Question

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

Answer 1

$R_{Fe}^{\prime }=R_{Fe}(1+{\alpha }_{Fe}\Delta \theta ),R_{cu}^{\prime }=R_{cu}(1+{\alpha }_{cu}\Delta \theta )$

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

$\Rightarrow R_{Fe}(1+{\alpha }_{Fe}\Delta \theta ),=R_{cu}(1+{\alpha }_{cu}\Delta \theta )$

$\Rightarrow 3.9[1+5\times {10}^{-3}(20-\theta \left)\right]=4.1[1+4\times {10}^{-3}(20-\theta \left)\right]$

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

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

$\Rightarrow 16.4(20-\theta )-19.5(20-\theta )=0.2\times {10}^3$

$\Rightarrow \theta -20=200$

$\Rightarrow \theta ={220}^{o}C$