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\begin{document}
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\heading{Train}
\auname{I.D. 064979768}
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{\bf The problem:}
An electric train is designed to use the magnetic field of the earth in order to move. Electric current flows from one side of the tracks to the other one, through a wire that goes between the train's wheels.
\begin{enumerate}
\item What current is needed in order to get a force of $10 \ kN$, if the perpendicular component of the field is $10 \ \mu T$, and the distance between the wheels is $3\ m$?
\item What is the power loss for every $1 \ \Omega$ of resistance?
\item Is it a realistic design of a train?
\end{enumerate}
\includegraphics{e_45_2_003_p0.jpg}
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\Dn
{\bf The solution:}
\Dn
1.
\begin{eqnarray}
F &=& 10 \ \mbox{kN} \\
B\hat x &=& 10 \ \mbox{$\mu$T} \\
l &=& 3 \ \mbox{m}
\end{eqnarray}
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since the field in constant:
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\begin{eqnarray}
\vec F = I\int d\vec l \times \vec B = I \left( \int d\vec l\right) \times \vec B = I(\vec r_2 - \vec r_1 ) \times \vec B
\end{eqnarray}
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We shall consider only the wire between the wheels, since the force along the train is canceled (opposite directions)
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\begin{eqnarray}
\vec r_2 - \vec r_1 &=& 3\hat y \\
|F| &=& |I(3\hat y) \times 10\cdot10^{-6} \hat x|
= 3 \cdot 10\cdot10^{ - 6} I = 10000 \ \mbox{N} \\
I &=& 3.33 \cdot 10^{8} \ \mbox{A}
\end{eqnarray}
2.
\begin{eqnarray}
R &=& 1\Omega \\
P &=& I^2 R = 1.11 \cdot 10^{17} \ \mbox{W}
\end{eqnarray}
3.
This kind of train is unreal because of the enormous loss of energy and it might be more dangerous to walk on the railway...
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\end{document}