Answer to Question #107014 in Electric Circuits for Abass ayomipo faruk

Question #107014
Suppose the resistance of a copper wire is 2ohms at a temperature of 100degre. Find the resistance of the wire at 0degre and 100degre(temperature coefficient of resistivity of copper wire is 0.00393)
1
Expert's answer
2020-03-31T09:19:03-0400

 The resistance of a normal metal at a temperature of "0 \\degree - 100 \\degree" of Celsius changes directly in proportion to the temperature according to the linear law:

(1) "R=R_0\\cdot(1+\\alpha \\Delta T)" , where "R_0" - the resistance at an temperature "T_0", "\\alpha" - temperature coefficient of resistivity, and "\\Delta T=T-T_0" the difference between actual temperature and initial one. For copper in our task we have "R_0=2 \\Omega" , "\\alpha=0.00393 K^{-1}, \\Delta T_1=0\\degree C-100\\degree C=-100 K" . When calculating the temperature difference, we took into account that "1\\degree C=1K", but the absolute temperature values in these scales of temperature (Celsius and Kelvin) are very different [°C] = [K] − 273.15 [1].

To find resistance at "T=0\\degree C" we substitute these quantities to (1)

"R_1=R_0\\cdot(1+\\alpha \\Delta T_1)=2\\Omega\\cdot(1+0.00393 K^{-1}\\cdot (-100K))=2\\Omega\\cdot(1-0.393)=1.21\\Omega"

At a temperature "100 \\degree" higher ("T_2=200\\degree C" )we get "\\Delta T_2=200\\degree C- 100\\degree C=100K" and

"R_2=R_0\\cdot(1+\\alpha \\Delta T_2)=2\\Omega\\cdot(1+0.00393 K^{-1}\\cdot (100K))=2.79\\Omega"

Answer: If the resistance of a copper wire is "2\\Omega" at a temperature of "100 \\degree C", the resistance of the wire at "0\\degree C" will be "1.21\\Omega" and at "200 \\degree C" will be "2.79\\Omega" .

[1] https://en.wikipedia.org/wiki/Kelvin


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