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Question
(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?
Question by OpenStax is licensed under CC BY 4.0.
Final Answer

a) $500000\textrm{C}^\circ$

b) $-250000\textrm{C}^\circ$

c) The sun's surface is about $6000^\circ\textrm{C}$, therefore a $500000\textrm{C}^\circ$ temperature rise is not reasonable. Constantan would change phase.

Absolute zero is $-273^\circ\textrm{C}$, so a $250000\textrm{C}^\circ$ temp decrease would require starting at a temperature so high that Constantan would be a plasma.

d) Assuming the termperature coefficient of resistivity is constant with changes in temperature is a false assumption.

Solution Video

OpenStax College Physics Solution, Chapter 20, Problem 39 (Problems & Exercises) (2:45)

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Video Transcript
This is College Physics Answers with Shaun Dychko. Constantine probably gets its name from the fact that its resistance does not change very much with temperature. Its temperature coefficient of resistivity is much smaller by three orders of magnitude than most other materials. It's going to be 0.002 times ten to the minus three per Celsius degree. So we're asked what temperature change should occur such that you nevertheless will double its resistance. So resistance is R naught times one plus alpha delta T and then we're told that R is two times R naught. So we substitute that in here and we get after we divide by <>R naught on both sides, we get one plus alpha delta t equals two. Then we subtract one from both sides and we get alpha delta t equals one. So the change in temperature is one over the temperature coefficient of resistivity after you divide both sides by alpha here. So that's one over 0.002 times ten to the minus three per Celsius degree which is 500,000 Celsius degrees which is a lot. In part B, it says suppose the resistance is one half of what it was before. So we substitute R naught over two in place of R and then divide both sides by R naught and we get one plus alpha delta t equals one half. Subtract one from both sides and we get alpha delta t is negative a half. Then divide both sides by alpha and you get delta t is negative one over two alpha. So that's negative one over two times 0.002 times ten to the minus three Celsius degrees which is negative 250,000 Celsius degrees. Well, let's consider how realistic those are. The sun's surface temperature is about 6,000 degrees Celsius and so a 500,000 Celsius degree temperature rise in part A is not reasonable because Constantine would certainly change its phase by then. It would probably become a plasma. Our answer for Part B is also unrealistic because absolute zero is negative 273 degrees Celsius so a 250,000 degree Celsius temperature decrease would require starting at a temperature that's so high the Constantine again would be a plasma and so this is not going to work in a terrestrial environment. So the false assumption here is that the Constantine's temperature coefficient of resistivity is constant with changes in temperature, whereas in fact, this alpha will in fact change as the temperature changes.