Question

Suppose the resistance of a copper wire is 2ohms at a temperature of 100degre. Find the resistance of the wire at 0degre and 100degre(temperature coefficient of resistivity of copper wire is 0.00393)

Answer 1

 The resistance of a normal metal at a temperature of $0 \degree - 100 \degree$ of Celsius changes directly in proportion to the temperature according to the linear law:

(1) $R=R_0\cdot(1+\alpha \Delta T)$ , where $R_0$ - the resistance at an temperature $T_0$, $\alpha$ - temperature coefficient of resistivity, and $\Delta T=T-T_0$ the difference between actual temperature and initial one. For copper in our task we have $R_0=2 \Omega$ , $\alpha=0.00393 K^{-1}, \Delta T_1=0\degree C-100\degree C=-100 K$ . When calculating the temperature difference, we took into account that $1\degree C=1K$, but the absolute temperature values in these scales of temperature (Celsius and Kelvin) are very different [°C] = [K] − 273.15 [1].

To find resistance at $T=0\degree C$ we substitute these quantities to (1)

$R_1=R_0\cdot(1+\alpha \Delta T_1)=2\Omega\cdot(1+0.00393 K^{-1}\cdot (-100K))=2\Omega\cdot(1-0.393)=1.21\Omega$

At a temperature $100 \degree$ higher ($T_2=200\degree C$ )we get $\Delta T_2=200\degree C- 100\degree C=100K$ and

$R_2=R_0\cdot(1+\alpha \Delta T_2)=2\Omega\cdot(1+0.00393 K^{-1}\cdot (100K))=2.79\Omega$

Answer: If the resistance of a copper wire is $2\Omega$ at a temperature of $100 \degree C$, the resistance of the wire at $0\degree C$ will be $1.21\Omega$ and at $200 \degree C$ will be $2.79\Omega$ .

[1] https://en.wikipedia.org/wiki/Kelvin