The air temperature in a town varied between 15 degrees Celsius and 40 degrees Celsius on a certain summer day. Find the percentage change in the resistance of an exposed aluminum wire in the town during the day. Use a value of 0.0040 inverse degrees Celsius for the temperature coefficient of resistance of aluminum.
In this example then, we have a town going through a change in temperature over the course of a summer day. And there’s an aluminum wire in the town that runs through that same change in temperature. There is a connection we know between the change in temperature of a conductor and that conductor’s ability to resist electrical current. We want to solve for the percentage change in the resistance of the aluminum wire. And to do that, we’ll connect the change in resistance with the change in temperature.
As a mathematical relation, we can write that the change in resistance of a resistor is equal to 𝑅 sub zero, a baseline resistance value, multiplied by its temperature coefficient of resistance multiplied by the change in temperature that it moves through. In our instance, we can rewrite Δ𝑅, the change in resistance, as a change from the baseline resistance value we’ll call 𝑅 sub zero.
Let’s choose to let capital 𝐷 be a variable that represents the decimal change of this resistance value. So comparing this product to Δ𝑅, we can see that 𝐷 corresponds to the change, Δ, and 𝑅 sub zero corresponds to 𝑅, the original or baseline resistance value. Knowing that, we can write that 𝐷 times 𝑅 sub zero is equal to the right-hand side of this expression. It’s equal to 𝑅 sub zero times 𝛼 sub 𝑎, the temperature coefficient of resistance of aluminum, times Δ𝑇, the change in temperature of the wire.
Looking at this equation, we see that 𝑅 sub zero is common to both sides, and so it cancels out. We’ve isolated 𝐷, which is the percent change of resistance, on the left-hand side. So now we’ll want to plug in for 𝛼 sub 𝑎 and Δ𝑇. 𝛼 sub 𝑎 is given as 0.0040 inverse degrees Celsius, and the temperature change goes from a maximum of 40 degrees Celsius to a minimum of 15 degrees Celsius. Multiplying these terms together, we find a result of 0.10.
But that’s not our final answer because we want to express our answer as a percentage. We’ll call that answer capital 𝑃, and it’s equal to 𝐷, the decimal version of our answer, multiplied by 100 percent, giving us a final answer of 10 percent. That’s the percent change in the resistance of this exposed aluminum wire over the course of this day.