The resistivity of a metal increases slightly with increased temperature. This can be expressed as ρ = ρ₀ [1 + α(T - T₀)], where T₀ is a reference temperature, usually 20°C, and α is the temperature coefficient of resistivity.
a. First find an expression for the current I through a wire of length L, cross-section area A, and temperature T when connected across the terminals of an ideal battery with terminal voltage ΔV. Then, because the change in resistance is small, use the binomial approximation to simplify your expression.
Your final expression should have the temperature coefficient a in the numerator.
b. For copper, α = 3.9 × 10^-3 °C^-1 . Suppose a 2.5-m-long, 0.40-mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at 20°C?
c. What is the rate, in A/°C, at which the current changes with temperature as the wire heats up?