ET2_Uebung_BEI/ET2_L_B12_A2.tex

\section{Drahtschleife}
Eine Drahtschleife $N=1$ wird von einem zeitlich veränderlichen Fluss durchsetzt.\\
$\Phi(t)=\Phi_0\cdot (1-e^{-t/T})$ mit $\Phi_0=60\cdot \power{10}{-6
}\,\volt\second$ und $T=1\,\milli\second$.
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Berechnen Sie die Spannung $u$ für $t=0{,}5\cdot T$!
\item Geben Sie die Polarität der Anschlussklemmen der Drahtschleife a-b für diesen Zeitpunkt an und begründen Sie ihre Angabe!
\end{enumerate}
\begin{align*}
	\begin{tikzpicture}[scale=1]
		\begin{scope}[>=latex,red,ultra thick,xshift=0,yshift=0] 			
			\draw (0,0) -- (0,1);
			\draw [dashed] (0,1) -- (0,2);				
		\end{scope}
		\begin{scope}[>=latex,blue,very thick,xshift=0,yshift=2cm] 
			\draw (0,0) ellipse (2cm and 1cm);		
		\end{scope}
		\begin{scope}[>=latex,blue,very thick,xshift=1.9cm,yshift=1.75cm] 
			\filldraw [black!0!] (0,0.02) rectangle (0.2,0.48);
			\draw [fill] (1pt,0) -- (1,0) circle (2pt) node [right] {b};
			\draw [fill] (1pt,0.5cm) -- (1,0.5) circle (2pt) node [right] {a}; 					
			\draw node at (1.5,0.25)[right] {$u$};	
		\end{scope}
		\begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] 			
			\draw [->] (0,2) -- (0,4) node [above] {$\Phi(t)$};				
		\end{scope}
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
\begin{align}
\intertext{Formeln:}
u&=-N\cdot \frac{d\Phi}{dt}
\end{align}
\begin{align*}
\intertext{Berechnung:}
\intertext{a) Spannung $u$ für $t=0{,}5\cdot T$}
u&=-N\cdot \frac{d\Phi}{dt} \qquad \text{mit $N=1$ }\\
&=-\frac{d\left(\Phi_0\cdot \left(1-e^{-\frac{t}{T}}\right)\right)}{dt}\\
&=-\Phi_0\cdot \frac{d}{dt} \left(1-e^{-\frac{t}{T}}\right)\\
&=-\Phi_0\cdot \Bigg[-e^{-\frac{t}{T}}\cdot \hspace{-.7cm}\underbrace{ \left(-\frac{1}{T}\right)}_{\mathrm{Nachdifferenzieren}}\hspace{-.5cm}\Bigg]\\
&\text{mit $t=0{,}5\cdot T\qquad T=1\,\milli\second$}\\
u&=-60\cdot \power{10}{-6}\,\volt\second\cdot \left[-e^{-0{,}5}\cdot \left(-\frac{1}{1\,\milli\second}\right)\right]=\uuline{-36{,}39\,\milli\volt}
\end{align*}
\begin{align*}
\intertext{b) Polarität der Spannung $u$ für $t=0{,}5\cdot T$}
\frac{d\Phi}{dt}&>0 \Rightarrow \text{Linke-Hand für Stromrichtung}\\
\end{align*}
\begin{align*}
\begin{tikzpicture}[scale=1]
\foreach \xs in {0} {	% Enter Start value of x label
\foreach \ys in {0} {	% Enter Start value of y label
\foreach \ii in {10} {	% Enter Number of Decades in x
\foreach \jj in {6} {	% Enter Number of Decades in y
\foreach \xe in {5} {	% Enter End value of x label
\foreach \ye in {60} {	% Enter End value of y label
		\foreach \i in {1,2,...,\ii} {
			\foreach \j in {1,2,...,\jj} {			
}}% End Log Grid
\draw[black!80!] (0,0) grid (\ii,\jj); % Draw Linear grid
