Many Formulas, One Structure

Explainer Article

Many Formulas, One Structure

Physics has many different formulas for the same constant. When each formula is decomposed into Planck-unit dimensions, the compound constants cancel — and every formula reduces to the same simple expression, regardless of which constants appeared in the starting formula. Fifteen constants, dozens of formulas, one rule.

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What this page shows
Each section below takes a well-known physical constant and lists its standard SI formulas — sometimes two, sometimes four entirely different expressions. Below each set of formulas, every constant is replaced by its Planck-unit expression. The intermediate Planck factors cancel, and every formula arrives at the same simple result: a product of Planck units and dimensionless ratios that translate unit-invariant physics into the arbitrary units we use for measurement and calculation. The cancellations are shown explicitly so you can verify each step.

The constants of physics — $\hbar$, $c$, $\varepsilon_0$, $\mu_0$, and so on — are composites of Planck units. Different formulas combine these composites in different ways, but the underlying Planck-unit building blocks are the same. When you substitute the Planck expressions and simplify, the compound factors cancel and a common structure emerges.

This is not a new physical claim. It is an algebraic identity — verifiable line by line. What makes it noteworthy is how cleanly and universally the cancellations work, and how the result isolates the physical content of each constant: a characteristic combination of Planck-scale quantities and the dimensionless ratios $\alpha$ and $m_{\mathrm{e}}/m_{\mathrm{P}}$.

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The one rule behind every equation on this page
Every universal constant is a package of Planck units — dressed, at most, with a geometric factor of $2\pi$ or $4\pi$ and the coupling ratio $\alpha$. Any formula relating the constants is therefore an identity among the same four Planck units, and it has no choice but to cancel down to the same universal form. The dozens of equations below are not dozens of separate facts. They are one rule, checked dozens of times.

The entire page in one sentence: physics quantifies nature with some fifteen “fundamental” constants, and every one of them reduces to a short product of the four Planck units ($l_{\mathrm{P}}$, $m_{\mathrm{P}}$, $t_{\mathrm{P}}$, $q_{\mathrm{P}}$), occasionally scaled by $\alpha$, the electron mass ratio $m_{\mathrm{e}}/m_{\mathrm{P}}$, and a factor of $2\pi$ or $4\pi$. Nothing else is needed. If you want the punchline before the proofs, the summary table at the bottom of the page collects all fifteen results in one place; the sections below show every cancellation that gets there.

This page uses the MathJax cancel extension for strikethrough notation. If the cancellation marks do not render, refresh the page.

$\require{cancel}$

The building blocks: six constants in Planck-unit form

Every decomposition on this page starts from the same set of substitutions. The universal constants that appear in the formulas below — $c$, $G$, $\hbar$, $\mu_0$, $\varepsilon_0$, and $e$ — can each be expressed as a product of four Planck units: $l_{\mathrm{P}}$ (length), $m_{\mathrm{P}}$ (mass), $t_{\mathrm{P}}$ (time), and $q_{\mathrm{P}}$ (charge).

$$\begin{aligned} c &= \frac{l_{\mathrm{P}}}{t_{\mathrm{P}}} \\[1.2em] G &= \frac{l_{\mathrm{P}}^3}{m_{\mathrm{P}}\, t_{\mathrm{P}}^2} \\[1.2em] \hbar &= \frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}} \qquad\qquad h \;=\; 2\pi\hbar \;=\; \frac{2\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}} \\[1.2em] \mu_0 &= \frac{4\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}}{q_{\mathrm{P}}^2} \\[1.2em] \varepsilon_0 \;=\; \frac{1}{\mu_0\, c^2} &= \frac{q_{\mathrm{P}}^2\, t_{\mathrm{P}}^2}{4\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^3} \\[1.2em] e^2 &= \alpha\, q_{\mathrm{P}}^2 \end{aligned}$$

The Planck units $l_{\mathrm{P}}$, $m_{\mathrm{P}}$, $t_{\mathrm{P}}$ are defined from $c$, $G$, and $\hbar$ — the three constants that set nature’s scales for length, mass, and time. The Planck charge $q_{\mathrm{P}} = \sqrt{4\pi\varepsilon_0\hbar c}$ extends the system to electromagnetism. The fine-structure constant $\alpha \approx 1/137.036$ is a dimensionless measure of the elementary charge relative to the Planck charge. The electron-to-Planck mass ratio $m_{\mathrm{e}}/m_{\mathrm{P}} \approx 4.185 \times 10^{-23}$ also appears in several results.

