Problem — heated copper wire

Problem

A copper wire at $20\ ^\circ$C has resistance $R_0 = 15\ \Omega$. The temperature coefficient of copper is $\alpha = 4{,}3\cdot 10^{-3}\ \text{K}^{-1}$. Calculate the wire’s resistance at $80\ ^\circ$C.

Solution

Law of variation with temperature — in metals resistance grows almost linearly with temperature: $R = R_0\,\bigl(1 + \alpha\,\Delta T\bigr)$ where $\Delta T$ is the deviation from the reference temperature ($20\ ^\circ$C).

Temperature rise: $\Delta T = 80 - 20 = 60\ \text{K}$

Substitution: $R = 15\,\bigl(1 + 4{,}3\cdot 10^{-3} \cdot 60\bigr) = 15\,(1 + 0{,}258) = 15 \cdot 1{,}258 \approx 18{,}9\ \Omega$

The resistance increases by $\sim 26\%$: as it heats up, the metal’s lattice vibrates more and hinders the motion of the electrons more.

$\ev{R \approx 18{,}9\ \Omega}$

Collegamenti

Argomenti: Circuiti elettrici Concetti: Legge di Ohm