diff --git a/v0.1/Formelsammlung.tex b/v0.1/Formelsammlung.tex
index ee893d858cea2fbee00d738c9fb30eb1f12fc984..100387798fca83fdf6dca110d95cf51f0efce7c5 100644
--- a/v0.1/Formelsammlung.tex
+++ b/v0.1/Formelsammlung.tex
@@ -1,4 +1,4 @@
-\chapter*{Formelsammlung}
+\chapter{Formelsammlung}
\begin{enumerate}[a)]
\item \textbf{Ladung/Ströme}
\begin{enumerate}[i)]
diff --git a/v0.1/ac_circuit.tex b/v0.1/ac_circuit.tex
index 41874b40139b4049150842633328a60af76e314d..ca140e10b97efab490f47e096170718bb14da668 100644
--- a/v0.1/ac_circuit.tex
+++ b/v0.1/ac_circuit.tex
@@ -7,7 +7,7 @@
Die Kapazität \index{Kapazität} $C$ (abhängig von der Geometrie, in
Einheiten Farad\index{Farad}) gibt das Verhältnis von Ladung $Q$ zu Spannung $U$ an
(z.B. Plattenkondensator mit Fläche $A$ und Plattenabstand $d$: $C=\epsilon_0\epsilon_r \frac{A}{d}$). Damit ist die Ladung $Q$ bei
- einer Spannung $U$ gegeben als $$Q=CU$$ und die Energie im elektrischen Feld $W_E=\frac{Q}{2C^2}$.
+ einer Spannung $U$ gegeben als $$Q=CU$$ und die Energie im elektrischen Feld des Kondensators $W_E=\frac{Q}{2C^2}$.
\item \textit{Induktivität}:\index{Induktivität} Die Induktivität $L$ (abhängig von der Geometrie, Einheit Henry\index{Henry}) gibt das Verhältnis von magnetischen Fluss $\phi_m$ zur
Stromstärke $I$ einer Spule an (z.B. Solenoidspule mit $N$ Windungen, Querschnittsfläche $A$ und
der L\"ange $\ell$ hat $L=\mu_0\mu_r\frac{AN^2}{\ell}$). Der magnetische Fluss $\phi_m$ ergibt sich bei einem Strom $I$ zu $$\phi_m = LI$$ und die induzierte Spannung $U_\mathrm{ind}$: $\dot \phi_m=L\dot I=-U_\mathrm{ind}$
diff --git a/v0.1/appendix.tex b/v0.1/appendix.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e1976b272b722104ace4058ddc6f7f1abfedc985
--- /dev/null
+++ b/v0.1/appendix.tex
@@ -0,0 +1,2 @@
+\chapter{Elektrisches Feld: Geladene Kugelfläche ohne Gauss'sches Gesetz}
+\end{appendices}
diff --git a/v0.1/chargedshell.tex b/v0.1/chargedshell.tex
new file mode 100644
index 0000000000000000000000000000000000000000..26ead0198220231732f23a674e02b43f8cc2a914
--- /dev/null
+++ b/v0.1/chargedshell.tex
@@ -0,0 +1,57 @@
+\chapter{Elektrisches Feld einer geladenen Kugelfläche: ohne Gauss'sches Gesetz}
+\label{appendix:chargedshell}
+Im Abschnitt \ref{section:gauss} haben wir das elektrische Feld in der Nähe einer geladenen Kugelschale
+mit dem Gauß'schen Gesetz bestimmt. Hier demonstrieren wir, dass das Ergebnis auch ohne das Gauß'sche Gesetz (allerdings mit mehr Aufwand) gefunden werden kann.
+
+Wir bestimmen das Feld an einem Punkt $P$, der sich im Abstand $r$ von dem Mittelpunkt $O$ einer Kugelschale mit Radius $R$ befindet. Wir nehmen zunächst an,
+dass der Punkt $P$ außerhalb der Kugelschale liegt, also $r>R$ gilt. Wir wählen den Ursprung des Koordinatensystems den Punkt $O$ und legen die $z$-Achse entlang
+der Richtung $\bar{OP}$, so dass $\mathbf{r} = r\mathbf{e}_z$.
+Auf der dünnen Kugelschale befindet sich die Gesamtladung $Q$ über die gesamte Fläche $4\pi R^2$ gleichförmig verteilt (siehe Abbildung~\ref{fig:spherecharge}).
+
+Um das elektrische Feld am Punkt $P$ auszuwerten, müssen wir über sämtliche Ladungen auf der Oberfläche integrieren. Aufgrund der Symmetrie ist
+das resultierende Feld $\mathbf{E}_P = E_P~\mathbf{e}_z$ (ähnlich wie beim Ring im Beispiel im Abschnitt~\ref{sec:efield}): Die Feldkomponenten in der Richtung senkrecht
+zur $z$-Richtung addieren sich zu Null, wenn wir über die Kugelsegmente integrieren.