These expressions are exact algebraic identities. Every decomposition below substitutes one or more of them into a standard formula and simplifies. When derived constants appear — such as the impedance of free space $Z_0$ or the von Klitzing constant $R_{\mathrm{K}}$ — their standard definitions are given so the chain of substitutions can be followed step by step.

$\alpha$ — Fine-structure constant

Three SI formulas
$$\alpha \;=\; \frac{e^2}{4\pi\varepsilon_0 \hbar c} \qquad \alpha \;=\; \frac{Z_0}{2\,R_{\mathrm{K}}} \qquad \alpha \;=\; \sqrt{\frac{2\,R_\infty\, h}{m_{\mathrm{e}}\, c}}$$

Three formulas built from entirely different constants. Each one reduces to the same thing: the pure number $\alpha$, with every Planck-unit factor cancelling exactly.

Definitions of derived constants used here: $Z_0 = \mu_0\, c$ (impedance of free space), $R_{\mathrm{K}} = h/e^2$ (von Klitzing constant), $R_\infty = m_{\mathrm{e}}\,\alpha^2\, c/(2h)$ (Rydberg constant).

Formula 1

$$\begin{aligned}
\frac{e^2}{4\pi\varepsilon_0 \hbar c} &= \frac{1}{4\pi}\,(\alpha\, q_{\mathrm{P}}^2) \left(\frac{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}\right) \left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}\right) \left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}}\right) \\[1.25em]
&= \alpha\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{l_{\mathrm{P}}^3}}{\cancel{l_{\mathrm{P}}^3}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}^2}}{\cancel{t_{\mathrm{P}}^2}}\;\frac{\cancel{q_{\mathrm{P}}^2}}{\cancel{q_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{\alpha}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\frac{Z_0}{2\,R_{\mathrm{K}}} &= \frac{1}{2}\left(\frac{\alpha\, q_{\mathrm{P}}^2\, t_{\mathrm{P}}}{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}\right) \left(\frac{4\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}\, q_{\mathrm{P}}^2}\right) \\[1.25em]
&= \alpha\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{l_{\mathrm{P}}^2}}{\cancel{l_{\mathrm{P}}^2}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}}\;\frac{\cancel{q_{\mathrm{P}}^2}}{\cancel{q_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{\alpha}
\end{aligned}$$

Formula 3

$$\begin{aligned}
\sqrt{\frac{2\,R_\infty\, h}{m_{\mathrm{e}}\, c}} &= \sqrt{\frac{2}{m_{\mathrm{e}}}\left(\frac{\alpha^2\, m_{\mathrm{e}}}{4\pi\, l_{\mathrm{P}}\, m_{\mathrm{P}}}\right) \left(\frac{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right) \left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}}\right)} \\[1.25em]
&= \sqrt{\alpha^2\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{m_{\mathrm{e}}}}{\cancel{m_{\mathrm{e}}}}\;\frac{\cancel{l_{\mathrm{P}}^2}}{\cancel{l_{\mathrm{P}}^2}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}}} \\[1.25em]
&= \;\boxed{\alpha}
\end{aligned}$$

!
Three paths, one result

The fine-structure constant is a pure number — a dimensionless ratio of the elementary charge to the Planck charge: $\alpha = e^2/(4\pi\varepsilon_0\hbar c)$. No matter which combination of electromagnetic and quantum constants you use to express it, the Planck-unit factors cancel completely, leaving only $\alpha \approx 1/137.036$.

$\mu_0$ — Magnetic constant

Three SI formulas

$$\mu_0 = \frac{2h\alpha}{e^2 c} \qquad \mu_0 = \frac{1}{\varepsilon_0 c^2} \qquad \mu_0 = Z_0^2\,\varepsilon_0$$

The third formula uses the impedance of free space $Z_0 = \mu_0\, c$.