\draw [<->,thick] (0,\jj+.2) node (yaxis) [above] {$\Phi\,[\micro\volt\second]$} |- (\ii+.2,0) node (xaxis) [right] {$t\,[\milli\second]$}; % Draw axes
\draw [thick] (-1.25,\jj+.2) node (yaxis) [above] {$u\,[\milli\volt]$} ; % Draw axes
\foreach \x in {\xs,.5,...,\xe}% x Axis Label:
\node [black,anchor=north] at(\x*2-\xs,0){$\x$};
\foreach \y in {\ys,10,...,\ye}% y Axis Label:
\node [red,anchor=east] at(0,\y/10-\ys){$\y$};
\foreach \y in {\ys,10,...,\ye}% y Axis Label:
\node [blue,anchor=east] at(-1,\y/10-\ys){$-\y$};
}}}}}}
\draw [black,thick,dashed] (0,0) -- (2,6) -- (2,0.3);
\draw [blue,thick,dashed] (1,3.64) -- (-.75,3.64) node [left]{$-36{,}39\,\milli\volt$};
\draw[color=red, very thick,domain=0:10]  plot[id=tau]  function{6*(1-exp(-x/2))};
\draw[color=blue, very thick,domain=0:10]  plot[id=tau]  function{6*(exp(-x/2))};
\draw [black]   node at (2,0)[above] {$T$};
\draw [red]   node at (2,3.5) [right]{$\Phi=\Phi_0\cdot \left(1-e^{-t/T}\right)$};
\draw [blue]   node at (2,2.5) [right]{$u=-\Phi_0\cdot \left(\frac{e^{-t/T}}{T}\right)$};
\draw node at (0,-1) [right]{Induzierte Spannung bei zunehmendem Fluss};
\end{tikzpicture}
\end{align*}
\begin{align*}
\intertext{\uline{Lenzsche Regel:}}
\text{Klemme a $\Rightarrow$ $+$}\\
\text{Klemme b $\Rightarrow$ $-$}\\
%F_{\sqcup\sqcap}\\ %-------------------Versuch mit Symbolen--------------
%\ding{43}\\
%\oplus\ominus\odot\\
%\circlearrowright\\
%\lightning\\
%\curvearrowright\\
%\leftthumbsup\\
%\SmallSquare\\
%\SmallTriangleUp\\ %-------------------Versuch mit Symbolen--------------
\end{align*}
\begin{align*}
	\begin{tikzpicture}[scale=1]
		\begin{scope}[>=latex,red,ultra thick,xshift=0,yshift=0] 			
			\draw (0,0) -- (0,1);
			\draw [dashed] (0,1) -- (0,2);
		\end{scope}
		\begin{scope}[>=latex,blue,very thick,xshift=0,yshift=2cm] 
			\draw (0,0) ellipse (2cm and 1cm);		
		\end{scope}
		\begin{scope}[>=latex,blue,very thick,xshift=1.9cm,yshift=1.75cm] 
			\filldraw [black!0!] (0,0.02) rectangle (0.2,0.48);
			\draw [fill] (1pt,0) -- (1,0) circle (2pt) node [below] {b $-$};
			\draw [fill] (1pt,0.5cm) -- (1,0.5) circle (2pt) node [above] {a $+$}; 					
			\draw node at (1.5,0.25)[right] {$u$};	
		\end{scope}
		\begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] 			
			\draw [->] (0,2) -- (0,4) node [above] {$\Phi(t)$};				
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=5cm,yshift=1.5cm,rotate=90] 			
			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};
			\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$U_{R}$}; 		
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=2cm]
			\draw [dashed] (0,.25)--(2,.5) (0,-.25)--(2,-.5);
			\draw [->,red] (0,.25) +(7.2:.75) -- +(7.2:1.25) node [above left]{$i$};	
		\end{scope}					
	\end{tikzpicture}
\end{align*}
\clearpage
}{}%