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I came across a question recently:

A wire has a resistance of 2.1 ohm at 300 K, and 2.7 ohm at 373 K. What is the temperature coefficient of resistance of the metal?

I tried by simply putting the formula R2 = R1 (1 + A(T2 - T1)), and got the answer as 2/511.
But when I did -
R2 / R1 = Ro (1 + A(T2 - To) / Ro (1 + A(T1 - To)
taking To as 273 K, I was getting 2/457.

This got me wondering, I can take any value of To and get a different answer. Why is this so, since I am simply taking out the ratio here?

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Temperature coefficient of resistance, $\alpha$, is most easily defined in terms of Celsius temperature, $\theta$. Thus$$R_\theta=R_0 (1 +\alpha \theta)$$ in which $R_0$ is the resistance at temperature 0°C and $R_\theta$ is the resistance at Celsius temperature $\theta$.

Your second attempt is equivalent to using the above definition and is, I believe, correct.

Note that $R_0$ has to be the resistance at temperature 0°C; it can't be the resistance at some other temperature of your choice.

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