Question

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.

Answer 1

The length of the wire is L=45cm=0.45m

Temperature linearly increases from T₁=6^oC to T₂=270^oC. Let at x=0 temperature is T₁=6^oC, and at x=L temperature is T₂=270^oC. The temperature as a function of distance is

The resistivity as a function of temperature is

where is the resistivity at temperature T_o=20^oC

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