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.

84.5°C

Given,

Resistance of iron wire, R_{Fe, i} at 20°C= 3.9 Ω

Resistance of Copper wire, R_{Cu, i} at 20°C = 4.1 Ω

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

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

Temperature coefficient of resistivity for copper, α_Cu=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 R_f is the final resistance after the change in temperature, R_i 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 T_f at which the resistance of both the wires will be same.

So, change in temperature ∆T is

Now the final resistance of iron wire, R_{Fe, f} at temperature T_f 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, R_{Cu, f} at temperature T_f 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 T_f from eqn.2 is

Or,

By substituting the known values, we get T_f as

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