Question

A particular application requires a 10 m length of aluminum wire to have a resistance 0.40 mΩ at 20∘C. If the resistance of the wire cannot vary by more than ± 7.5 % from its value at 20 ∘C, to what operating temperature range should it be restricted?

What would be the operating range if the wire were, instead, made of copper?

Answer 1

We will use the following expression:

$\dpi{80} \fn_phv \frac{\Delta R}{R_0}=\alpha \Delta T$

Since the variation cannot be more than 7.5% from its value at 20ºC, this can be expressed as

$\dpi{80} \fn_phv \frac{\Delta R}{R_0}= 0.075=\alpha\Delta T$

The temperature coefficient of resistance for the aluminum wire at 20ºC is 0.00429. So we can solve for T

$\dpi{80} \fn_phv \Delta T=\frac{0.075}{0.00429}=17.482$ ºC

So the operating range is 2.51 ºC to 37.48 ºC.

The  temperature coefficient of resistance of the cooper at 20ºC is 0.0039. The difference in temperature allowed is:

$\dpi{80} \fn_phv \Delta T=\frac{0.075}{0.0039}=19.23$

And the operating range for the cooper is from 0.769 ºC to 39.23 ºC

[EDITED]

According to other sources the temperature coefficient for the cooper at 20ºC can also be: 0.00386 (a) , 0.00393 (b) . The corresponding operating ranges are:

a) From 0.569 ºC to 39.43 ºC

b) From 0.916 ºC to 39.08 ºC

Please note that the results may vary depending on what textbook is used in your class or what value for the temperature coefficient of the cooper at 20ºC are you using. The procedure should be the same.