Question

The resistance of an iron wire and a copper wire at 20°C are 3.9 Ω and 4.1 Ω, respectively. At what temperature will the resistance be equal? Temperature coefficient of resistivity for iron is 5.0 × 10–3 K–1 and for copper, it is 4.0 × 10–3 K–1. Neglect any thermal expansion.

Answer 1

The resistance be equal at the temperature T is 84.5 °C .

Explanation:

Resistance of the iron wire at 20°C = 3.9 Ω

Resistance of the copper wire at 20°C = 4.1 Ω

Temperature coefficient ($\alpha$) for iron (fe) = $5.0 \times 10^{-3} \mathrm{K}^{-1}$

Temperature coefficient ($\alpha$) for copper (cu) = $4.0 \times 10^{-3} \mathrm{K}^{-1}$

Step 1:

Let's Consider,  

Resistance(R) of iron wire at T °C = $R_{F e}$'

Resistance(R) of copper wire at T °C = $R_{C u}$'

Step 2:

We know:

$\mathrm{R}=\mathrm{R}_{0}(1+\alpha \Delta \mathrm{T}) \mathrm{R}=\mathrm{R}_{0}(1+\alpha \Delta \mathrm{T})$

Here, ΔT = T – 20 where T= Temperature.

Resistance of iron = $\mathrm{R}_{\mathrm{fe}}^{\prime}=\mathrm{R}_{\mathrm{fe}}[1+\alpha \mathrm{Fe}(\mathrm{T}-20)]$

Resistance of copper = $\mathrm{R}_{\mathrm{Cu}}^{\prime}=\mathrm{R}_{\mathrm{Cu}}[1+\alpha \mathrm{Cu}(\mathrm{T}-20)]$

Step 3:

By equating above two statements & after substituting values we get

$\begin{aligned}&[1+\alpha F e(T-20)] R_{F e}[1+\alpha C u(T-20)] R_{C U}\\&\Rightarrow[1+5 \times 10-3(T-20)] 3.9=[1+4 \times 10-3(T-20)] 4.1\\&\Rightarrow 3.9 \times 5 \times 10-3(T-20)+3.9=4.1 \times 4 \times 10-3(T-20)+4.1\end{aligned}$

$\begin{array}{l}{\Rightarrow 4.1 \times 4 \times 10-3(T-20)-3.9 \times 5 \times 10-3(T-20)=-4.1+3.9} \\{\Rightarrow 16.4(T-20)-19.5(T-20)=-0.2 \times 103} \\{\Rightarrow(T-20)(-3.1)=-0.2 \times 103} \\{\Rightarrow T-20=64.5 \Rightarrow}\end{array}$

   T=84.5 °C