Mathematical and Physical Journal
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# Problem P. 5062. (October 2018)

P. 5062. From experimental results we know that the resistance of conductors depends on their temperature. For some alloys this temperature coefficient of resistance is negative, whilst for some others it is positive. Hence, by connecting pieces of wires of different alloys, one can create a wire whose resistance is independent of the temperature in a wide temperature range. In the table below, for two alloys manganin and constantan, the resistance values $\displaystyle r$ of unit-length wires measured at $\displaystyle 0~{}^\circ$C, and their temperature coefficient of resistance $\displaystyle \alpha$ are given.

 $\displaystyle r\;[\Omega/\rm m]$ $\displaystyle \alpha\;[1/{}^\circ\rm C]$ constantan 6.3 $\displaystyle -5{.}0\cdot 10^{-5}$ manganin 5.3 $\displaystyle +1{.}4\cdot 10^{-5}$

What length of wires made from manganin and constantan should be connected in series in order to gain an equivalent resistance of $\displaystyle 5{.}0~\Omega$ which is independent of the temperature?

(3 pont)

Deadline expired on November 12, 2018.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Sorosan kapcsolva $\displaystyle \ell_1$ hosszúságú konstantán- és $\displaystyle \ell_2$ hosszúságú manganinvezetéket az eredő ellenállásuk $\displaystyle T$ Celsius-fok hőmérsékleten

$\displaystyle R(T)=\ell_1r_1(1+\alpha_1 T)+\ell_2r_2(1+\alpha_2 T)\equiv 5{,}0~\Omega.$

Ez az összefüggés akkor teljesül széles hőmérséklet-tartományban, ha

$\displaystyle \ell_1 r_1+\ell_2 r_2=5{,}0~\Omega, \qquad \text{és}\qquad \ell_1r_1 \alpha_1+\ell_2r_2 \alpha_2=0,$

vagyis (a megadott számadatok mellett) $\displaystyle \ell_1\approx17 ~\rm cm$ és $\displaystyle \ell_2\approx 74~\rm cm$.

### Statistics:

 39 students sent a solution. 3 points: Arhaan Ahmad, Bekes Barnabás, Csépányi István, Gál Péter Levente, Girus Kinga, Havasi Márton, Horváth 999 Anikó, Jánosik Máté, Keltai Dóra, Kertész Balázs, Lipták Gergő, Nagyváradi Dániel, Ocskó Luca, Rozgonyi Gergely, Rusvai Miklós, Schneider Anna, Sugár Soma, Tanner Norman, Telek Dániel, Tiefenbeck Flórián, Toronyi András, Turcsányi Máté, Virág Levente. 2 points: Merkl Levente, Tóth Lilla Eszter . 0 point: 13 students. Not shown because of missing birth date or parental permission: 1 solutions.

Problems in Physics of KöMaL, October 2018