Formula 1

$$\begin{aligned}
\frac{2h\alpha}{e^2 c} &= 2\alpha\left(\frac{2\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}}\right)\left(\frac{1}{\alpha\, q_{\mathrm{P}}^2}\right)\left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}}\right) \\[1.25em]
&= 4\pi\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{t_{\mathrm{P}}^2}{q_{\mathrm{P}}^2}\;\frac{\cancel{\alpha}}{\cancel{\alpha}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{F_{\mathrm{P}}}{I_{\mathrm{P}}^2}} \quad \text{N/A}^2
\end{aligned}$$

Formula 2

$$\begin{aligned}
\frac{1}{\varepsilon_0 c^2} &= \left(\frac{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}\right)\left(\frac{t_{\mathrm{P}}^2}{l_{\mathrm{P}}^2}\right) \\[1.25em]
&= 4\pi\;\frac{l_{\mathrm{P}}\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{t_{\mathrm{P}}^2}{q_{\mathrm{P}}^2}\;\frac{\cancel{l_{\mathrm{P}}^2}}{\cancel{l_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{F_{\mathrm{P}}}{I_{\mathrm{P}}^2}} \quad \text{N/A}^2
\end{aligned}$$

Formula 3

$$\begin{aligned}
Z_0^2\,\varepsilon_0 &= \left(\frac{16\pi^2\, m_{\mathrm{P}}^2\, l_{\mathrm{P}}^4}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^4}\right)\left(\frac{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}\right) \\[1.25em]
&= 4\pi\;\frac{l_{\mathrm{P}}\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{t_{\mathrm{P}}^2}{q_{\mathrm{P}}^2}\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{l_{\mathrm{P}}^3}}{\cancel{l_{\mathrm{P}}^3}}\;\frac{\cancel{t_{\mathrm{P}}^2}}{\cancel{t_{\mathrm{P}}^2}}\;\frac{\cancel{q_{\mathrm{P}}^2}}{\cancel{q_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{F_{\mathrm{P}}}{I_{\mathrm{P}}^2}} \quad \text{N/A}^2
\end{aligned}$$

!
One structure: Planck force per Planck current squared

All three formulas for $\mu_0$ can be expressed as $4\pi\, F_{\mathrm{P}} / I_{\mathrm{P}}^2$. The magnetic constant encodes one structural relationship — a ratio of Planck force to the square of the Planck current — dressed in different combinations of compound constants.

$Z_0$ — Impedance of free space

Three SI formulas

$$Z_0 = \mu_0\, c \qquad Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} \qquad Z_0 = \frac{2\alpha\, h}{e^2}$$

Formula 1

$$\begin{aligned}
\mu_0\, c &= \left(\frac{4\pi\, l_{\mathrm{P}}\, m_{\mathrm{P}}}{q_{\mathrm{P}}^2}\right)\left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right) \\[1.25em]
&= 4\pi\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}}\;\frac{t_{\mathrm{P}}}{q_{\mathrm{P}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}} \quad \text{V/A}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\sqrt{\frac{\mu_0}{\varepsilon_0}} &= \sqrt{\left(\frac{4\pi\, l_{\mathrm{P}}\, m_{\mathrm{P}}}{q_{\mathrm{P}}^2}\right)\left(\frac{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}\right)} \\[1.25em]
&= \sqrt{16\pi^2\;\frac{E_{\mathrm{P}}^2}{q_{\mathrm{P}}^2}\;\frac{t_{\mathrm{P}}^2}{q_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}} \quad \text{V/A}
\end{aligned}$$

Formula 3

$$\begin{aligned}
\frac{2\alpha\, h}{e^2} &= 2\alpha\left(\frac{2\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}}\right)\left(\frac{1}{\alpha\, q_{\mathrm{P}}^2}\right) \\[1.25em]
&= 4\pi\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}}\;\frac{t_{\mathrm{P}}}{q_{\mathrm{P}}}\;\frac{\cancel{\alpha}}{\cancel{\alpha}} \\[1.25em]
&= \;\boxed{4\pi\;\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}} \quad \text{V/A}
\end{aligned}$$

!
Planck voltage per Planck current

The impedance of free space can be expressed as $4\pi$ times the ratio of Planck voltage to Planck current — $4\pi\, Z_{\mathrm{P}}$, where $Z_{\mathrm{P}} = V_{\mathrm{P}}/I_{\mathrm{P}} \approx 29.98\;\Omega$ is the Planck impedance.

