Note III. Explanation of Temperature Coefficients.

1. Coefficient of variation of resistance at constant mass and free expansion with the temperature.

If R1 and R2 are the resistances measured at the temperatures t1 and t2 of a uniform wire, between two potential points rigidly fixed to the wire when the current flows parallel to the axis of the wire, the coefficient of variation of resistance at constant mass and free expansion for the temperature t1, α1, is defined by the formula:
R2 = R1 [1 + α1 (t2 - t1)]
2. Coefficient of variation of the volume resistivity with the temperature.

If ρ represents the volume resistivity of the wire, i.e., if the resistance R of the wire is equal to ρ (l/S) (l = length of wire, S = section) and if, for the temperature t1, the coefficient of variation of volume resistivity with the temperature is represented by β1, the following is obtained:
ρ2 = ρ1 [1 + β1 (t2 - t1)]
If γ represents the coefficient of linear expansion of the metal, the following is approximately correct:
β1 = α1 + γ
3. Coefficient of variation of the mass resistivity with the temperature.

If δ represents the mass resistivity, i.e., if the resistance R of the wire is equal to δ (l²/m), l being its length and m its mass, and if the coefficient of the variation of the mass resistivity with the temperature for the temperature t1 is represented by β'1, the following is obtained:
δ2 = δ1 [1 + β'1 (t2 - t1)]
Giving the approximate formula:
β1 = α1 - 2γ

I. Standard Annealed Copper

The following shall be taken as normal values for standard annealed copper:

(1) At a temperature of 20°C, the volume resistivity of standard annealed copper is:
1/58 = 0.017241... ohm square millimetre per metre ([i]ohm mm²/m[/i]).
(2) At a temperature of 20°C, the density of standard annealed copper is:
8.89 grammes per cubic centimetre ([i]g/cm³[/i]).
(3) At a temperature of 20°C, the coefficient of linear expansion of standard annealed copper is:
0.000017 per degree Centigrade.
(4) At a temperature of 20°C, the coefficient of variation of the resistance with temperature of standard annealed copper, measured between two potential points rigidly fixed to the wire (the metal being allowed to expand freely), is:
0.00393 = 1/254.45... per degree Centigrade.
(5) As a consequence, it follows from (1) and (2) that at a temperature of 20°C, the mass resistivity of standard annealed copper is:
1/58 × 8.89 = 0.15328... ohm gramme per metre per metre.

Note II. The constants at 0°C of standard annealed copper deduced from the values given above for 20°C are the following:

(1) Density at 0°C:
8.90 g/cm³
(2) Coefficient of linear expansion per degree Centigrade:
0.000017
(3) Volume resistivity at 0°C:
1.5881 microhm centimetres
(4) Coefficient at 0°C of variation of volume resistivity:
0.00428 per degree Centigrade
(5) Coefficient at 0°C of variation of resistance (at constant mass and free expansion), measured between two potential points rigidly fixed to the wire:
1/234.45 = 0.00426 per degree Centigrade