+
+\[
+ E_P = \frac{1}{4\pi\epsilon_0}~\int \mathrm{d}A \frac{Q}{4\pi R^2} \frac{(\mathbf{r}-R\mathbf{e}_{r'})\cdot \mathbf{e}_z}{|\mathbf{r}-R\mathbf{e}_{r'}|^3},
+\]
+wobei das Flächenelement $\mathrm{d}A = R^2\mathrm{d}\phi \sin\vartheta\mathrm{d}\vartheta$ und $\vartheta$ der Winkel zwischen $\mathbf{e}_{r'}$ und $\mathbf{e}_z$ ist.
+Wir erhalten nach Integration über $\phi$ und Ausmultiplizieren von Zähler und Nenned:
+\[
+E_P = \frac{Q}{8\pi\epsilon_0} \int\limits_0^\pi \mathrm{d}\vartheta \sin\vartheta \frac{r-R\cos\vartheta}{(r^2+R^2-2Rr\cos\vartheta)^{3/2}}.
+\]
+Wir substituieren $z=r^2+R^2-2Rr\cos\vartheta$, so dass $\mathrm{d}z=2Rr\sin\vartheta~\mathrm{d}\vartheta$ und
+$R\cos\vartheta = (r^2+R^2-z)/2r$:
+\[
+E_P = \frac{Q}{32\pi\epsilon_0 Rr^2} \int\limits_{(R-r)^2}^{(R+r)^2} \mathrm{d}z \frac{r^2-R^2+z}{z^{3/2}}.
+\]
+Wir werten die Stammfunktionen an den Grenzen aus:
+\[
+E_P = \frac{Q}{32\pi\epsilon_0 Rr^2}\left. \left( 2 z^{1/2} - 2\frac{r^2-R^2}{z^{1/2}} \right) \right|_{(R-r)^2}^{(R+r)^2}
+\]
+so dass wir mit
+\[
+ E_P = \frac{Q}{16\epsilon_0 Rr^2} \left(R+r - |R-r| - \left(\frac{(r-R)(r+R)}{R+r} - \frac{(r-R)(r+R)}{|R-r|}\right)\right)
+\]
+eine Fallunterscheidung machen müssen zwischen $r>R$ und $r<R$.
+Für $r>R$ ist $|R-r| = r-R$, so dass
+\[
+E_P (r>R) = \frac{Q}{16\pi \epsilon_0 Rr^2} 4R = \frac{Q}{4\pi \epsilon_0 r^2},
+\]
+und für $r<R$ mit $|R-r| = R-r$:
+\[
+E_P(r<R) = \frac{Q}{16\pi\epsilon_0 Rr^2} (R+r-R+r - (r-R +r+R)) = 0.
+\]
+Damit ist das Ergebnis aus dem Abschnitt~\ref{sec:anwendunggauss} bestätigt: Außerhalb einer geladenen Kugelschale mit Gesamtladung $Q$ fällt das
+elektrische Feld genauso ab, als ob sich im Mittelpunkt eine Punktladung $Q$ befinden würde. Innerhalb der Kugelschale verschwindet das Feld.
+
+\begin{figure}
+\parbox{0.7\linewidth}{
+\includegraphics[width=1.\linewidth]{figures/drawing_sphere.png}}
+\parbox{0.28\linewidth}{
+\caption{Zur Bestimmung des elektrischen Feldes im Punkt $P$ wird über die geladene Kugelfläche mit Radius $a$ integriert.
+\label{fig:spherecharge}}
+}
+\end{figure}
+
+
diff --git a/v0.1/elektrostatik.tex b/v0.1/elektrostatik.tex
index 3b5e2da4115b36de195c2e2c9a3157555f79e1d2..f23c8cf9e245f567c5f34838102a3709022c44ad 100644
--- a/v0.1/elektrostatik.tex
+++ b/v0.1/elektrostatik.tex
@@ -200,6 +200,7 @@ Das Superpositionsprinzip lässt sich in dieser Weise auch durch ein Volumeninte
\section{Das elektrische Feld}
+\label{sec:efield}
Wir haben mit dem Coulomb-Gesetz \ref{eq:coulomb1} ein neues Kraftgesetz gefunden
(vgl. Gravitationskraft, s. Physik I).
Ein Kraftgesetz erlaubt es uns, die Kräfte zwischen zwei Körpern anzugeben. Wir hatten hierbei durch die Notation $\mathbf{F}_{1,2}$ angedeutet, dass die Kraft auf die Ladung $Q_2$ ursächlich durch
@@ -238,8 +239,8 @@ beliebigen Ladungsverteilung
\lambda = \frac{Q}{2\pi a}.
\]
Wir teilen das elektrische Feld am Punkt
-$P$ im Abstand $x$ in zwei Komponenten auf $E_x=|\mathbf{E}~\cos\theta$ und $E_\perp$. Wenn wir entlang dem Ring
-integrieren bekommen wir für
+$P$ im Abstand $x$ in zwei Komponenten auf $E_x=|\mathbf{E}|~\cos\theta$ und $E_\perp$. Wenn wir entlang dem Ring
+integrieren, bekommen wir für
\[
E_\perp = \frac{1}{4\pi\epsilon_0}\int d\ell \frac{\lambda}{r^2}=0
\]
@@ -721,7 +722,7 @@ ist proportional zur Ladung in dem umschlossenen Volumen.}
\[
\mathbf{E}(R) = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^2} \mathbf{e}_r.
