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Value of α (alpha) at Different Temperatures

So far we did not make any distinction between values of α (alpha) at different temperatures. But it is found that value of α (alpha) itself is not cönstant, but depends on the initial temperature on which the increment in resistance is based. When the increment is based on the resistance measured at 00C, then α has the value of α0 (alpha nol) any other initial temperature t0C, value of α is αt , and so on. It should be remembered that, for any conductor, α0 has the maximum value. 


Suppose a conductor of resistance R0 at 00C (point A in Fig. above) is heated to toC (point B). Its resistance Rt after heating is given by

 Rt  = R0 (1 + α0 t)                                         … eq (1)


where α0 is the temperature-coeffcient at 00C.

Now, suppose that we have a conductor of resistance Rt at temperature t0C. Let this conductor be cooled from toC to 00. Obviously, now the initial point is B and the final point is A. The final resistance R0is given in terms of the initial resistance by the following equation

                                       R0  = Rt [1 + αt  (-t)] 

                                            = Rt (1 - αt t)                                             .. eq (2)

 From Equation above, we have αt = (Rt – R0) / (Rt t )  

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

αt =  α0 / (1 + α0 . t )                        …. Eq (3)

in general, let

α1  = tempt-coeff. at t1 0C

α2 = tempt-coeff. at t 0C

then orm equation (3) we get

αt = α0 / (1 + α0 . t1) or 1/α1  =  (1 + α0 . t1 ) / α0 

sunstracting one from the other, we get

(1/ α2 ) -  (1/ α1 ) =  ( t2 – t1 ) 

or 

(1/ α2 ) = (1/ α1 ) + ( t2 – t1 )

 or  

α2 = 1/ [ (1/ α1 ) + ( t2 – t1 ) ]

                

Valur of α for coppor at different temperatures are given in the table below

 different value of α or copper

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

In case R0 is not given, the relation between the known resitance R1 at t1 oC and the unknown resistance R2 at t2oC can be found as follows:

R2 = R0(1 + α0t2) and R1 = R0(1 + α0t1) ...... eq (4)

The eq (4) expression can be simplified by a little approximation as follows

R2/R1 = (1 + α0t2) (1 + α0t1)-1

          =  (1 + α0t2) (1 - α0t1)

          =  1 + α0 (t2 – t1)

R2 = R1 [1 + α0 (t2 – t1)]


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