Video: Calculating the Length Decrease of an Object that Decreases in Temperature

The temperature of a tungsten wire is reduced from 25°C to −15°C . Find the percent decrease in the wire’s resistance. Use a value of 0.0045°C⁻¹ for the temperature coefficient of resistance of tungsten.

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Video Transcript

The temperature of a tungsten wire is reduced from 25 degrees Celsius to negative 15 degrees Celsius. Find the percent decrease in the wire’s resistance. Use a value of 0.0045 inverse degrees Celsius for the temperature coefficient of resistance of tungsten.

In this example then, we have a wire made out of tungsten. And we’re told that the temperature of this wire is decreased and that the wire’s resistance, as a result of the temperature decrease, also goes down. We know just how much the temperature changed but we don’t yet know, and want to solve for, how much the resistance decreased.

Since we’re talking about a temperature change and a resistance change, there’s a particular relationship that relates these two quantities that we’ll find useful. That relationship says that the change in resistance of a resistor, Δ𝑅, is equal to a baseline or a reference resistance value multiplied by the temperature coefficient of resistance of that material multiplied all by its change in temperature.

Looking back to our problem statement, we see we want to solve for the percent decrease in the wire’s resistance. In other words, we wanna focus in on Δ𝑅. We can write the term Δ𝑅 as the product of two other terms. First, Δ𝑅 is related to the reference temperature 𝑅 sub zero. 𝑅 sub zero represents a baseline amount. And then, it’s modified by an amount, capital 𝐷, which we’ll let represent the decimal change in the resistance of this wire. So the capital 𝐷 then is an expression of our Δ and 𝑅 sub zero is an expression of 𝑅.

Having rewritten Δ𝑅 this way, we can now equate it to the right side of this equation. And we find that 𝑅 sub zero times capital 𝐷 is equal to 𝑅 sub zero times 𝛼 times Δ𝑇. And we see that the reference resistance 𝑅 sub zero cancels from both sides. What we have then is an expression for the decimal decrease in the wire’s resistance. That’s part way to our solution of the percent decrease.

What we’ll do now is plug in for 𝛼 and Δ𝑇. 𝛼, the temperature coefficient of resistance of tungsten, is given as 0.0045 inverse degrees Celsius. And the change in temperature of this resistor is 40 degrees Celsius. That 40 is based on the difference between negative 15 degrees Celsius and positive 25 degrees Celsius. Remember, we’re going over from positive to negative. We see the units in this expression cancel one another out. And when we multiply these two values, we find a result of 0.18. But that’s not our final answer because we want our answer as a percent not a decimal.

To make the conversion to solve for what we’ll call capital 𝑃, which is the percent decrease in the wire’s resistance, we’ll take capital 𝐷 and we’ll multiply it by 100 percent. That gives us 18 percent and that is the result we’re asked for. By the temperature of the tungsten wire going down 40 degrees Celsius, its resistance also decreases by 18 percent.

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