That model doesn't make any sense.
The **absolute** change in resistance of an object
cannot depend solely on the temperature:
it depends on the size and shape of the object, too.

But it might make sense for the **relative** or **fractional**
change in resistance of an object to depend solely on its temperature.
In other words, the resistance might increase by 1 percent for every
increase of 1 degree Celsius;
so a teeny wire would still have a teeny resistance (just a teeny bit larger
than before, in ohms), and a long cable would still have a big resistance
(a lot larger than before, in ohms).

Mathematically speaking, a relative or fractional dependency can be written like this:

[ ] resistance(T) = resistance(To) [ 1 + A * (T - To) ] [ ]

This is slightly different than the earlier formula -- see?

In this case, if we look at the change in resistance (or resistance) as a function of change in temperature, we find

resistance(T) - resistance(To) resistance(To) * A = ------------------------------- T - To

And now, if you were to plot resistance versus temperature, the slope
of the graph would be **resistance(To) * A**.
This slope would NOT the be same at all temperatures,
because it depends on the resistance at the reference point.

Question 5: What would the units of this coefficient be?

Once again, the graphs below show close-ups of the behavior of resistance
as a function of temperature, near reference temperatures
of **To = 20** and **To = 40** Celsius.
Use the new coefficient (expressing fractional change in resistance
with temperature), assuming a value **A** of exactly 0.02.
Draw marks at the resistance values one degree above and below
the reference temperature on each graph, and use them to
draw a line on each graph showing resistance as a function of
temperature.

Question 6: Are the slopes of the two graphs the same?

Assume real copper behaves according to the above equation, with a coefficient of A = 0.02. Consider again the four very different objects:

- a short copper wire of length 10 cm, with resistance R = 0.000 01 ohm at T = 20 degrees
- a long copper wire of length 10 m, with resistance R = 0.001 ohm at T = 20 degrees
- a translatantic copper cable, with resistance R = 100 ohm at T = 20 degrees
- a solid copper sphere, with resistance R = 0.000 000 001 ohm at T = 20 degrees

Question 7: Calculate the resistance of each of the 4 objects at T = 21 degrees.

Question 8: Does that make any sense?

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.