A copper wire has a resistance of 0.500 ohms at 20.0 degrees Celsius and an iron wire has a resistance of 0.525 ohms at the same temperature. The temperature coefficient of resistance of copper is 0.00390 inverse degrees Celsius and the temperature coefficient of resistance of iron is 0.00650 inverse degrees Celsius. At what temperature do the resistances of the wires equal each other?
In this example then, we have a copper wire and an iron wire that we’re comparing. These two wires have different resistances which are given to us in the problem statement and they also change based on temperature differently. That’s what this value the temperature coefficient of resistance is. It tells us just how the resistance of an object changes with temperature.
Knowing that these two wires have different resistances and different temperature coefficients of resistance, we want to figure out at what temperature do the resistance values equal one another. There is a mathematical relationship that connects some of the terms that have come up in this problem statement. Let’s start by looking at that.
This equation we’ve written here is an equation for resistance as a function of the resistor’s temperature 𝑇. It tells us that that resistance value of a resistor is equal to some reference resistance 𝑅 sub zero multiplied by the quantity one plus 𝛼, where 𝛼 is the temperature coefficient of resistance for the material multiplied itself by the temperature 𝑇 of the material minus our reference temperature 𝑇 sub zero.
Clearly, there are a lot of terms in this equation. But overall, we can recall that what it does is to calculate the resistance based on temperature. And in this exercise, we’re told that the resistances of these two wires are equal to one another at some temperature 𝑇. That means we can write out 𝑅 as a function of 𝑇 for both copper and for iron and then set those equations equal to one another because we’re told to assume the resistances are the same.
So here’s how we’ll do that. On the left-hand side of this expression here, we’ve written out the equation for the resistance of copper as the function of temperature. Notice that we’ve left our temperature capital 𝑇 unknown since that’s what we want to solve for. We’ve set this resistance of the copper wire equal to the resistance of the iron wire when it’s at that same temperature capital 𝑇.
Notice that we’ve used different values for the resistances of these two wires and we’ve also used different values for their temperature coefficients of resistance. But as far as the reference temperature 𝑇 sub zero, that’s the same in each case.
There’s a lot going on here, but recall that what we want to do is solve for this variable, capital 𝑇. It’s at that temperature capital 𝑇 that the resistances of the wires are equal. What we want to do then is algebraically rearrange this expression to solve for capital 𝑇.
To simplify the process of doing that, there is a small substitution we can make. Let’s let the quantity capital 𝑇 minus 𝑇 sub zero be represented by capital 𝑋. That way, when we go through our calculations, there will be one last term to think about. And of course, we’ll have to remember this correction once we get to the end.
Now, with that substitution made, what we want to do is solve for capital 𝑋 in this equation. To start doing that, let’s multiply through using our different resistor values: 𝑅 sub Cu and 𝑅 sub Fe. This gives us four total terms, two on each side of our equation. And if we subtract 𝑅 sub Fe from both sides and then if we subtract 𝑅 sub Cu times 𝛼 sub Cu times 𝑋 from both sides of the equation and then after factoring out the quantity 𝑋 from the right side of this expression, we can divide both sides of our equation by the quantity in this parenthesis.
And once that’s done, we now have reached our goal of finding an expression for 𝑋. At this point though, let’s recall that it’s not capital 𝑋 we want to solve for, but actually the temperature which we’ve called capital 𝑇 and that 𝑋 is equal to this temperature minus some reference temperature we’ve called 𝑇 sub zero.
So really, all this on the left side is equal to capital 𝑇 minus 𝑇 sub zero or if we add 𝑇 sub zero to both sides, we now have what we set out to get in the first place, an expression for the temperature capital 𝑇 at which the resistances of the wires are equal.
Now, it’s a matter of carefully plugging in the appropriate values for the different terms in this expression. We can find those values in our problem statement. 𝑅 Sub Cu is 0.500 ohms and 𝑅 sub Fe is 0.525 ohms. And then, 𝛼 sub Fe, the temperature coefficient of resistance of iron, is 0.00650 inverse degrees Celsius. And 𝛼 sub Cu is the temperature coefficient of resistance of copper, which is given as 0.00390 inverse degrees Celsius.
And then, lastly, we’ll plug in for the reference temperature 𝑇 sub zero. That reference temperature is 20.0 degrees Celsius. With all those values plugged in, we’re ready to calculate capital 𝑇.
When we do, we find a result of 2.91 degrees Celsius. That’s the temperature at which the resistances of the wires are equal.