In: Physics
Copper wire has a resistivity ρ = 1.7 × 10-8 Ω⋅m when at 20°C and it has a temperature coefficient α = 3.9 × 10-3 K-1. A solid cylinder of copper of length L = 45 cm and diameter D = 1.5 mm has one end held at T1 = 6°C and the other end is held at T2 = 270°C. The temperature increases linearly between the two ends of the cylinder. Determine the total resistance in milliohms.
The length of the wire is L=45cm=0.45m
Temperature linearly increases from T1=6oC to T2=270oC. Let at x=0 temperature is T1=6oC, and at x=L temperature is T2=270oC. The temperature as a function of distance is
The resistivity as a function of temperature is
where is the resistivity at temperature To=20oC
Resistivity as a function of distance is
The diameter of the copper wire is D=1.5mm
The radius of the copper wire is R=D/2=0.75mm=0.75x10-3m
The cross sectional area of the wire is
Consider the wire to be made of small elements of length dx. Consider one such element at position x, the resistance of the element is
Using (2)
The wire is the series combination of these small elements, the total resistance of the wire can be calculated by integrating over the length of the wire
Substituting values