Fields That ClickEpisode 4 / Strength Is a Number Without Units — α ≈ 1/137

Episode 3 asked "how far does it reach?" This time — "how strong is it?" The 1/137 that starred in our sister series "Cosmology That Clicks" comes back, now in the language of fields

Strength Is a Number Without Units "How many newtons strong is the electromagnetic force?" — this question actually has no answer. The "strength" of a force
doesn't depend on charge, distance, or units; it condenses into a single number without units, α ≈ 1/137.

Tools you need: division, comparing exponents, c from Episode 2 and ℏ from Episode 3 This episode's number: α = e²/(4πε₀ℏc) ≈ 1/137

In Episode 3 we read how far a force reaches (its range). What's left is how strong a force is. But the moment you try to measure "strength" naively, you hit a wall. The Coulomb force can be any number of newtons you like, depending on the charges and the distance — halve the distance and it goes up fourfold; switch the units to dynes and the number changes too. Can we express "strength" in a form that depends on neither charge, nor distance, nor units? One of the most beautiful answers in twentieth-century physics is exactly this. The strength of the electromagnetic force condenses into a single number, obtained by combining \(e,\ \varepsilon_0,\ \hbar,\ c\) so exquisitely that the units cancel completely — the fine-structure constant \(\alpha\approx1/137\). This episode reads, using ratios alone, why the strength of a force turns into "a single number without units," and what comes into view when you line up the four forces by that number.

01Can You Measure "Strength" in Newtons? — The Trap of Units

The Coulomb force between two electrons is \(F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r^2}\). Can we call this \(F\) "the strength of the electromagnetic force"? No. Depending on the distance \(r\) it can be any number of newtons, and if you change the system of units the number changes too. \(F\) is a number about the situation, not the strength of the force itself. To talk about a physical "strength," you need a pure number that depends on neither the situation nor the units. A quantity that carries units depends on the meters and seconds we chose — the point we repeated in Episodes 1 and 2, "a quantity with units is stage machinery," now comes into play head-on.

02Erasing the Units — the Single Number Called α

Let's line up the basic quantities that appear in electromagnetism. The charge \(e\), the property of the vacuum \(\varepsilon_0\), the quantum \(\hbar\) (Episode 3), and the speed of light \(c\) (Episode 2). Combine them so that the units cancel exactly, and ——

The fine-structure constant — the strength of the electromagnetic force (no units)
$$\alpha=\frac{e^2}{4\pi\varepsilon_0\,\hbar c}\approx\frac{1}{137.036}\approx0.0073$$

When you assemble the numerator and denominator, the units vanish completely. What remains is the pure number \(1/137\). Whether you measure in meters or in inches, whether a Martian measures it, it's the same 1/137. This is the true body of "the strength of the electromagnetic force," independent of situation and of units.

That one point — that there are no units — is decisive. The "\(3\)" in \(3\) meters depends on the convention of your ruler, but the \(1/137\) of \(\alpha\) depends on nobody's convention. A bare number chosen by the universe. That's why physicists have spent a century asking "why 137?" — the values of \(c\) or \(e\) are products of units, but \(\alpha\) is a genuine mystery worth asking about.

A handshake with the sister series "Cosmology That Clicks" This \(\alpha\) was the star of the entire sister series "Cosmology That Clicks." Its backbone there, \(c\cdot t=\text{constant}\), ran along the line "as long as \(\alpha\) stays invariant, it leaves no trace in observations" — the claim that even if you swap out the interpretation of the units-carrying \(c\), as long as the dimensionless \(\alpha\) doesn't move, the physics is the same. Here in "Fields That Click," we reread that \(\alpha\) as the strength of the bond between field and charge. The same 1/137: cosmology sees it as an "invariant fingerprint," field theory sees it as "coupling strength" — two faces of a single number.

03What Does α Measure? — the "Effectiveness" of Trading One Photon

The physical meaning of \(\alpha\), in the language of fields, is this — the effectiveness with which a charged particle trades one quantum of the electromagnetic field (a photon). The likelihood of the "single handshake" in which an electron absorbs or emits one photon is \(\sqrt{\alpha}=e\) (in natural units), and as a probability it is \(\alpha\approx1/137\).

Try it — the cost of one "handshake" $$\text{amplitude of one handshake}\ \sim\ \sqrt{\alpha}=\sqrt{1/137}\approx\frac{1}{11.7}$$

Every time an electron trades one photon, the probability "shrinks" by \(\alpha\approx1/137\). So in electromagnetic calculations, the more handshakes there are, the smaller the terms get — \(\alpha,\ \alpha^2,\ \alpha^3\dots\) — and a few handshakes give you almost the whole answer (perturbation works). That \(\alpha\) is much smaller than 1 — this "smallness" is the reason electromagnetism can be computed with precision. If \(\alpha\) were close to 1, handshakes would matter without end and it would be unmanageable (that's the strong force: more below).

