# Inferred Absolute Temperature

Since temperature can have such a pronounced effect on the resistance of a conductor, it is important that we have some method of determining the resistance at any temperature within operating limits.
Fig. 1: Effect of temperature on the resistance of copper
An equation for this purpose can be obtained by approximating the curve (in Fig. 1 ) by the straight dashed line that intersects the temperature scale at $-234.5℃$. Although the actual curve extends to absolute zero ($-273.15℃$, or $0 K$), the straight-line approximation is quite accurate for the normal operating temperature range. At two temperatures $T_1$ and $T_2$, the resistance of copper is $R_1$ and $R_2$, respectively, as indicated on the curve.
Using a property of similar triangles, we may develop a mathematical relationship between these values of resistance at different temperatures. Let x equal the distance from $-234.5℃$ to $T_1$ and y the distance from $-234.5℃$ to $T_2$, as shown in Fig. 1 From similar triangles, $${x \over R1} = {y \over R2}$$ $$\bbox[5px,border:1px solid grey] {{234.5 + T1 \over R1} = {234.5 + T2 \over R2}} \tag{1}$$ The temperature of -234.5℃ is called the inferred absolute temperature (Ti) of copper. For different conducting materials, the intersection of the straight-line approximation occurs at different temperatures. A few typical values are listed in Table 1. The minus sign does not appear with the inferred absolute temperature on either side of Eq. (1) because x and y are the distances from $-234.5℃$ to $T_1$ and $T_2$, respectively, and therefore are simply magnitudes.
Eq. (1) can easily be adapted to any material by inserting the proper inferred absolute temperature. It may therefore be written as follows: $$\bbox[5px,border:1px solid grey] {{|T_i| + T1 \over R1} = {|T_i| + T2 \over R2}} \tag{2}$$ where $|T_i|$ indicates that the inferred absolute temperature of the material involved is inserted as a positive value in the equation. In general, therefore, associate the sign only with $T_1$ and $T_2$. The temperature of -234.5℃ is called the inferred absolute temperature (Ti) of copper. For different conducting materials, the intersection of the straight-line approximation occurs at different temperatures. A few typical values are listed in Table 1.
Table.no.1: Inferred absolute temperatures (Ti).
Example 1: If the resistance of a copper wire is 50 Ω at 20℃, what is its resistance at $100℃$ (boiling point of water)?
Solution: According to Eq. (1): $${234.5℃ + 20℃ \over 50 Ω} = {234.5℃ + 100℃ \over R_2}$$ $$R_2 = {(50 Ω)(334.5℃) \over 254.5℃} = 65.72 Ω$$