Fields That ClickEpisode 2 / Why c ── speed is the square root of stiffness ÷ mass

In Episode 1 we said news "walks." So then ── what decides that walking speed?

Why c ── speed is the square root of stiffness ÷ mass A plucked string's sound, the ripples on water, light ── the speed of transmission is set by one and the same skeleton.
The square root of the ratio of stiffness to mass. And when a field's "mass" pins to zero, the ratio hits the ceiling c.

Tools you need: division and square roots, plus Episode 1's "only talks to its neighbors" This episode's ratio: v/c = √(stiffness/mass) = √(k/m)

In Episode 1 we said that force travels through a field, and since a field only talks to its neighbors, news "walks" at the speed of light $c$. So a naive question ── what decides that walking speed? Why 300,000 km per second, no faster and no slower? This episode's answer is disappointingly universal. The sound of a string, the waves of an earthquake, light ── the speed of transmission can all be written with the same single formula ── $v=\sqrt{\text{stiffness}/\text{mass}}$. The stiffer, the faster; the heavier, the slower. That's all. And when you build this "stiffness" and "mass" for the electromagnetic field, $c=1/\sqrt{\varepsilon_0\mu_0}$ just falls out. The why of "for some reason it travels at the speed of light" turns, this episode, into inevitability.

01The speed of the telephone game ── how hard it pulls, and how hard it is to move

Pluck the left end of a taut string. A dip runs to the right. Let's break down this "speed at which the dip runs" as a game of telephone. When one point dips, it tries to pull its neighbor up. This restoring strength is the stiffness $k$ (for a string, the tension). But the neighbor has an unwillingness to move. This is the mass $m$ (for a string, its density). Stiffness urges "pass it to the neighbor fast," and mass hits the brakes with "don't rush." The result of this tug-of-war is the speed of transmission ──

The wave speed ── the skeleton common to all waves

$$v=\sqrt{\dfrac{\text{stiffness}}{\text{mass}}}=\sqrt{\dfrac{k}{m}}$$

Stiffness $k$ is the numerator (bigger = faster), mass $m$ is the denominator (bigger = slower). And both sit inside a square root ── quadruple the stiffness and the speed only doubles.

Why a square root, in one line. Stiffness $k$ sets the "restoring force," mass $m$ sets "how hard it is to accelerate." The speed of vibration (angular frequency) is $\omega=\sqrt{k/m}$ ── the familiar spring pendulum. The wave speed is the speed at which that vibration transfers to the neighbor, so it wears the same $\sqrt{k/m}$ face. The $\sqrt{k/m}$ of a spring pendulum and the $\sqrt{k/m}$ of a wave speed are the same thing.

02Stiffer is faster, heavier is slower ── check it around you

With this one formula, you can read every "difference in speed" around you.

Let's try it ── the same formula appears again and again with different faces

A wave traveling on a string (tension T, linear density μ)

$$v=\sqrt{\dfrac{T}{\mu}}\quad(\text{stiffness}=T,\ \text{mass}=\mu)$$

Sound traveling through air or a solid (bulk modulus K, density ρ)

$$v=\sqrt{\dfrac{K}{\rho}}\quad(\text{stiffness}=K,\ \text{mass}=\rho)$$

Tightening a guitar string ($T\uparrow$) raises the pitch because $v\uparrow$. A thick string ($\mu\uparrow$) sounds low because $v\downarrow$. Sound runs through iron about 15 times faster than through air because iron is "stiff (large $K$) and, in the ratio, light." The names differ, but the skeleton is always $v=\sqrt{\text{stiffness}/\text{mass}}$.

So far, waves in visible matter. Now at last ── to the speed of light, running through a vacuum that should hold nothing.

03The speed of light too has the same skeleton ── c = 1/√(ε₀μ₀)

The electromagnetic field also has "stiffness" and "mass." The electric field has a reluctance to store up charge, and this is the electrical stiffness (the reciprocal of the vacuum permittivity $\varepsilon_0$, i.e. $1/\varepsilon_0$). The magnetic field has a stickiness = inertia resisting changes in current, and this is the magnetic mass (the vacuum permeability $\mu_0$). Put into the same skeleton ──

The speed of the electromagnetic wave (light)

$$c=\sqrt{\dfrac{\text{electric stiffness}}{\text{magnetic mass}}}=\sqrt{\dfrac{1/\varepsilon_0}{\mu_0}}=\dfrac{1}{\sqrt{\varepsilon_0\mu_0}}$$

Stiffness $=1/\varepsilon_0$, mass $=\mu_0$. Even letter for letter, it is the same as the string's $\sqrt{T/\mu}$.

