Q. 143.7( 6 Votes )

The resistances of an iron wire and a copper wire at 20°C and 3.9 Ω and 4.1 Ω respectively. At what temperature will the resistances be equal? Temperature coefficient of resistivity for iron is 5.0 × 10–3 K–1 and for copper it is 4.0 × 10–3 K–1. Neglect any thermal expansion.

Answer :

84.5°C


Given,


Resistance of iron wire, RFe, i at 20°C= 3.9 Ω


Resistance of Copper wire, RCu, i at 20°C = 4.1 Ω


Initial temperature of both the wires, Ti= 20°C


Temperature coefficient of resistivity for iron, αFe=5.0 × 10–3 K–1


Temperature coefficient of resistivity for copper, αCu=4.0 × 10–3 K–1


Formula used,


For most of the conducting materials, the relation connecting the resistance with the change in temperature can be represented as,



Where Rf is the final resistance after the change in temperature, Ri is the initial temperature, α is the Temperature coefficient of resistivity and ∆T is the change in temperature from the initial temperature of the material.


Solution:


We have the expression connecting the change in temperature and the resistance of the material.


First, let us assume that the final temperature is Tf at which the resistance of both the wires will be same.


So, change in temperature ∆T is



Now the final resistance of iron wire, RFe, f at temperature Tf can be written based on eqn.1 as,



Or by substituting the known values, we can write it as,



Similarly, the final resistance of iron wire, RCu, f at temperature Tf can be written based on eqn.1 as,



Or by substituting the known values, we can write it as,



At the final temperature, it is given that the resistance of both the wires are same. So we can equate eqn.3 and eqn.4.


So,



Or,



By solving,



Or,



Or,



We know that ∆T is the difference between final and initial temperature, and hence Tf from eqn.2 is



Or,



By substituting the known values, we get Tf as



Hence the temperature in which the resistances are equal is 84.5°C


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