$c\,R_\infty$ — Rydberg frequency

Four SI formulas

$$c\,R_\infty = c\,R_\infty \qquad c\,R_\infty = \frac{\alpha^2\, m_{\mathrm{e}}\, c^2}{2h} \qquad c\,R_\infty = \frac{E_{\mathrm{h}}}{2h} \qquad c\,R_\infty = \frac{e^4\, m_{\mathrm{e}}}{8\,\varepsilon_0^2\, h^3}$$

Here $R_\infty = m_{\mathrm{e}}\,\alpha^2\, c/(2h)$ is the Rydberg constant and $E_{\mathrm{h}} = \alpha^2\, m_{\mathrm{e}}\, c^2$ is the Hartree energy.

Formula 1: $c \cdot R_\infty$

$$\begin{aligned}
c\,R_\infty &= \left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{\alpha^2\, m_{\mathrm{e}}}{4\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\alpha^2\, m_{\mathrm{e}}}{4\pi\, m_{\mathrm{P}}\, t_{\mathrm{P}}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{t_{\mathrm{P}}}} \quad \text{Hz}
\end{aligned}$$

Formula 2: $\alpha^2 m_{\mathrm{e}} c^2 / (2h)$

$$\begin{aligned}
\frac{\alpha^2\, m_{\mathrm{e}}\, c^2}{2h} &= \frac{\alpha^2\, m_{\mathrm{e}}}{2}\left(\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\right)\left(\frac{t_{\mathrm{P}}}{2\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{\alpha^2}{4\pi}\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{1}{t_{\mathrm{P}}}\;\frac{\cancel{l_{\mathrm{P}}^2}}{\cancel{l_{\mathrm{P}}^2}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{t_{\mathrm{P}}}} \quad \text{Hz}
\end{aligned}$$

Formula 3: $E_{\mathrm{h}} / (2h)$

$$\begin{aligned}
\frac{E_{\mathrm{h}}}{2h} &= \left(\frac{\alpha^2\, m_{\mathrm{e}}\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{m_{\mathrm{P}}\, t_{\mathrm{P}}^2}\right)\left(\frac{t_{\mathrm{P}}}{4\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{\alpha^2}{4\pi}\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{1}{t_{\mathrm{P}}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{l_{\mathrm{P}}^2}}{\cancel{l_{\mathrm{P}}^2}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{t_{\mathrm{P}}}} \quad \text{Hz}
\end{aligned}$$

Formula 4: $e^4 m_{\mathrm{e}} / (8\varepsilon_0^2 h^3)$

$$\begin{aligned}
\frac{e^4\, m_{\mathrm{e}}}{8\,\varepsilon_0^2\, h^3} &= \frac{m_{\mathrm{e}}}{8}\left(\alpha^2\, q_{\mathrm{P}}^4\right)\left(\frac{16\pi^2\, l_{\mathrm{P}}^6\, m_{\mathrm{P}}^2}{t_{\mathrm{P}}^4\, q_{\mathrm{P}}^4}\right)\left(\frac{t_{\mathrm{P}}^3}{8\pi^3\, l_{\mathrm{P}}^6\, m_{\mathrm{P}}^3}\right) \\[1.25em]
&= \frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{t_{\mathrm{P}}}\;\frac{\cancel{q_{\mathrm{P}}^4}}{\cancel{q_{\mathrm{P}}^4}}\;\frac{\cancel{l_{\mathrm{P}}^6}}{\cancel{l_{\mathrm{P}}^6}}\;\frac{\cancel{m_{\mathrm{P}}^2}}{\cancel{m_{\mathrm{P}}^2}}\;\frac{\cancel{t_{\mathrm{P}}^3}}{\cancel{t_{\mathrm{P}}^4}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{t_{\mathrm{P}}}} \quad \text{Hz}
\end{aligned}$$

!
Four formulas, one frequency

Four different combinations of constants — involving $c$, $R_\infty$, $m_{\mathrm{e}}$, $h$, $e$, and $\varepsilon_0$ — all reduce to the same expression: $(\alpha^2/4\pi)(m_{\mathrm{e}}/m_{\mathrm{P}})(1/t_{\mathrm{P}})$. The Rydberg frequency can be expressed as the Planck frequency $1/t_{\mathrm{P}}$, scaled by two dimensionless ratios: the square of the charge coupling ($\alpha^2$) and the electron-to-Planck mass ratio.