Let's plug in numbers. Two constants measurable on a lab bench, with no connection to light ── $\varepsilon_0=8.85\times10^{-12}$, $\mu_0=1.26\times10^{-6}$ ──

Let's try it ── the speed of light comes out of the electric and magnetic constants alone $$c=\dfrac{1}{\sqrt{(8.85\times10^{-12})(1.26\times10^{-6})}}\approx 3.0\times10^{8}\ \text{m/s}$$

Using no light at all, the speed computed from an electricity experiment and a magnetism experiment alone was the speed of light itself, which had been measured beforehand. This is Maxwell's trembling line of 1862 ── "This speed agrees exactly with the speed of light. We can hardly avoid the conclusion that light is an electromagnetic wave." The true identity of light was the electromagnetic field's game of telephone.

Correspondence table ── "stiffness/mass" survives even when you switch fields String: stiffness $T$, mass $\mu$. Sound: stiffness $K$, mass $\rho$. Electromagnetic field: stiffness $1/\varepsilon_0$, mass $\mu_0$. ── The field changes, but the form of the ratio $v=\sqrt{\text{stiffness}/\text{mass}}$ alone stays invariant. The constants with units ($T,\mu,\varepsilon_0,\mu_0$) are stage sets; what decides the physics is the ratio. Episode 1's "quantities with units are stage sets" is at work here too.

04Let's play with it ── stiffness and mass change the speed

The figure below is a race. The top lane is a "spring chain" whose stiffness $k$ and mass $m$ you can freely change. The bottom lane is the reference: light traveling through a vacuum (speed $c$, unchangeable). They start at the same time ── which reaches the finish line first?

Raise the stiffness $k$ with the slider and the spring chain's pulse gets faster. Raise the mass $m$ and it gets slower. The displayed speed ratio is exactly $v/c=\sqrt{k/m}$. When $k=m$ (the default), the ratio is 1 ── it runs neck and neck with light. When stiffness and mass balance, that field's wave has the same speed as light. "Why c" is: "c is that field's own $\sqrt{k/m}$, set by the electromagnetic field's stiffness and mass."

Figure: a race between the spring chain's wave (top, adjustable stiffness k / mass m) and light traveling through a vacuum (bottom, reference c). The speed ratio is v/c = √(k/m). Raise k for faster, m for slower. Whoever reaches the finish line first is "faster"

stiffness k 1.00 mass m 1.00

spring chain's wave (v = c·√(k/m)) light (vacuum) = reference c

05Why a "ceiling" ── reading c dimensionlessly

Speed itself (m/s) has units. Strip the units and turn it into a ratio, and this episode's star appears ── compared to light, how much slower is that wave?

The slowness ratio ── refractive index n (unitless)

$$n=\dfrac{c}{v}\;(\geq 1)$$

In a vacuum (the electromagnetic field with nothing added), $n=1$ ── this is the ceiling. In glass or water, the electrons are shaken by the field and move along with it, adding extra "mass" so that $v1$. Light bends (refracts) in glass because this $n$ is greater than 1.

Here is the heart of "the ceiling." A medium can only add extra mass (the inertia of the moving charges). Add mass and $v$ drops, $n$ rises ── so in a vacuum, i.e. when the extra mass is zero, the speed pins to its maximum $c$. When nothing is added, it is fastest. Episode 6's Higgs field and next episode's mass are, in fact, both stories of "adding mass to a field" ── a field given mass becomes slower than $c$, and even its reach shrinks. Conversely, a massless field (the photon) hits the ceiling of the ratio, runs at exactly $c$, and reaches the infinite beyond. The why of "for some reason light-speed" was the inevitability that "the field's mass is zero, so $\sqrt{k/m}$ hits the ceiling."

◇ ◇ ◇

The honest line ── where it's rigorous, and where it becomes a "metaphor"

$v=\sqrt{\text{stiffness}/\text{mass}}$ is a rigorous skeleton genuinely common to all waves ── strings, sound, electromagnetic waves, water waves (it comes from the wave equation). $c=1/\sqrt{\varepsilon_0\mu_0}$, the refractive index $n=c/v\geq1$, that a medium lowers $v$, that the photon is massless and gives $c$ ── all established physics.

Two points, honestly, though. ① The electromagnetic field is not a substance called the ether quivering. The 19th century imagined it so, but experiment (Michelson–Morley) denied it. $\varepsilon_0,\mu_0$ are properties of the vacuum = the field itself, and "stiffness/mass" is a rigorous metaphor for the field (mathematically isomorphic to an electric circuit's $v=1/\sqrt{LC}$) ── take it that way. ② This episode explained "why that wave gives $c$," but "why $c$ is this universe's absolute speed limit (no information can exceed $c$)" is a deeper matter of relativity = the structure of spacetime, for another time. Here, get a firm grip on "a massless field has its $\sqrt{k/m}$ reach the maximum."