So \(\alpha\) is the indicator of how strongly field and charge grip each other. The larger it is, the more strongly they couple; the smaller, the more weakly. The "strength" of a force was, all along, this coupling strength.

04Play With It — Lining Up the Four Forces, Dimensionlessly

A "strength" with the units erased lets us compare forces on the same footing. Each of the four forces has a corresponding dimensionless coupling. The figure below lines up those strengths on a logarithmic ruler (one tick = a factor of ten). Farther right is stronger.

The strong force has \(\alpha_s\sim1\) (far right), electromagnetism is \(\alpha\approx1/137\), and the bare coupling of the weak force sits between them. And gravity alone lies far off to the left — weaker by orders of magnitude. Gravity's dimensionless coupling is set by the particle's mass, \(\alpha_G=(mc^2/E_{\text{Planck}})^2\). Slide the particle's mass from the electron up to the proton and the gravity marker moves right, but it stays dozens of orders of magnitude away from the other forces. For two electrons, gravity is about \(10^{-43}\) of electromagnetism. See this staggering gulf with your own eyes.

Figure: the four forces' dimensionless couplings compared on a log axis (one tick = ×10). Farther right is stronger. The gravity marker moves with particle mass: α_G=(mc²/E_Planck)². Even moving between the electron (0.511 MeV) and the proton (938 MeV), gravity stays dozens of orders of magnitude away from the other forces
strong / electromagnetic / weak (couplings fixed) gravity (moves with particle mass)

05Why Does the Weakest Force, Gravity, Rule the Universe?

Here a puzzle arises. Gravity is the weakest force, just \(10^{-43}\) of electromagnetism. And yet planets, stars, galaxies — the large-scale structure of the universe is ruled by gravity. Why does the weakest win? The answer lies in the fact that \(\alpha\) was a number without units, plus one more thing — the way things add up.

A distance-independent ratio — gravity ÷ electricity for two electrons
$$\frac{F_{\text{grav}}}{F_{\text{elec}}}=\frac{G m_e^2}{\,e^2/4\pi\varepsilon_0\,}=\frac{\alpha_G}{\alpha}\approx 2.4\times10^{-43}$$

Both thin out as \(1/r^2\) (Episode 5), so the \(r\) cancels cleanly — this ratio is a pure number independent of distance. Up close or across galaxies, for two electrons gravity is \(10^{-43}\) of electricity.

So why can it win? Electric force has positive and negative signs, and matter is usually neutral (equal amounts of + and −). So in a large lump the electric forces almost entirely cancel to zero. Gravity's "charge," on the other hand, is just mass, with a single sign. No cancellation happens; mass only ever piles up. The gravity of Earth's \(10^{50}\) atoms simply all adds together and pins us to the ground. Even a weakness of \(10^{-43}\), piled up \(10^{50}\) times over, wins by orders of magnitude. A force that is weak but always adds up beats a force that is strong but cancels out, winning by reversal at scale. This is the trick behind "the weakest force, gravity, rules the universe." Strength (\(\alpha\)) and the presence or absence of sign — the product of these two properties decides how the universe looks.

◇ ◇ ◇
The honest line — how far is "1/137" really a single number?

That \(\alpha=e^2/4\pi\varepsilon_0\hbar c\approx1/137.036\) is dimensionless and represents the coupling strength of the electromagnetic interaction; that \(\alpha_G=Gm^2/\hbar c=(m/M_{\text{Planck}})^2\) is the dimensionless gravitational coupling of two masses; that for two electrons \(F_{\text{grav}}/F_{\text{elec}}\approx2.4\times10^{-43}\) is independent of distance; and that in neutral matter the electric forces cancel while gravity accumulates and prevails macroscopically — all of these are established physics.

But two caveats. ① \(\alpha\approx1/137\) is the low-energy value; in fact it slowly changes (runs) with the fineness (energy) at which you measure — at high energy it is about \(1/128\). The four couplings are not fixed values; they move with energy, and there are hints that at high energy they approach a single point (unification — see the sister series "Force That Clicks" Episode 8 / "Cosmology That Clicks" Episode 6). The values in the figure are a low-energy snapshot. ② How you present the "strength" of the weak force is subtle: its bare coupling \(\alpha_w\sim1/30\) is comparable to electromagnetism, yet because its carrier W is heavy and short-ranged (Episode 3) it looks weak in everyday life — "strength (\(\alpha\))" and "effectiveness (range × strength)" are different things. In the figure we placed the bare coupling. Lining gravity up in the same row via \(\alpha_G\) is also a convenience; gravity is not a gauge coupling in the same sense as the other three.

