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