Re: R = V/I with real Physicists

[brian whatcott <inet@INTELLISYS.NET>]

At 13:11 5/10/00 -0500, Gary Karshner wrote:
>        I finally have time to put my foot into this mess.

(I think this is physicist talk for "Now I'm going to show you how
it should be done."    :-)

> William Lichten in his book "Data and Error Analysis" gives as an example
> the current verses voltage relationship for a tungsten lamp from an applied
> voltage of 30 mV to 5.3 volts. The log-log graph starts out with a slope of
> about one (where Ohm's law is approximately correct) and then at higher
> voltages it shifts over to about a 5/3 power law relationship where
> radiation cooling becomes important. It is a nice example of how different
> relationships are at work depending on the physical conditions of the system.


I did not have Lichten's book to hand, but I do have Langmuir's 1916 values
for the resistivity of tungsten.
I ran them through the usual non linear regression package to find
that I could exclude the parameter P0 from the following simple model
without loss.

   1: Title "Fit rho = p0 + p1*temp^p2 for Tungsten";
   2: Variables temp,rho;
   3: Parameter p0;
   4: Parameter p1;
   5: Parameter p2;
   6: Constrain p2 = .1,5;
   7: Function rho =  p1*temp^p2;
   8: Plot;
   9: Rplot;
  10: Data; //         Langmuir's 1916 Data  K  vs ohm.cm
      293   5.51
     1000  25.3
     1500  41.4
     2000  59.4
     3000  98.9
     3500 118

Beginning computation...
Stopped due to: Both parameter and relative function convergence.


   ----  Final Results  ----

NLREG version 4.1
Copyright (c) 1992-1998 (shareware) Phillip H. Sherrod.

Fit rho = p0 + p1*temp^p2 for Tungsten
Number of observations = 6
Maximum allowed number of iterations = 500
Convergence tolerance factor = 1.000000E-010
Stopped due to: Both parameter and relative function convergence.
Number of iterations performed = 25
Final sum of squared deviations = 1.1597560E+000
Final sum of deviations = -1.1800469E-001
Standard error of estimate = 0.53846
Average deviation = 0.309042
Maximum deviation for any observation = 0.857288
Proportion of variance explained (R^2) = 0.9999  (99.99%)
Adjusted coefficient of multiple determination (Ra^2) = 0.9998  (99.98%)
Durbin-Watson test for autocorrelation = 2.626


             ----  Descriptive Statistics for Variables  ----

 Variable    Minimum value   Maximum value    Mean value     Standard dev.
----------  --------------  --------------  --------------  --------------
      temp             293            3500        1882.167         1210.83
       rho            5.51             118          58.085        43.30075


                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t
Prob(t)
----------  -------------  ----------------  --------------  ---------
-------
        p1              1     0.00504356717    0.0004214331      11.97
0.00028
        p2              1        1.23339889      0.01048428     117.64
0.00001


                  ----  Analysis of Variance  ----

  Source     DF   Sum of Squares    Mean Square    F value   Prob(F)
----------  ----  --------------  --------------  ---------  -------
Regression     1        9373.617        9373.617   32329.62  0.00001
Error          4        1.159756        0.289939
Total          5        9374.777


I conclude that Langmuir's data are consistent with the following fit:

 Resistivity of Tungsten (ohm.cm) = 0.0050 (+/- 0.0004) *
Temperature(kelvin)^1.23(+/- 0.01)

However, it is easy to see that the exponent 1.23 is very far from
5/3 = 1.67 so I reran the six data pairs in two halves to test the
dependency on radiation (fitting three data pairs with two parameters
 is not tremendously respectable, I realise...)
   Here are some results for the three high temperature pairs:

                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t
Prob(t)
----------  -------------  ----------------  --------------  ---------
-------
        p1              1     0.00548511899     0.001441022       3.81
0.16355
        p2              1         1.2230087      0.03269746      37.40
0.01702

The exponent slightly falls further...     1.22 +/- 0.03

> Ohms law like almost all laws trying to describe the physical properties
> of matter is an approximation at best....
> Gary Karshner
>
> St. Mary's University
> San Antonio, Texas
> KARSHNER@STMARYTX.EDU

As a sanity check, I modeled the following compilation of various
investigators' results for platinum.

Data; //    Platinum compilation of Dewar, Fleming, Niccoli,
      //    Nernst,Pirrani             K  vs ohm.cm

69.9     2.44
175.5   6.87
273    10.96
293    10
373    14.85
673     26
1073   40.3
1273   47
1473   52.7
1673   58
1873   63
                   ----  Calculated Parameter Values  ----

 Parameter  Initial guess   Final estimate   Standard error      t
Prob(t)
----------  -------------  ----------------  --------------  ---------
-------
        p1              1      0.0688244668      0.01218674       5.65
0.00031
        p2              1       0.908475201      0.02440486      37.23
0.00001

This leads to the following model for platinum:

Resistivity = [0.07 +/- 0.01] * Temp abs kelvin ^ [0.91 +/- 0.02]

I conclude that glib physical models with a simple power law dependency are
likely to be seriously misguided.

[I think that was engineer talk for "This is how it REALLY is" ]
  :-)



brian whatcott <inet@intellisys.net>
Altus OK