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

Given:-

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

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

Temperature coefficient α_Fe for iron = 5.0 × 10^–3 K^–1

Temperature coefficient α_Cu for copper = 4.0 × 10^–3 K^–1

Let

The temperature at which the resistance be equal = T

Resistance of iron wire at T °C = R_Fe'

Resistance of copper wire at T °C = R_Cu'

We know:-

\[R =  R_0 (1 + \alpha ∆ T)\]

Here, ΔT = T - 20

\[\Rightarrow R_{Fe} ' = R_{Fe} \left[ 1 + \alpha_{Fe} \left( T - 20 \right) \right]\]

\[ R_{Cu} ' = R_{Cu} \left[ 1 + \alpha_{Cu} \left( T - 20 \right) \right]\]

\[ R_{Fe} ' = R_{Cu} '\]

\[ \Rightarrow R_{Fe} \left[ 1 + \alpha_{Fe} \left( T - 20 \right) \right] = R_{Cu} \left[ 1 + \alpha_{Cu} \left( T - 20 \right) \right]\]

\[ \Rightarrow 3 . 9 \left[ 1 + 5 \times {10}^{- 3} \left( T - 20 \right) \right] = 4 . 1 \left[ 1 + 4 \times {10}^{- 3} \left( T - 20 \right) \right]\]

\[ \Rightarrow 3 . 9 + 3 . 9 \times 5 \times {10}^{- 3} \left( T - 20 \right) = 4 . 1 + 4 . 1 \times 4 \times {10}^{- 3} \left( T - 20 \right)\]

\[ \Rightarrow 4 . 1 \times 4 \times {10}^{- 3} \left( T - 20 \right) - 3 . 9 \times 5 \times {10}^{- 3} \left( T - 20 \right) = - 4 . 1 + 3 . 9\]

\[ \Rightarrow 16 . 4 \left( T - 20 \right) - 19 . 5 \left( T - 20 \right) = - 0 . 2 \times {10}^3 \]

\[ \Rightarrow \left( T - 20 \right)\left( - 3 . 1 \right) = - 0 . 2 \times {10}^3 \]

\[ \Rightarrow T - 20 = 64 . 5\]

\[ \Rightarrow T = 84 . 5^\circ K\]