Not Quite Pi

Harys Dalvi

August 2021

In my AP Physics C E&M class, and on the exam, we were given a sheet with the equations and constants we needed. By the time the exam came, I had most of the equations and some of the constants memorized anyways; not because I tried to memorize them, but just through repeated exposure.

But there was one equation I saw somewhere else that I never had to use and never understood, but that fascinated me. $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$ Before even understanding this equation, it's immediately clear that it puts the “electromagnetic” in “electromagnetic wave”. It has the speed of an electromagnetic wave in a vacuum $c$, and the constants for both electricity ($\epsilon_0$) and magnetism ($\mu_0$). It unites quantities in an elegant and surprising way that reminds me of Euler's famous $e^{i \pi}+1=0$.

But with the new redefinition of SI units, this equation feels a little strange to me. This is because while $c$ is an exactly defined quantity, both $\epsilon_0$ and $\mu_0$ must now be determined experimentally [1]. At first, it's confusing for an exactly defined value $c$ to be based on two experimental and uncertain quantities.

As I looked into the redefinition more, I found an especially annoying change now that $\mu_0$ had to be determined experimentally. In this post, I'll talk about the 2019 SI redefinition, its sad implications for the magnetic constant $\mu_0$, and how we can explain the changes.

What's Weird?

It's possible to derive this equation from Maxwell's equations, which involve both $\epsilon_0$ and $\mu_0$ [2]. But like I said, in the SI system of units, $c$ is the more fundamental quantity at exactly 299,792,458 m/s. $\epsilon_0$ and $\mu_0$ must be determined experimentally.

I understand why they did the redefinition — they wanted the definitions to depend on unchanging physical constants that are the same throughout space and time. After all, this seems logical when you scrutinize the old method. There used to be a little platinum cylinder in Paris that weighed exactly a kilogram because SI said so. That sounds fine, but if the cylinder collects a little dust, or a few atoms falls off the cylinder, suddenly the mass of a kilogram changes for scientists all over the world. That's not just a hypothetical; the mass has varied slightly over time [3].

So that part of the redefinition is all good. But the redefinition had an unfortunate side effect that enraged my inner perfectionist. Before the redefinitions, we could say $$\mu_0 = 4 \pi \times 10^{-7} \ \text{H}/\text{m}$$ Now we have to say “$\mu_0$ is equal to approximately $4 \pi \times 10^{-7} \ \text{H}/\text{m}$ with a relative standard uncertainty of $2.3 \times 10^{-10}$.” I hate that: it's so close to $4 \pi$, but not quite there. It's so close that it can't be a coincidence. So why is it so close but not quite there?

Why ϵ₀ doesn't bother me

Let's go back to the equation from the beginning. $$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$ We can solve for $\epsilon_0$ to get $$\epsilon_0 = \frac{1}{\mu_0 c^2}$$ So in a system of units where we have exact values for both $\mu_0$ and $c$, we can also find an exact value for $\epsilon_0$. The thing is, our “exact value” would be

$$\frac{1}{(4\pi \times 10^{-7})(299792458)^2} = \frac{10000000}{359502071494727056\pi} \frac{\text{F}}{\text{m}}$$

This is technically an “exact value”, but it's not very convenient. That's why we've always used the decimal form for $\epsilon_0$, with or without the SI redefinitions. Either way, it's about $8.85 \times 10^{-12} \ \text{F}/\text{m}$.

In the new SI system, we can find $\epsilon_0$ by finding $\mu_0$ experimentally and then using this equation with the speed of light. The problem is not with $\epsilon_0$, but with $\mu_0$.

Why μ₀ does bother me

To understand why $\mu_0$ is now as frustrating as it is, we first need to understand why it was as perfect as it was.