Practice problems (solvable with just this episode's formulas)
  1. Why is it fitting that \(\alpha=e^2/4\pi\varepsilon_0\hbar c\), being "a number without units," is "the true body of strength"? State it in contrast with the units-carrying force \(F\) (in newtons).
    Show the answer
    \(F\) is a "quantity about the situation" whose number changes with distance and system of units, but \(\alpha\) has its units erased, so it is the same 1/137 regardless of distance, units, or observer. Being a pure number independent of situation, it can serve as the indicator of the force's own strength.
  2. The probability of one electromagnetic "handshake" is \(\alpha\approx1/137\). About how much smaller is the contribution of two handshakes? Why can electromagnetism be computed precisely?
    Show the answer
    Two handshakes give \(\alpha^2\approx(1/137)^2\approx5\times10^{-5}\). With each additional round the terms shrink by another factor of \(\alpha\), so it converges in a few rounds (perturbation works). Because \(\alpha\ll1\), it can be computed precisely.
  3. Estimate the proton's \(\alpha_G=(mc^2/E_{\text{Planck}})^2\) using \(mc^2=938\) MeV and \(E_{\text{Planck}}=1.22\times10^{22}\) MeV.
    Show the answer
    \((938/1.22\times10^{22})^2=(7.7\times10^{-20})^2\approx5.9\times10^{-39}\). Larger than the electron's \(1.75\times10^{-45}\) (because the proton is heavier), yet still 36 orders of magnitude below electromagnetism's 1/137.
  4. Gravity is only \(10^{-43}\) of electromagnetism, so why can it dominate stars and galaxies? Give two reasons.
    Show the answer
    (1) Gravity's source (mass) has a single sign, so it doesn't cancel — it only piles up. (2) Since matter is neutral, electric forces almost entirely cancel to zero at macroscopic scales. Hence in a large lump, weak-but-accumulating gravity beats strong-but-cancelling electric force by reversal.

Episode 4 SummaryStrength Is the Single Number 1/137 — the True Body With Units Erased

A force's "strength" cannot be measured in newtons (that's a situation-dependent quantity that changes with distance and units). Combine \(e,\varepsilon_0,\hbar,c\) so the units erase, and the strength of the electromagnetic force condenses into the pure number \(\alpha=e^2/4\pi\varepsilon_0\hbar c\approx1/137\). Because there are no units, it is the same for whoever measures it and in whatever units — a bare number chosen by the universe, and the star of the sister series "Cosmology That Clicks" too. Physically it is the effectiveness with which a charged particle trades one photon, and because \(\alpha\ll1\), electromagnetism can be computed precisely.

Having erased the units, we can line up the four forces on the same footing: strong \(\sim1\), electromagnetic \(1/137\), weak (bare coupling) in between, gravity far below (\(10^{-43}\) of electromagnetism for electrons). Moreover \(F_{\text{grav}}/F_{\text{elec}}\) has both going as \(1/r^2\), so the \(r\) cancels — a pure number independent of distance. That the weakest force, gravity, still rules the universe is because mass has a single sign and doesn't cancel but piles up. The strength of a force, too, is in the end a single dimensionless number — the fourth step in reading fields by ratios.

This document is Episode 4 of the "Fields That Click" series, reading for physics-loving high-schoolers and undergraduates. That the fine-structure constant \(\alpha=e^2/4\pi\varepsilon_0\hbar c\approx1/137.036\) is dimensionless and gives the strength of the electromagnetic coupling; that \(\alpha\) is the coupling strength at the vertex (photon emission/absorption) and serves as the small parameter of the perturbation expansion; that the dimensionless gravitational coupling is given by \(\alpha_G=Gm^2/\hbar c=(m/M_{\text{Planck}})^2\) (\(M_{\text{Planck}}c^2\approx1.22\times10^{22}\) MeV), giving \(\alpha_G\approx1.75\times10^{-45}\) for a pair of electrons and \(F_{\text{grav}}/F_{\text{elec}}\approx2.4\times10^{-43}\) (distance-independent); and that in neutral matter electromagnetic forces cancel while gravity accumulates and prevails macroscopically — all of these are established standard physics. That \(\alpha\) is a renormalization-group-dependent quantity that runs with energy scale, with \(1/137\) being the low-energy limit (about \(1/128\) at \(M_Z\)); that the weak and strong couplings run too; and that the bare weak coupling \(\alpha_w\sim1/30\) is comparable to electromagnetism, its "weakness" coming mainly from the short range due to the W mass (Episode 3) — these are stated explicitly in the "honest line" in the body. The figure's weak/strong/electromagnetic values are representative low-energy figures, and lining gravity up in the same row via \(\alpha_G\) is a convenience for comparison (gravity is not a gauge interaction of the Standard Model). — To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden). Adjacent episodes: Episode 3, Range / Table of contents.

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the particle-mass slider moves gravity's dimensionless coupling (electron ↔ proton). "Show the answer" opens each solution.