A resistor made of nichrome wire is used in an application, where the resistor’s resistance must not change more than 5.00 percent from its value at a temperature of 25.0 degrees Celsius. Find the highest temperature at which the resistor can be used. Use a value of 0.0004 inverse degrees Celsius for the temperature coefficient of resistance of nichrome.
So then, we have a wire made of nichrome material and we have a condition for the performance of this wire. Our condition is that the resistance of this wire can’t change more than 5.00 percent compared to the resistance of the wire when it’s at a temperature of 25.0 degrees Celsius.
The idea then is we’ll start at that temperature and we’ll go up as far as we possibly can until our resistance value is at our maximum change allowed, 5.00 percent. We want to solve for the highest temperature which corresponds to this percent change in resistance. We’re dealing then with a resistor that changes temperature. And as a result of that temperature change, its resistance changes.
These two quantities are connected by a mathematical relationship we can recall. The change in a resistor’s resistance Δ𝑅 is equal to a baseline or reference resistance value 𝑅 sub zero multiplied by 𝛼 the temperature coefficient of resistance multiplied by the change in temperature of the material.
In our case, the material is nichrome. And we have 𝛼 the temperature coefficient of resistance for nichrome given to us in the problem statement. That’s good news. But before we use that value for 𝛼, let’s consider Δ𝑅, the change in this wire’s resistance.
Whenever we talk about a change, we want to be aware of a change from what? In other words, what was our baseline? In this case, we’ve taken a measurement or a reading of the resistance of the wire at a particular temperature, 25.0 degrees Celsius. That as a matter of fact is 𝑅 sub zero, our reference resistance.
And based on this reference, we have a condition which is that we can never deviate more than 5.00 percent above or below this reference value. If we write 5.00 percent as a decimal, that would look like 0.05. When we multiply that by our baseline reading 𝑅 sub zero, that’s our change constraint for the resistance of the wire.
We can replace in fact Δ𝑅 with 𝑅 sub zero times 0.05. 𝑅 sub zero replaces the resistance 𝑅 and 0.05 replaces our Δ or our change. Now, 0.05 times 𝑅 sub zero is itself equal to something. It’s equal to the right-hand side of this expression. It equals 𝑅 sub zero times 𝛼 sub 𝑛, the temperature coefficient of resistance of nichrome, multiplied by Δ𝑇, the change in temperature of the resistor.
Looking at this equation, notice that 𝑅 sub zero, the reference resistance, cancels out from both sides. When we consider what we want to solve for in the remaining expression, we know it will have something to do with Δ𝑇. It’s not Δ𝑇 directly, but Δ𝑇 will help us solve for the highest possible temperature of the resistor within our constraint.
So let’s isolate Δ𝑇 by dividing both sides by 𝛼 sub 𝑛. Having done that, we’re now ready to plug in for 𝛼 sub 𝑛 the value of 0.0004 inverse degrees Celsius. And when we calculate this fraction, that is when we solve for Δ𝑇, we find a result of 125 degrees Celsius. Now, that’s not our answer; it’s our change in temperature from our original baseline value. Recall that that baseline was 25.0 degrees Celsius.
To find then our maximum possible temperature, we’ll call it 𝑇 sub max. We’ll take Δ𝑇 and we’ll add to it 25.0 degrees Celsius. 125 plus 25.0 gives us 150 degrees Celsius. That’s the maximum possible temperature this resistor can have and still have a resistance not deviating more than 5.00 percent from its value at the given temperature.