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In my textbook, a relationship is plotted between the resistance of the material and the temperature, and the temperature coefficient is defined as the slope of that graph divided by an arbitrary resistance $R_1$ on the graph. Does that mean that the coefficient varies depending on the temperature I choose to divide the slope by?

If so, how does it stand as a valid reference to the material’s resistance growth rate with temperature?

If not, then where did I go wrong with this train of thought?

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The temperature coefficient is often defined defined as $\alpha = \dfrac{R_{\rm T}-R_0}{R_0 \, T}$
where the temperatures are in degree Celsius and the reference resistance $R_0$ is measured at $0^\circ \rm C$.

Which gives the equation $R_{\rm T} = R_0(1 + \alpha\, T)$ ie a linear relationship between resistance and temperature.

As you have pointed out the relationship is not linear and so a better relationship like $R_{\rm T} = R_0(1 + \alpha\, T + \beta \, T^2 + . . . .)$ can be used for large temperature variations.
This implies that other constants like $\beta$ must be given as well as the temperature coefficient of resistance $\alpha$ as is shown in this reference for a platinum resistance thermometer.

So it does depend on the accuracy to which you are working and the range of temperatures.
If you are given just $\alpha$ with a stated reference temperature then the implication is that as the temperature deviation from the reference temperature increases there will be a corresponding reduction in the accuracy of resistance or temperature as a result of using the first order equation.

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  • $\begingroup$ I believe that the OP is asking how alpha, when defined as it is defined, gives a valid reference to the growth rate of resistance with temperature, rather than it’s range of validity. Could you elaborate on that please, sir? $\endgroup$
    – SNB
    Jun 29, 2018 at 16:27
  • $\begingroup$ In short, how is defining temperature coefficient as the slope of that graph divided by an arbitrary resistance more useful than regarding the slope itself as a measure of the rate of growth with temperature? This seems to be the way in which all temperature coefficients are defined, judging by this wiki en.m.wikipedia.org/wiki/Temperature_coefficient $\endgroup$
    – SNB
    Jun 29, 2018 at 16:55
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    $\begingroup$ @SNB The problem is that the graph of resistance against temperature is not a straight line. So your idea of using the gradient would require a reference temperature and the dimensions of the conductor. $\endgroup$
    – Farcher
    Jun 29, 2018 at 18:23
  • $\begingroup$ We need the dimensions of the conductor because it expands with temperature, is it? $\endgroup$
    – SNB
    Jun 30, 2018 at 10:15
  • $\begingroup$ @SNB Imagine you had a copper wire and measured its change of resistance with temperature at a given temperature. Now I have an entirely different piece of copper wire and measure its change of resistance with temperature at the same given temperature. In general I will get an entirely different result from you unless by chance the resistances of the wire (which depend on the dimensions of the wire) are the same. $\endgroup$
    – Farcher
    Jun 30, 2018 at 10:23
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Yes,temperature coefficient of resistance depends on temperature to some extent. At small temperature changes resistance versus temperature graph is linear which implies temperature coefficient is independent of temperature. At higher temperature the graph is nonlinear which implies temperature coefficient is now not independent of temperature.

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