Reader S. Cho wrote in with the following question:

Hello! I have read your book, Everyday Calculus. It is very interesting and gives me a knowledge for math. I realized that math is all around us. But I have two questions in the book. First, I met the formula “r(V)=k sqrt(P_0 l) over V” in chapter 1. However I couldn’t find this formula in any other books. Could you give me the source of it or a reference book? Second, does P_0 mean the electric power? I’m waiting for your reply. Sincerely yours.

The formula $r(V)=k\frac{\sqrt{P_0l}}{V}$ comes from the combination of the three formulas:

$P_0=IV$,         $R=\rho \frac{l}{\pi r^2}$,          $V=IR$.

The first formula relates the voltage ($V$) and current ($I$) produced by a power source ($P_0$). The second is Pouillet’s Law, which relates a power line’s resistance to the flow of electric current $R$ to the power line’s length $l$ and radius $r$ (I’ve assumed the power line is a long cylinder). The constant $\rho$ is called the electrical resistivity. The last formula is Ohm’s Law.

Now, if we solve Ohm’s Law for $I$ and plug that into the first equation we get

$P_0=\frac{V^2}{R}$.

Then, using Pouillet’s Law we get

$P_0=\frac{V^2(\pi r^2)}{\rho l}$.

Solving this for $r^2$ yields

$r^2=\frac{\rho}{\pi}\frac{P_0l}{V^2}$.

Taking the square root of both sides and letting $k=\sqrt{\rho/\pi}$ gives

$r=k\frac{\sqrt{P_0l}}{V}$.

In the first chapter of Everyday Calculus I explain how this equation spelled doom for Thomas Edison. His early power grid operated at a fixed voltage of 110 Volts, which meant that as the length of the power line ($l$) increased, the power line’s radius ($r$) got bigger too. To avoid large power lines hanging over pedestrians (clearly a danger for anyone walking underneath them) Edison therefore had to build his power plants very close to his customers. This represented a huge cost, and limited the early usage of electricity. But the same equation also tells us that we can avoid this issue by using much larger voltages, which is what’s done today. These high voltages coming out of the power plants–in some cases nearly 1 million Volts–are reduced to the familiar 120 Volts we see in homes using transformers, which are the result of other neat physical principles (and math) at work.