Chapter 27, Problem 53EAP

The resistivity of a metal increases slightly with increased temperature. This can be expressed as $\rho ={\rho }_0\left[1+a\left(T-T_0\right)\right]$ , where T_ois a reference temperature, usually $\text{2}0°\text{C}$ , and a is the temperature coefficient of resistivity.

1. First find an expression for the current / 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 $\Delta \text{V}$ . 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.
2. For copper, $\alpha =3.9\times 10-3°{\text{C}}^{\text{-1}}$ . Suppose a $\text{2}.\text{5}$ -m-long, $0.\text{4}0$ -mm-diameter copper wire is connected across the terminals of a 1.5 V ideal battery. What is the current in the wire at $\text{2}0°\text{C}$?
3. What is the rate, in A/°C, at which the current changes with temperature as the wire heats up?