The value so close to $4\pi \times 10^{-7}$ is, of course, not a coincidence. It has to do with the redefinition of the Ampere, one of the base SI units. Previously, the Ampere was defined as “the constant current which, if maintained in two straight parallel conductors of infinite length of negligible circular cross section and placed one metre apart in a vacuum, would produce between these conductors a force equal to $2 \times 10^{-7}$ newton per metre of length.” [4]

Let's consider this (very hypothetical) old definition more carefully. There are two wires one meter apart, each with a current of one Ampere. We can use Ampère's law to find the magnetic field of each wire. $$\oint\limits \mathbf{B} \cdot d \mathbf{l} = \mu_0 I$$ Where $\mathbf{B}$ is the magnetic field, and $I$ is the current enclosed by the path we are looking at. We will look at a circular path around the wire with radius $r$. Then the enclosed current $I$ is one Ampere, and the line integral becomes easy since the magnetic field is constant along the path. $$\oint d \mathbf{l} = 2\pi r$$ $$B = \frac{\mu_0 I}{2\pi r}$$ Now that we have the magnitude of the magnetic field $B$, we can find the force on the wire. Normally the force on a wire with current $I$ and magnetic field $B$ is $$F = I \ell B$$ In this case, $\ell = \infty$, but we can still solve for the force per unit length, which is what the definition is looking for: $F/\ell = IB$. Plugging in what we got for $B$ gives $$\frac{F}{\ell} = \frac{\mu_0 I^2}{2\pi r}$$ Remember, what we're really interested in right now is $\mu_0$: we want to find why it has such a neat value. Let's solve for it. $$\mu_0 = \frac{2\pi r (F/\ell)}{I^2}$$ If we look back at the definition, we see that $F/\ell = 2 \times 10^{-7} \ \text{N}/\text{m}$, $r = 1 \ \text{m}$, and $I = 1 \ \text{A}$. Let's plug this all in. $$\mu_0 = 4\pi \times 10^{-7} \ \text{H}/\text{m}$$ There's the number we all know and love.

All that was assuming the old definition of the Ampere. Currently, the Ampere is defined so that the elementary charge $e$ is equal to exactly $1.602175534 \times 10^{-19} \ \text{A} \cdot \text{s}$. But to avoid confusion, the redefined SI Ampere is of course close to the old value for the Ampere, but not quite there. That's why the new value for $\mu_0$ is close to $4\pi \times 10^{-7} \ \text{H}/\text{m}$, but not quite there.

We can also see from this equation that the value for $\mu_0$ depends on some other SI base units, such as the kilogram (because of the Newton), second (again because of the Newton), and the meter. The second and meter didn't change because they were already defined from physical constants: the second is defined based on the radioactive decay of Cs-113, and the meter is defined with the second by setting an exact value in m/s for the speed of light. The kilogram, on the other hand, did change: we no longer depend on the International Prototype of the Kilogram, that platinum cylinder. This is an additional source of change for $\mu_0$.

To put it another way, the force between two infinitely long wires 1 m apart with 1 A currents is no longer exactly $2 \times 10^{-7} \ \text{N}/\text{m}$, because both a Newton and an Ampere aren't quite equal to the same thing as before.

How I consoled myself

The loss of an exact value for $\mu_0$ is sad, but let's look at the bigger picture.

Humans have tried to standardize units for millenia. Ancient Egyptians used their arms and minds to come up with a standardized cubit for measurement, our first recorded measurement as a species. The Indus River civilization is well-known for its uniform standardization of weights using decimal and binary relations. Today in America we have pounds, inches, and gallons left over from British times.

In the context of this history of unit standardization, almost as old as history itself, it's a huge milestone we've reached in 2019: now all our units are finally defined based on physical constants, or by “the Universe rather than the Earth” to put it metaphorically. Regardless of small imperfections like this (and the mole, another topic that annoys me with the redefinition) this is a huge accomplishment.

Also, electricity and magnetism can be united with relativity. Although I haven't learned exactly how yet, I see that this means magnetism isn't as fundamental as it seems from the AP electricity and magnetism curriculum alone: it's really a consequence of more fundamental interactions. For me, this means the constant for magnetism doesn't need such a special value. Things like the speed of light or elementary charge are more fundamental and more deserving of exact values.

And most important of all, I'm not in a situation where the difference matters. I still type $4\pi \times 10^{-7}$ into my calculator when I work with $\mu_0$. And when I do, even though I know I'm technically not quite right, I shrug and say “close enough, at least it looks neat on the screen.”

References

The International System of Units (SI) (U.S. Department of Commerce) ^
Maxwell's equations (University of New South Wales) ^
This is why physicists just completely redefined the kilogram (Matt Reynolds, Wired) ^
Ampere (Encyclopædia Britannica) ^