# How can such an equation for the temperature of a *black body* be valid in this case?

I am currently studying The Art of Electronics, third edition, by Horowitz and Hill. Exercise 1.6 says the following:

C. Power in resistors
The power dissipated by a resistor (or any other device) is $$\P = IV\$$. Using Ohm’s law, you can get the equivalent forms $$\P = I^2 R\$$ and $$\P = V^2 / R\$$.
Exercise 1.6. Optional exercise: New York City requires about $$\10^{10}\$$ watts of electrical power, at 115 volts (this is plausible: 10 million people averaging 1 kilowatt each). A heavy power cable might be an inch in diameter. Let’s calculate what will happen if we try to supply the power through a cable 1 foot in diameter made of pure copper. Its resistance is $$\0.05 \ \mu \Omega\$$ ($$\5 \times 10^{−8}\$$ ohms) per foot. Calculate (a) the power lost per foot from “$$\I^2R\$$ losses," (b) the length of cable over which you will lose all $$\ 10^{10} \$$ watts, and (c) how hot the cable will get, if you know the physics involved ($$\ \omega = 6 \times 10^{-12} \text{W}/\text{K}^4 \text{cm}^2 \$$). If you have done your computations correctly, the result should seem preposterous. What is the solution to this puzzle.

We managed to get the answer to (a) here as $$\ 3.8 \times 10^8 \ \text{W}/\text{ft} \$$. I then get $$\ \dfrac{10^{10} \text{W}}{3.8 \times 10^8 \ \text{W}/\text{ft}} = 26.32 \ \text{ft} \$$ for (b).

Now, I'm trying to solve (c). I'm referring to this document, which claims that, to calculate the heat dissipated by the cable, we can use the Stefan-Boltzmann equation. However, according to the Wikipedia article, this equation describes the power radiated from a black body in terms of its temperature. How can such an equation for the temperature of a black body be valid in this case?

## EDIT

• You just assume it's valid at this stage and see what sort of answer you get. – user_1818839 Jan 2 at 16:46
• The document you linked ("this document") has nowhere in its text the phrase "heat dissipated". – Andy aka Jan 2 at 16:49
• @BrianDrummond But that doesn't answer my question. That GitHub document is not an official solutions manual (there is no solutions manual available for this edition of the textbook) – it's a crowdsourced solutions manual. So just because someone has claimed that that is the correct solution, does not mean that it is actually the correct solution. So I think it is a valid question as to whether the Stefan-Boltzmann equation is valid for this case. – The Pointer Jan 2 at 16:51
• @Andyaka The solution for (c) of exercise 1.6 in the document indeed has the phrase "heat dissipated": "To calculate the heat dissipated by the cable, ..." – The Pointer Jan 2 at 16:53
• Related (really!) to this question – TimWescott Jan 2 at 16:53