$hc\,R_\infty$ — Rydberg energy

Two SI formulas

$$hc\,R_\infty = hc\,R_\infty \qquad hc\,R_\infty = \tfrac{1}{2}\,m_{\mathrm{e}}\,\alpha^2\, c^2$$

Formula 1

$$\begin{aligned}
hc\,R_\infty &= \left(\frac{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{\alpha^2\, m_{\mathrm{e}}}{4\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}}\right) \\[1.25em]
&= \tfrac{1}{2}\,\alpha^2\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{\cancel{\pi}}{\cancel{\pi}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\tfrac{1}{2}\,\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\tfrac{1}{2}\,m_{\mathrm{e}}\,\alpha^2\, c^2 &= \tfrac{1}{2}\,m_{\mathrm{e}}\,\alpha^2\left(\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\right)\frac{m_{\mathrm{P}}}{m_{\mathrm{P}}} \\[1.25em]
&= \tfrac{1}{2}\,\alpha^2\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2} \\[1.25em]
&= \;\boxed{\tfrac{1}{2}\,\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

$R_\infty$ — Rydberg constant

Two SI formulas

$$R_\infty = \frac{m_{\mathrm{e}}\,\alpha^2\, c}{2h} \qquad R_\infty = \frac{m_{\mathrm{e}}\, e^4}{8\,\varepsilon_0^2\, h^3\, c}$$

Formula 1

$$\begin{aligned}
\frac{m_{\mathrm{e}}\,\alpha^2\, c}{2h} &= \frac{\alpha^2\, m_{\mathrm{e}}}{2}\left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{t_{\mathrm{P}}}{2\pi\, m_{\mathrm{P}}\, l_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{\alpha^2}{4\pi}\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{1}{l_{\mathrm{P}}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{l_{\mathrm{P}}}} \quad \text{m}^{-1}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\frac{m_{\mathrm{e}}\, e^4}{8\,\varepsilon_0^2\, h^3\, c} &= \frac{m_{\mathrm{e}}}{8}\left(\alpha^2\, q_{\mathrm{P}}^4\right)\left(\frac{16\pi^2\, l_{\mathrm{P}}^6\, m_{\mathrm{P}}^2}{t_{\mathrm{P}}^4\, q_{\mathrm{P}}^4}\right)\left(\frac{t_{\mathrm{P}}^3}{8\pi^3\, l_{\mathrm{P}}^6\, m_{\mathrm{P}}^3}\right)\left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\alpha^2}{4\pi}\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{1}{l_{\mathrm{P}}}\;\frac{\cancel{16}}{\cancel{16}}\;\frac{\cancel{\pi^2}}{\cancel{\pi^2}}\;\frac{\cancel{m_{\mathrm{P}}^2}}{\cancel{m_{\mathrm{P}}^2}}\;\frac{\cancel{l_{\mathrm{P}}^6}}{\cancel{l_{\mathrm{P}}^6}}\;\frac{\cancel{t_{\mathrm{P}}^4}}{\cancel{t_{\mathrm{P}}^4}}\;\frac{\cancel{q_{\mathrm{P}}^4}}{\cancel{q_{\mathrm{P}}^4}} \\[1.25em]
&= \;\boxed{\frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{l_{\mathrm{P}}}} \quad \text{m}^{-1}
\end{aligned}$$

$a_0$ — Bohr radius

Two SI formulas

$$a_0 = \frac{\hbar}{\alpha\, m_{\mathrm{e}}\, c} \qquad a_0 = \frac{4\pi\varepsilon_0\,\hbar^2}{m_{\mathrm{e}}\, e^2}$$

Formula 1

$$\begin{aligned}
\frac{\hbar}{\alpha\, m_{\mathrm{e}}\, c} &= \frac{1}{\alpha\, m_{\mathrm{e}}}\left(\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{1}{\alpha}\left(\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\right)l_{\mathrm{P}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{1}{\alpha}\;\frac{\bar{\lambda}_{\mathrm{C}}}{l_{\mathrm{P}}}\; l_{\mathrm{P}} \;=\; \frac{\bar{\lambda}_{\mathrm{C}}}{\alpha}} \quad \text{m}
\end{aligned}$$

Using $m_{\mathrm{P}}/m_{\mathrm{e}} = \bar{\lambda}_{\mathrm{C}}/l_{\mathrm{P}}$, the reduced Compton wavelength–mass duality.

