Problem 17: Find the resistance at 50^oC of copper wire 2mm in diameter and 3m long.

Solution:

Step: 1 Overview

Dependence of resistance upon temperature:
The electrical resistance of most of the metals increase with rise in temperature. Experimentally it has been observed that the change in the resistance of a metallic conductor with the change in resistance in nearly linear over a wide range of temperature above and below 0^oC. If the resistor at 0^oC is R_o, then resistor at any temperature (measured in Celsius scale) will be:

R_t=R_o(1+αt)

Where t is the temperature at which we want to calculate resistance, and α (alpha) is the constant known as co-efficient of resistance and its value is different for different substances.

Step: 2 Calculation

Given:

ρ_copper= 1.6 x 10^-8
α_copper= 0.0039
L= 3m
d= 2mm
r=d/2= 1mm = 0.001m
t= 50^o

Solution:
In order to find resistance at 20^oC, first we need to find resistance at 0^oC, which can be found by the formula:

R_o=ρL/A

First we need to calculate area, which is equal to:

A= 3.14r²
A= (3.14)(0.001)²
A= (3.14)(0.000001)
A= 3.14x10^-6

Now putting value of area in above formula:

R_o=ρL/A
R_o=(1.6x10^-8)(3)/(3.14x10^-6)
R_o=0.0152Ω

We know that:

R_t=R_o(1+αt)
R_t=(0.0152)(1+(0.0039)(50))
R_t=(0.0152)(1+0.195)
R_t=(0.0152)(1.195)
R_t=0.0182Ω (Ans)