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\begin{document}
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\heading{resistors cube}
\auname{I.D. 066101528}
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{\bf The problem:}
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In the resistors cube you see in the sketch all resistors are equal and
their resistance is $R$.
\begin{enumerate}
\item Find the equivalent resistance between points a and b.
\item How does the equivalent resistance changes if we connect point 4 with point 5
with a resistor R ?
\end{enumerate}
\includegraphics[width=0.4\hsize]{e_42_1_024_p0.jpg}
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\Dn
{\bf The solution:}
\Dn
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1. We can easily see that the cube is symmetrical for a current to junction 4 and 5
therefore the potential is equal in those two points, same about points 3 and 6.
If the potential is equal it means that we can connect these two junctions into one and
create the electrical circuit you see in sketch 1.
\includegraphics[scale=0.75]{e_42_1_024_s_p1.jpg}
We simplify circuit 1 by connecting all the resistors in parallel into one resistor:
\[
\frac{1}{{R_t }} = \frac{1}{R} + \frac{1}{R} \Rightarrow R_t = \frac{R}{2}
\]
now we have a circuit like in sketch 2.
\includegraphics[scale=0.75]{e_42_1_024_s_p2.jpg}
We simplify circuit 2 by connecting all the resistors in row into one resistor:
\[
R_t = \frac{R}{2} + R + \frac{R}{2} = 2R
\]
now we have a circuit like in sketch 3.
\includegraphics[scale=0.75]{e_42_1_024_s_p3.jpg}
We simplify circuit 3 by connecting all the resistors in parallel into one resistor:
\[
\frac{1}{{R_t }} = \frac{2}{R} + \frac{1}{{2R}} \Rightarrow R_t = \frac{{2R}}{5}
\]
now we have a circuit like in sketch 4.
\includegraphics[scale=0.75]{e_42_1_024_s_p4.jpg}
We simplify circuit 4 by connecting all the resistors in row into one resistor:
\[
R_t = \frac{R}{2} + \frac{{2R}}{5} + \frac{R}{2} = \frac{{7R}}{5}
\]
now we have a circuit like in sketch 5.
\includegraphics[scale=0.75]{e_42_1_024_s_p5.jpg}
We simplify circuit 5 by connecting all resistors in parallel into one resistor:
\[
\frac{1}{{R_t }} = \frac{1}{R} + \frac{5}{{7R}} \Rightarrow R_t = \frac{{7R}}{{12}}
\]
we found the equivalent resistance between points a and b to be
\[
R_t = \frac{{7R}}{{12}}
\]
2. Of course, the equivalent resistance would not change - that is because the potential between
those two points is zero, therefore, there would be no current and the resistance would not change!
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