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. An equation for this purpose can be obtained by approximating the curve (in Fig.no.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 T1 and T2, the resistance of copper is R1 and R2, 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 T1 and y the distance from -234.5℃ to T2, as shown in Fig.No.1 From similar triangles, $$ {x \over R1} = {y \over R2}$$ $$\bbox[5px,border:1px solid blue] {\color{blue}{{234.5 + T1 \over R1} = {234.5 + T2 \over R2}}} \tag{1}$$
Fig.no.2: Similar Triangles have equal ratio of sides.
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 T1 and T2, 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: $$\bbox[5px,border:1px solid blue] {\color{blue}{{|T_i| + T1 \over R1} = {|T_i| + T2 \over R2}}} \tag{2}$$ 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 T1 and T2. 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): $$ {234.5℃ + 20℃ \over 50 Ω} = {234.5℃ + 100℃ \over R2}$$ $$R2 = {(50 Ω)(334.5℃) \over 254.5℃} = 65.72 Ω$$
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