektrische Widerstand $R$ bestimmt wieviel Leistung notwendig ist damit
ein bestimmter elektrischer Strom flie\ss{}t:

$$ R = \frac{P}{I^2}, \bk R = \frac{\rho l}{A}, \bk\bk \left[R\right] = \unit{\Omega} = \unit[1]{Ohm} $$

$\rho$ = spezifischer Widerstand = eine Materialkonstante

$l$ = L\"ange des Leiters \\
$A$ = Querschnitt des Leiters

\vspace{2mm}
Kehrwert des Widerstands = Leitwert $G$, \\ $\left[G\right] = \unitfrac{1}{\Omega} = \unit{S} = \unit[1]{Siemens}$
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Ohmsches Gesetz und \\ elektrische Spannung}
\vspace*{5mm}

Die Spannung $U$ ist jene physikalische Gr\"o\ss{}e die den Strom gegen den
Widerstand durch den Leiter ``treibt''.

Das Ohmsche Gesetzt beschreibt den Zusammenhang zwischen Spannung, Widerstand und Strom:

$$ U = R \cdot I, \bk\bk \left[U\right] = \unit{V} = \unit{\Omega \cdot A} = \unit[1]{Volt} $$
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Kirchhoffsche Gesetze}
\vspace*{10mm}
{\bf Knotenregel, Kirchhoffsches Stromgesetz:} \\
\bk Die Summe aller Str\"ome in einem Knoten ist Null.

\vspace*{5mm}
{\bf Maschenregel, Kirschhoffsches Spannungsgesetz:} \\
\bk Die Summe aller \"uberwundenen Spannungen in einer Masche ist Null.
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Serien- und Parallelschaltung \\ von Widerst\"anden}
\vspace*{5mm}
{\bf Serienschaltung von Widerst\"anden (Spannungsteiler):} \\
\bk Die Spannung teilt sich proportional zu den Widerstandswerten \\
\bk Die Widerstandwerte addieren sich

\vspace*{5mm}
{\bf Parallelschaltung von Widerst\"anden (Stromteiler):} \\
\bk Der Strom teilt sich proportional zu den Leitwerten \\
\bk Die Leitwerte addieren sich
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Sicherheitseinrichtungen \\ bei Elektrizit\"at}
\vspace*{5mm}
{\bf FI (Fehlerstromschutzschalter):} \\
Misst Stromdifferenz zwischen Zu- und Ableitung. \\
L\"ost aus wenn Stromdifferenz zu gross wird (d.h. wenn der Strom woandershin fliesst).

\vspace*{5mm}
{\bf Sicherung:} \\
Misst den Strom und l\"ost aus, wenn der Strom einen festgelegten Schwellwert \"uberschreitet.

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Elektromagnetismus}
Eine bewegte Ladung (Strom) erzeugt ein Magnetfeld $\overrightarrow{B}$ \\
(Rechte-Hand-Regel, Rechtsschraubregel).

Ein ver\"anderliches $\overrightarrow{B}$-Feld erzeugt eine elektromagnetische Welle.

$\overrightarrow{B}$-Felder \"uben eine Kraft auf bewegte Ladungen aus (Lorentzkraft).

\vspace*{5mm}
Ruhende Ladungen erzeugen ein $\overrightarrow{E}$-Feld.

$\overrightarrow{E}$-Felder \"uben eine Kraft auf ruhende und bewegte Ladungen aus (Coulombkraft).
\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{karte}{Transformator}
Ein Transformator ist ein System aus 2 magnetisch gekoppelten Spulen mit
unterschiedlicher Windungszahl $N_1$, $N_2$.

Durch magnetische Induktion wird Leistung von der Prim\"arspule auf die
Sekund\"arspule \"ubertragen.

$$ \frac{N_1}{N_2} = \frac{U_1}{U_2} = \frac{I_2}{I_1}, \bk\bk P = U \cdot I = \text{const} $$

\end{karte}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}