The resistances of an iron wire and a copper wire at 20C are 3.9 Ω and 4.1 Ω respectively. At what temperature will the resistances 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.

Sol. R’ base Fe = R base Fe (1 + ∝ base Fe ∆θ), R’ base cu = R base cu (1 + ∝ base Cu ∆θ) R’ base Fe = R base Cu ⇒ R base Fe (1 + ∝ base Fe ∆θ), R base cu (1 + ∝ base Cu ∆θ) ⇒ 3.9 [1 + 5 * 10^-3 (20 - θ)] = 4.1 [1 + 4 * 10^-3 (20 - θ)] ⇒ 3.9 + 3.9 * 5* 10^-3 (20 - θ) = 4.1 + 4.1 * 4* 10^-3 (20 - θ) ⇒ 4.1 * 4 * 10^-3 (20 - θ) – 3.9 * 5 * 10^-3 (20 - θ) = 3.9 – 4.1 ⇒ 16.4 (20 - θ) – 19.5(20 - θ) = 0.2 * 10^3 ⇒ (20 - θ) (-3.1) = 0.2 * 10^3 ⇒ θ - 20 = 200 ⇒ θ = 220°C.