Practice problems (solvable with only this episode's formulas)

A string's tension $T$ was quadrupled. By what factor does the wave speed $v=\sqrt{T/\mu}$ change?
See the answer
$\sqrt{4}=2$ times. Since it's inside a square root, quadruple stiffness only doubles the speed. Read the other way: to merely double the speed, the tension need only be $\sqrt2$ times.
In the spring chain, stiffness $k$ was doubled and mass $m$ was multiplied by 8. By what factor does the speed ratio $v/c=\sqrt{k/m}$ change from the original?
See the answer
$\sqrt{(2)/(8)}=\sqrt{1/4}=1/2$. Half the original speed. Even making it stiffer (faster), if you make it heavier (slower) by more, the net is slower ── what matters is only the ratio of $k$ to $m$.
A certain glass has refractive index $n=1.5$. By what factor is the speed of light $v$ in the glass, relative to vacuum? Why does it become less than 1?
See the answer
$v=c/n=c/1.5\approx0.67\,c$. About 2/3 of vacuum. In glass, electrons shaken by the field move along with it, adding extra "mass (inertia)," so $v1$. A concrete case of "add mass, get slower."
"Only in a vacuum is $n=1$, and that is the maximum speed." Explain this fact in one line in the language of $v=\sqrt{\text{stiffness}/\text{mass}}$.
See the answer
A medium can only add "extra mass" to the field, and adding mass lowers $v$. So in a vacuum, which adds nothing (extra mass zero), $v$ is maximal = $c$, $n=1$. "The ceiling" means "the state of zero mass."

Episode 2 summarySpeed is √(stiffness/mass) ── c is the ratio pinned at its ceiling

What decides the speed of the news that "walked" in Episode 1? The answer is universal: the speed of any wave $=\sqrt{\text{stiffness}/\text{mass}}=\sqrt{k/m}$. Stiffness in the numerator (faster), mass in the denominator (slower), both inside a square root. String $\sqrt{T/\mu}$, sound $\sqrt{K/\rho}$, electromagnetic field $\sqrt{(1/\varepsilon_0)/\mu_0}=1/\sqrt{\varepsilon_0\mu_0}=c$ ── the names differ but the skeleton is one. Maxwell got $c$ from the electric and magnetic constants alone and saw that light is an electromagnetic wave.

Strip the units into a ratio and you get the refractive index $n=c/v\geq1$. A medium can only add extra mass to the field, so $v$ drops and $n>1$. Hence in a massless vacuum the speed pins to its maximum $c$. The why of "for some reason it travels at the speed of light" was the inevitability that "the carrier (photon) has zero mass, so $\sqrt{k/m}$ hits the ceiling." The unit-bearing $T,\varepsilon_0,\mu_0$ are stage sets; what decides the physics is the ratios $\sqrt{k/m}$ and $n$ ── the second step in reading a field through ratios.

Next time ── Episode 3 "Mass is the field's brake ── range is ℏ/mc." This time we said a massless field reaches infinity at $c$. So what happens when the carrier is heavy? A heavy field, while running, runs out of breath and decays, and its reach shrinks (Yukawa). That reach = range is a length set by the mass $m$: $\lambda=\hbar/mc$. Why electromagnetism reaches the edge of the universe while the weak force works only within a range smaller than an atomic nucleus ── that difference too can be read with just one brake called mass and the ratio of range to distance. On to the story of "a field given mass."

This document is Episode 2 of the "Fields That Click" series, a reading piece for high-schoolers and undergrads who love physics. That the phase speed of a wave derived from the wave equation takes the form $v=\sqrt{\text{restoring coefficient}/\text{inertia}}$ (string $v=\sqrt{T/\mu}$, sound $v=\sqrt{K/\rho}$, electromagnetic wave $c=1/\sqrt{\varepsilon_0\mu_0}$), that Maxwell derived the speed of light from the electromagnetic constants and established the electromagnetic theory of light, that the refractive index $n=c/v\geq1$ and the phase speed drops in a medium, and that the photon has zero rest mass and propagates at $c$ in a vacuum, are all established standard physics. The picture treating the electromagnetic field as a mechanical medium called the "ether" has been denied since the Michelson–Morley experiment, and $\varepsilon_0,\mu_0$ are properties of the vacuum (the field) itself. The "stiffness/mass" correspondence is a metaphor based on the mathematical isomorphism with a transmission line's $v=1/\sqrt{LC}$ ($1/\varepsilon_0\!\leftrightarrow\!1/C$, $\mu_0\!\leftrightarrow\!L$); that $c$ is itself the absolute limit of information transfer originates from special relativity (the structure of spacetime) and is beyond the reach of this episode's mechanical explanation. The figure is a schematic of the pulse propagation speed ratio $v/c=\sqrt{k/m}$, not a numerical solution of the lattice's rigorous equations of motion. ── To print, use your browser's "Print" → "Save as PDF" (in the print version the sliders and answers are frozen or hidden). Adjacent episodes: Episode 1 Delay / Contents.

Print / make PDF: ⌘+P (Ctrl+P on Windows). On screen, the stiffness k and mass m sliders change the speed of the spring chain's wave (= the speed ratio to light √(k/m)). "See the answer" opens the solutions.