Formula 2

$$\begin{aligned}
\frac{4\pi\varepsilon_0\,\hbar^2}{m_{\mathrm{e}}\, e^2} &= \frac{4\pi}{m_{\mathrm{e}}}\left(\frac{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}\right)\left(\frac{l_{\mathrm{P}}^4\, m_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\right)\left(\frac{1}{\alpha\, q_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{1}{\alpha}\left(\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\right)l_{\mathrm{P}}\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{l_{\mathrm{P}}^3}}{\cancel{l_{\mathrm{P}}^3}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}^2}}{\cancel{t_{\mathrm{P}}^2}}\;\frac{\cancel{q_{\mathrm{P}}^2}}{\cancel{q_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{\frac{\bar{\lambda}_{\mathrm{C}}}{\alpha}} \quad \text{m}
\end{aligned}$$

$R_{\mathrm{K}}$ — von Klitzing constant

Two SI formulas

$$R_{\mathrm{K}} = \frac{\mu_0\, c}{2\alpha} \qquad R_{\mathrm{K}} = \frac{2\pi\hbar}{e^2}$$

Formula 1

$$\begin{aligned}
\frac{\mu_0\, c}{2\alpha} &= \frac{1}{2\alpha}\left(\frac{4\pi\, l_{\mathrm{P}}\, m_{\mathrm{P}}}{q_{\mathrm{P}}^2}\right)\left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{2\pi}{\alpha}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}}\;\frac{t_{\mathrm{P}}}{q_{\mathrm{P}}}\;\frac{\cancel{2}}{\cancel{2}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\frac{2\pi}{\alpha}\;\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}} \quad \text{V/A}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\frac{2\pi\hbar}{e^2} &= 2\pi\left(\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{1}{\alpha\, q_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{2\pi}{\alpha}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}}\;\frac{t_{\mathrm{P}}}{q_{\mathrm{P}}} \\[1.25em]
&= \;\boxed{\frac{2\pi}{\alpha}\;\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}} \quad \text{V/A}
\end{aligned}$$

$E_{\mathrm{h}}$ — Hartree energy

Four SI formulas

$$E_{\mathrm{h}} = \alpha^2 m_{\mathrm{e}}\, c^2 \qquad E_{\mathrm{h}} = \frac{e^2}{4\pi\varepsilon_0\, a_0} \qquad E_{\mathrm{h}} = 2hc\,R_\infty \qquad E_{\mathrm{h}} = \frac{\hbar^2}{m_{\mathrm{e}}\, a_0^2}$$

Here $a_0 = \hbar/(\alpha\, m_{\mathrm{e}}\, c)$ is the Bohr radius and $R_\infty = m_{\mathrm{e}}\,\alpha^2\, c/(2h)$ is the Rydberg constant.

Formula 1: $\alpha^2 m_{\mathrm{e}} c^2$

$$\begin{aligned}
\alpha^2 m_{\mathrm{e}}\, c^2 &= \alpha^2\, m_{\mathrm{e}}\left(\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\right)\frac{m_{\mathrm{P}}}{m_{\mathrm{P}}} \\[1.25em]
&= \alpha^2\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2} \\[1.25em]
&= \;\boxed{\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

Formula 2: $e^2 / (4\pi\varepsilon_0\, a_0)$

$$\begin{aligned}
\frac{e^2}{4\pi\varepsilon_0\, a_0} &= \frac{1}{4\pi\, a_0}\left(\alpha\, q_{\mathrm{P}}^2\right)\left(\frac{4\pi\, l_{\mathrm{P}}^3\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}\right) \\[1.25em]
&= \alpha\;\frac{l_{\mathrm{P}}}{a_0}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{q_{\mathrm{P}}^2}}{\cancel{q_{\mathrm{P}}^2}} \\[1em]
&\quad \left[\;\frac{l_{\mathrm{P}}}{a_0} = \alpha\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\right] \\[1em]
&= \;\boxed{\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

Formula 3: $2hc\,R_\infty$

$$\begin{aligned}
2hc\,R_\infty &= 2\left(\frac{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{l_{\mathrm{P}}}{t_{\mathrm{P}}}\right)R_\infty \\[1em]
&\quad \left[\;R_\infty = \frac{\alpha^2}{4\pi}\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)\frac{1}{l_{\mathrm{P}}}\;\right] \\[1em]
&= \alpha^2\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{m_{\mathrm{P}}\, l_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\;\frac{\cancel{4\pi}}{\cancel{4\pi}}\;\frac{\cancel{l_{\mathrm{P}}}}{\cancel{l_{\mathrm{P}}}}\;\frac{\cancel{t_{\mathrm{P}}}}{\cancel{t_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

