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\date{\today{}}
\author{Thomas Kuster}
\title{UWIS, Physik II, Lösung Serie 5}

\begin{document}
\markright{UWIS, Physik II, Lösung Serie 5}
\pagestyle{myheadings}

\maketitle

\section{Metallflugzeug}

\begin{eqnarray*}
s_{AB} & = & 20 \mbox{m Spannweite (von A nach B)}\\
v & = & 50 \mbox{ms$^{-1}$ Geschwindigkeit}\\
B & = & 43 \mbox{$\mu$T} = 4.3 \cdot 10^{-5} \mbox{T Magnetfeld senkrecht zur Flugrichtung}\\
U_{AB} & = &  \mbox{V induzierte Spannung von A nach B}
\end{eqnarray*}

\begin{eqnarray}
F_L & = & q v B \sin{\varphi} \label{F_L}\\
F_E & = & E q \label{F_E}\\
\end{eqnarray}

\begin{eqnarray}
0 = F_L + F_E \overset{\mbox{(\ref{F_L}) \& (\ref{F_E})}}{=} q v B \underbrace{\sin{\varphi}}_{v \perp B \Rightarrow 1} + E q \Rightarrow v B + E = 0 \Rightarrow E =-v B \label{E=-vB}
\end{eqnarray}
$$
U_{AB} = E s_{AB} \overset{\mbox{(\ref{E=-vB})}}{=} -v B s_{AB} = -50 \mbox{ms$^{-1}$ } 4.3 \cdot 10^{-5} \mbox{T } 20 \mbox{m} =
$$
$$
= -0.043 \mbox{m$^2$s$^{-1}$Vsm$^{-2}$} \Rightarrow \abs{U_{AB}}= 0.043\mbox{V}
$$

\section{Rechteckige Spule}

\begin{eqnarray*}
a & = & 0.4 \mbox{m Seitenlänge der Spule}\\
b & = & 0.3 \mbox{m Seitenlänge der Spule}\\
\vec{A} & = & \begin{pmatrix}0.4\\0\\0\end{pmatrix}\mbox{m} \times \begin{pmatrix}0\\0.3\\0\end{pmatrix}\mbox{m} = \begin{pmatrix}0\\0\\0.12\end{pmatrix}\mbox{m$^2$ Flächennormale}\\
N & = & 12 \mbox{ Windungen der Spule}\\
I & = & 3 \mbox{A Strom in der Spule}\\
\vec{B} & = & \begin{pmatrix}0.3\\0\\0.4\end{pmatrix} \mbox{T homogenes Magnetfeld}
\end{eqnarray*}

\begin{eqnarray*}
\vec{m} & = & N I \vec{A} \label{m}\\
\vec{M} & = & \vec{m} \times \vec{B} \label{mB}\\
\end{eqnarray*}

\subsection{Magnetisches Moment der Spule ($\vec{m}$)}
$$
\mbox{(\ref{m})} \Rightarrow \vec{m} = N I \vec{A} = 12 \cdot 3\mbox{A} \begin{pmatrix}0\\0\\0.12\end{pmatrix}\mbox{m$^2$} = \begin{pmatrix}0\\0\\4.32\end{pmatrix} \mbox{Am$^2$}
$$

\subsection{Drehmoment der Spule ($\vec{M}$)}

$$
\mbox{(\ref{m}) \& (\ref{mB})} \Rightarrow \vec{M} = \vec{m} \times \vec{B} = \begin{pmatrix}0\\0\\4.32\end{pmatrix} \mbox{Am$^2$} \times \begin{pmatrix}0.3\\0\\0.4\end{pmatrix} \mbox{T} =
$$
$$
\begin{pmatrix}0\\1.296\\0\end{pmatrix}\mbox{Cs$^{-1}$ m$^2$ JC$^{-1}$sm$^{-2}$} = \begin{pmatrix}0\\1.296\\0\end{pmatrix}\mbox{J} = \begin{pmatrix}0\\1.296\\0\end{pmatrix}\mbox{Nm}
$$

\section{Ferromagnetismus}
\begin{eqnarray*}
\frac{n}{l} & = & 50 \mbox{Windungen cm$^{-1}$} = 5000 \mbox{Windungen m$^{-1}$}\\
I & = & 2 \mbox{A Strom in der Spule}\\
B & = & 1.72 \mbox{T Magnetfeld total}\\
\mu_0 & = & 4\pi \cdot 10^{-7} \mbox{VsA$^{-1}$m$^{-1}$}
\end{eqnarray*}

\subsection{Angelegtes Feld ($B_0$)}
$$
B_0 \approx \frac{\mu_0 n I}{l} = 4\pi \cdot 10^{-7}\mbox{VsA$^{-1}$m$^{-1}$} 5000\mbox{m$^{-1}$} 2\mbox{A} = 0.0126\mbox{T}
$$

\subsection{Magnetisierung ($M$)}
$$
B = B_0 + \mu_0 M \Rightarrow M = \frac{B - B_0}{\mu_0} = \frac{1.72 \mbox{T} - 0.0126\mbox{T}}{4\pi \cdot 10^{-7} \mbox{VsA$^{-1}$m$^{-1}$}} =
$$
$$
1358732 \mbox{Vsm$^{-2}$AmV$^{-1}$s$^{-1}$} \approx 1.359 \cdot 10^6\mbox{Am$^{-1}$}
$$

\subsection{Permeabilität ($\mu$)}
$$
B = \mu B_0 \Rightarrow \mu = \frac{B}{B_0} = \frac{1.72\mbox{T}}{0.0126\mbox{T}} \approx 137
$$

\section{Induktion}
\begin{eqnarray*}
B & = & 0.6 \mbox{T Magnetfeld}\\
v & = & 8 \mbox{ms$^{-1}$ Geschwindigkeit des Metallstab}\\
l & = & 0.15 \mbox{m Metallschienenabstand}\\
R & = & 25\Omega \mbox{ Widerstand}
\end{eqnarray*}

\subsection{Spannung ($U$)}
$$
U = -\frac{B \Delta A}{\Delta t} = -B \frac{l \Delta s}{\Delta t} = -B l v = - 0.6\mbox{T} \cdot 0.15\mbox{m} \cdot 8 \mbox{ms$^{-1}$} =
$$
$$
-0.72 \mbox{Vsm$^{-2}$m$^2$s$^{-1}$} = -0.72\mbox{V}
$$
\subsection{Strom ($I$)}
$$
U = R I \Rightarrow I = \frac{U}{R} = \frac{-0.72\mbox{V}}{25\Omega} = 0.0288\mbox{A} = 28.8\mbox{mA}
$$
\subsection{Kraft ($F_B$)}
$$
F = I l B \underbrace{\sin{\varphi}}_{=1} = 0.0288\mbox{A} \cdot 0.15\mbox{m} \cdot 0.6 \mbox{T} = 0.002592\mbox{AmT} =
$$
$$
0.002592\mbox{N} = 2.592\mbox{mN}
$$

\subsection{Leistung ($P$)}
$$
P = U I = 0.72\mbox{V} \cdot 0.0288 \mbox{A} = 0.020736 \mbox{JC$^{-1}$Cs$^{-1}$} = 20.7\mbox{mW}
$$

\end{document}