\]
- Das Flächenelement in Kugelkoordinaten $d\mathbf{A} = R^2\sin\vartheta d\theta d\phi \mathbf{e}_r$ ist nach Konvention für geschlossene Flächen so ausgerichtet, dass der Normalenvektor
+ Das Flächenelement in Kugelkoordinaten $d\mathbf{A} = R^2\sin\vartheta d\vartheta d\phi \mathbf{e}_r$ ist nach Konvention für geschlossene Flächen so ausgerichtet, dass der Normalenvektor
nach außen zeigt ($\mathbf{n}=\mathbf{e}_r$). Wir können das Integral sofort auswerten:
\[
\Phi_E = \oint d\mathbf{A}\cdot \mathbf{E} = R^2 \frac{Q}{4\pi\epsilon_0 R^2} \underbrace{\int\limits_{0}^{2\pi}d\phi\int\limits_{0}^\pi d\vartheta \sin\vartheta}_{=2\pi\cdot 2=4\pi}=\frac{Q}{\epsilon_0}
@@ -758,6 +759,7 @@ die wir integrieren.
\subsection{Anwendungen Gauß'sches Gesetz}
+\label{sec:anwendunggauss}
Das Gauß'sche Gesetz ermöglicht es prinzipiell, das elektrische Feld ohne genaue
Kenntnis der Ladungsverteilung zu bestimmen. Der elektrische Fluss bzw. die
Ladung innerhalb einer geschlossenen Fläche genügt, um in bestimmten Fällen
@@ -803,6 +805,22 @@ Insbesondere kann mit einer sinnvoll gewählten Oberfläche
\end{example}
+\begin{example}[Geladene Kugelschale]
+ Wir betrachten eine Kugelschale mit Radius $R$, die die Ladung $Q$ trägt. Wir berechnen analog zum vorigen Beispiel
+ den Fluß und setzen den Fluß gleich mit der eingeschlossenen Ladung:
+ \[
+ \oint \mathrm{d}\mathbf{A}\cdot E = 4\pi r^2 E = \frac{Q}{\epsilon_0},
+ \]
+ für $r>R$. Damit ergibt sich das Feld zu
+ \[
+ E(r) = \frac{Q}{4\pi\epsilon_0 r^2},
+ \]
+ also identisch zu dem Feld einer Punktladung im Mittelpunkt der Kugelschale. Für $r<R$ ist der Fluß null, weil sich innerhalb der Integrationsfläche
+ keine Ladungen befinden.
+Im Anhang \ref{appendix:chargedshell}, bestimmen wir zum Vergleich das elektrische Feld ohne
+Anwendung des Gauß'schen Gesetzes.
+\end{example}
+
\begin{example}[Homogen geladene unendlich ausgedehnte Platte]
Abbildung~\ref{fig:gauss12} rechts: Wir nehmen an, dass die Platte
mit einer konstanten Flächenladungsdichte $\sigma$ aufgeladen ist. Aufgrund
diff --git a/v0.1/figures/drawing_sphere.png b/v0.1/figures/drawing_sphere.png
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new file mode 100644
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diff --git a/v0.1/maxwell_gleichungen.tex b/v0.1/maxwell_gleichungen.tex
index 93d418e800fec1d3060941f136fdfc5efd372ba4..658ce1f2a2bebf566e7272e47fc0b44202bef7fa 100644
--- a/v0.1/maxwell_gleichungen.tex
+++ b/v0.1/maxwell_gleichungen.tex
@@ -13,7 +13,7 @@
\begin{equation}
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\end{equation}
-\textbf{ Amp\`ere-Maxwell-Gleichung} (siehe auch Abschnitt~\ref{section:maxwell-strom}):
+\textbf{Amp\`ere-Maxwell-Gleichung} (siehe auch Abschnitt~\ref{section:maxwell-strom}):
\begin{equation}
\nabla \times \mathbf{B} = \mu_0 \mathbf{j} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}.
\end{equation}
diff --git a/v0.1/skript.tex b/v0.1/skript.tex
index 5de08878c0109ffd0566b43ea705bcef806eb56e..a4f46567e940527de26ffc6cb945027ea98d59ad 100644
--- a/v0.1/skript.tex
+++ b/v0.1/skript.tex
@@ -2,6 +2,7 @@
\usepackage[bitstream-charter]{mathdesign}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
+\usepackage[toc,page]{appendix}
\usepackage{enumerate}
\usepackage{bm}
\usepackage{color}
@@ -144,6 +145,9 @@ Konsultieren Sie gegebenenfalls auch die empfohlenen Lehrb\"ucher.\\
\input{em_waves}
\input{geometrical_optics}
\input{wellenoptik}
-\input{Formelsammlung}
\printindex
+\begin{appendices}
+\input{Formelsammlung}
+\input{chargedshell}
+\end{appendices}
\end{document}