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\begin{document}
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\heading{Electric current}
\auname{I.D. 200475846}
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{\bf The problem:}
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Find the resistance between the bases of a circle conic with radii $a$ and $2a$, with a height $L$ and filled with material with a conductivity $\sigma$.
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\Dn
{\bf The solution:}
\Dn
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1. The equation we shall use to find the resistance is: $dR = \frac{dx}{S(x)\sigma}$. $\sigma$ is given to us, so to find the total resistance we need to find the area $S(x)$.
In order to find the radius of the cone as a function of $L$ we shall use basic geometry and we will come up with the expression:
\begin{eqnarray}
r(x) = \frac{a (L+x)}{L}.
\end{eqnarray}
\Dn
Now all we need to do is to substitute $S(x) = \pi r^2(x)$ into $dR = \frac{dx}{S(x)\sigma}$.
\begin{eqnarray}
dR = \frac{L^2 dx}{\sigma \pi a^2 (L+x)^2}
\end{eqnarray}
Now we shall perform the integral. The integral boundaries are from $0$ to $L$.
\begin{eqnarray}
R = \int_0^L{\frac{L^2 dx}{\sigma \pi a^2 (L+x)^2}} = \frac{L}{2 \sigma \pi a^2}
\end{eqnarray}
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