Find the change in temperature required to decrease the resistance of a carbon resistor by 25 percent. Use a value of negative 0.0005 inverse degrees Celsius for the temperature coefficient of resistance of carbon.
Here, we want to solve for a change in temperature; we can call it Δ𝑇. And perhaps, the first thing that stands out from this problem statement is that our temperature coefficient of resistance for this material — carbon — is negative. What does that mean?
Well, normally, if you heated up a resistor, then its resistance would increase; that is, for a resistor 𝑅 as its temperature goes up, its resistance goes up as well. It’s harder to flow current through it. But with this particular temperature coefficient of resistance is saying is that actually as you increase the temperature of carbon, its resistance goes down. This is opposite what we’d usually expect, but it’s a special property of carbon.
Now, recall that we said we wanted to solve for Δ𝑇, which is a change in temperature which corresponds to a decrease in the resistance of a carbon resistor by 25 percent, we have then a change in temperature, we’ve called that Δ𝑇, and we also have a corresponding change in resistance; we can call that Δ𝑅.
It turns out that Δ𝑇 and Δ𝑅 are related to one another through an equation. We can write that the change in resistance value of a given resistor is equal to a reference resistance of that resistor multiplied by its temperature coefficient of resistance 𝛼 all times the change in temperature of the resistor Δ𝑇.
If we apply this relationship to our special case, we can work from left to right to substitute in for the terms in this equation, which we can solve for or are given. First, Δ𝑅, which is the change in our resistor value, we know that that’s equal to 25 percent change. And in particular, it’s a decrease that as we’ve lost 25 percent of the original resistance.
If we call that original resistance our reference resistance 𝑅 sub zero, then that change in overall resistance can be represented as negative 0.25 times 𝑅 sub zero. Notice that 0.25 is the decimal representation of a 25 percent change. And our minus sign is there because it’s a decrease.
Moving on to the right-hand side of our equation, we don’t know what 𝑅 sub zero, our reference resistance, is. But we don’t have to because it appears on both sides of our equation. So it cancels out.
𝛼, the temperature coefficient of resistance of carbon, is given to us; that’s a negative 0.0005 inverse degrees Celsius. And Δ𝑇 as we said is what we want to solve for. To do that, let’s divide both sides of this equation by 𝛼 and then plug in for the given value of 𝛼.
Before we calculate this fraction to solve for Δ𝑇, notice two things: first the minus signs cancel out; our change in temperature then will be a positive number and also the overall units of this fraction are degrees Celsius.
Knowing that, when we calculate it, we find a nice round number of 500 degrees Celsius. That’s how much the temperature of carbon would have to go up in order for the resistance of carbon to go down by 25 percent.