Value of α at Different Temperatures ( In terms of Electrical )

So far we did not make any distinction between values of a at different temperatures. But it is found that value of a itself is not constant but depends on the initial temperature on which the increment in resistance is based. When the increment is based on the resistance measured at O°C, then α has the value of ${\alpha }_0$.At any other initial temperature t° C,value of α is αt, and so on. It should be remembered that, for any conductor, ${\alpha }_0$ has the maximum value.

Suppose a conductor of resistance R0 at 0 °C (point A in Figure) is heated to t °C (point B). Its resistance Rt after heating is given by

$Rt=R0 ( 1+{\alpha }_0 t )$ _ _ _ _ _ _ _ (1)

where α0 is the temperature-coefficient at 0°C.

Now, suppose that we have a conductor of resistance Rt, at temperature t°C.. Let this conductor be cooled from t°C.to 0°C.. Obviously, now the initial point is B and the final point is A. The final resistance R0 is given in tenns of the initial resistance by the following equation$$

$R0=Rt [1+\alpha t(-t)]=Rt (1-\alpha t\times t)$ _ _ _ _ _ _ _ _ (2)

From Eq. (ii) above, we have $\alpha t=\frac{Rt-R0}{Rt\times t}$

Substuting the value of Rt from Eq. (1), we get

$\alpha t=\frac{R0 ( 1 +\alpha 0 t)-R0}{R0 ( 1 +\alpha 0 t)\times t}$

$so, \; \alpha t \; =\frac{\alpha 0}{1+\alpha 0 \mathrm{t }}$         _ _ _ _ _ _ _ (3)

In general, let αt =temp. coeff. at t1°C;  α2 = tempt. coeff. at t2°C.  Then from Eq. (3) above, we get

$\; \alpha 1 \; =\frac{\alpha 0}{1+\alpha 0 \mathrm{t1 }}$ or $\frac{1}{\alpha 1}=\frac{1+\alpha 0 \mathrm{t1}}{\alpha 0}$

Similarly,

$\frac{1}{\alpha 2}=\frac{1+\alpha 0 \mathrm{t2}}{\alpha 0}$

So, 

$\frac{1}{\alpha 1}-\frac{1}{\alpha 2}=(t2-t1) or \frac{1}{\alpha 2}=\frac{1}{\alpha 1}+ ( t2-t1) or \alpha 2=\frac{1}{1/\alpha 1 + ( t2-t1 )}$

Values of α for copper at different temperatures are given in Table below

Temp in C	0	5	10	20	30	40	50
α	0.00427	0.00418	0.00409	0.00393	0.00378	0.00364	0.00352

In view of the dependence of α on the initial temperature, we may define the temperature coefficient of resistance at a given temperature as the charge in resistance per ohm per degree centrigrade change in temperaturefrom the given temperature.

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