Question

The resistances of an iron wire and a copper wire at 20 ${}^{\circ }$ are 3.9 $\Omega$ and 4.1 $\Omega$ respectively. At what temperature will the resistances be equal ? Temperature coefficient of resistivity for iron is 5.0 ${\times }^{-3}{K}^{-1}$ and for copper it is 4.0 $\times {10}^{-3}{K}^{-1}$. Neglect any thermal expansion.

Answer 1

$R_{Fe}^{\prime }=R_{Fe}\times (1+{\alpha }_{Fe}\times △\theta )$

$R_{Cu}^{\prime }=R_{Cu}(1+{\alpha }_{cu}\times △\theta )$

$R_{Fe}^{\prime }=R_{Cu}^{\prime }$

$\Rightarrow R_{Fe}\times (1+{\alpha }_{Fe\times △\theta })$

$\Rightarrow 3.9[1+5\times {10}^{-3}(20-\theta \left)\right]$

=$\Rightarrow 3.9+3.9\times 5\times {10}^{-3}(20-\theta )$

$\Rightarrow 4.1\times 4\times {10}^{-3}(20-\theta )-3.9\times 5\times {10}^{-3}(20-\theta )$

= $4.1-3.9$ $\Rightarrow 16.4(20-\theta )-19.5(20-\theta )$

=$0.2\times {10}^3$

$\Rightarrow (20-\theta )(-3.1)=0.2\times {10}^3$

$\Rightarrow \theta -20=64.5$

$\Rightarrow \theta ={84.5}^{\circ }C$