At 20^ C , an iron wire and a copper wire have resistances of 3.9 and 4.1 , respectively. Determine the temperature at which their resistances become equal. The temperature coefficient of resistivity is 5.0 10⁻³ K ⁻¹ for iron and 4.0 10⁻³ K ⁻¹ for copper. Ignore any effects of thermal expansion.

Step-by-step solution

Let the initial temperature be T₀ = 20^ C and the final temperature be T . The resistance of a wire at temperature T is given by R = R₀[1 + (T - T₀)] . For the iron wire: R_ Fe = 3.9 [1 + 5.0 10⁻³ (T - 20)] For the copper wire: R_ Cu = 4.1 [1 + 4.0 10⁻³ (T - 20)] Equating the two resistances: 3.9 [1 + 5.0 10⁻³ (T - 20)] = 4.1 [1 + 4.0 10⁻³ (T - 20)] 3.9 + 19.5 10⁻³ (T - 20) = 4.1 + 16.4 10⁻³ (T - 20) (19.5 - 16.4) 10⁻³ (T - 20) = 4.1 - 3.9 3.1 10⁻³ (T - 20) = 0.2 T - 20 = 0.2 3.1 10⁻³ = 200 3.1 64.5^ C T = 20 + 64.