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 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):
$$ {234.5℃ + 20℃ \over 50 Ω} = {234.5℃ + 100℃ \over R_2}$$
$$R_2 = {(50 Ω)(334.5℃) \over 254.5℃} = 65.72 Ω$$
Do you want to say or ask something?