Question

Current through a heated metal wire

A 39.5 m long copper wire at a temperature of 22.9 ^oC has a radius of 0.275 mm. If a potential difference of 11.8 V is applied across the length of the wire, determine the current in the wire.

If the wire is heated to 90.6 ^oC and the 11.8 V potential difference is maintained, what is the resulting current in the wire?

Answer 1

At T₀= 20°C copper has an electrical resistivity of ρ₀ = 1.68×10⁻⁸ Ω∙m
Its temperature coefficient as α = 3.9×10⁻³ K⁻¹

The resistivity at 22.9°C is
ρ = ρ₀ ∙ (1 + α∙(T - T₀) = 1.68×10⁻⁸ Ω∙m ∙ (1 + 3.9×10⁻³ K⁻¹ ∙ (2.9K) ) = 1.7×10⁻⁸ Ω∙m
The resistance of the wire is:
R = ρ∙L/A = ρ∙L/(π∙R²)
= 1.7×10⁻⁸ Ω∙m ∙ 39.5m /(π ∙ (0.275×10⁻³m)²) = 2.83 Ω
Current can be from Ohm's law:
I = V / R = 11.8V / 2.83 Ω = 4.17 A

At 90.6°C
ρ = 1.68×10⁻⁸ Ω∙m ∙ (1 + 3.9×10⁻³ K⁻¹ ∙ 70.6K ) = 2.14×10⁻⁸ Ω∙m
=> R = 2.14×10⁻⁸ Ω∙m ∙ 39.5m /(π ∙ (0.275×10⁻³m)²) = 3.56 Ω

I = 11.8V/3.56 Ω = 3.31 A