Formula 4: $\hbar^2 / (m_{\mathrm{e}}\, a_0^2)$

$$\begin{aligned}
\frac{\hbar^2}{m_{\mathrm{e}}\, a_0^2} &= \left(\frac{l_{\mathrm{P}}^4\, m_{\mathrm{P}}^2}{t_{\mathrm{P}}^2}\right)\frac{1}{m_{\mathrm{e}}\, a_0^2} \\[1.25em]
&= \frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\;\frac{l_{\mathrm{P}}^2}{a_0^2}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2} \\[1em]
&\quad \left[\;\frac{l_{\mathrm{P}}^2}{a_0^2} = \alpha^2\;\frac{m_{\mathrm{e}}^2}{m_{\mathrm{P}}^2}\;\right] \\[1em]
&= \alpha^2\;\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{\cancel{m_{\mathrm{e}}}}{\cancel{m_{\mathrm{e}}}}\;\frac{\cancel{m_{\mathrm{P}}}}{\cancel{m_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\alpha^2\left(\frac{m_{\mathrm{e}}}{m_{\mathrm{P}}}\right)E_{\mathrm{P}}} \quad \text{J}
\end{aligned}$$

!
Four formulas, one energy

The Hartree energy — whether derived from $m_{\mathrm{e}} c^2$, from Coulomb’s law at the Bohr radius, from the Rydberg constant, or from the kinetic energy at the Bohr orbit — can be expressed as $\alpha^2 (m_{\mathrm{e}}/m_{\mathrm{P}})\, E_{\mathrm{P}}$. It is the Planck energy, scaled by the square of the charge coupling and the electron mass ratio.

$\kappa$ — Quantum of circulation

Two SI formulas

$$\kappa = \frac{\pi\hbar}{m_{\mathrm{e}}} \qquad \kappa = \frac{e}{m_{\mathrm{e}}}\,\Phi_0$$

Here $\Phi_0 = h/(2e)$ is the magnetic flux quantum.

Formula 1

$$\begin{aligned}
\frac{\pi\hbar}{m_{\mathrm{e}}} &= \frac{\pi}{m_{\mathrm{e}}}\left(\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right) \\[1.25em]
&= \;\boxed{\pi\left(\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\right)\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}}} \quad \text{m}^2\text{/s}
\end{aligned}$$

Formula 2

$$\begin{aligned}
\frac{e}{m_{\mathrm{e}}}\,\Phi_0 &= \frac{1}{m_{\mathrm{e}}}\left(\sqrt{\alpha}\, q_{\mathrm{P}}\right)\left(\frac{\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{\sqrt{\alpha}\, q_{\mathrm{P}}\, t_{\mathrm{P}}}\right) \\[1.25em]
&= \pi\;\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\;\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}}\;\frac{\cancel{\sqrt{\alpha}}}{\cancel{\sqrt{\alpha}}}\;\frac{\cancel{q_{\mathrm{P}}}}{\cancel{q_{\mathrm{P}}}} \\[1.25em]
&= \;\boxed{\pi\left(\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\right)\frac{l_{\mathrm{P}}^2}{t_{\mathrm{P}}}} \quad \text{m}^2\text{/s}
\end{aligned}$$

$\varepsilon_0$ — Electric constant

SI formula

$$\varepsilon_0 = \frac{1}{\mu_0\, c^2}$$

$$\begin{aligned}
\frac{1}{\mu_0\, c^2} &= \left(\frac{t_{\mathrm{P}}^2\, q_{\mathrm{P}}^2}{4\pi\, l_{\mathrm{P}}\, m_{\mathrm{P}}}\right)\left(\frac{t_{\mathrm{P}}^2}{l_{\mathrm{P}}^2}\right) \\[1.25em]
&= \frac{1}{4\pi}\;\frac{t_{\mathrm{P}}^2}{l_{\mathrm{P}}\, m_{\mathrm{P}}}\;\frac{q_{\mathrm{P}}^2}{l_{\mathrm{P}}^2}\;\frac{\cancel{t_{\mathrm{P}}^2}}{\cancel{t_{\mathrm{P}}^2}} \\[1.25em]
&= \;\boxed{\frac{1}{4\pi\, F_{\mathrm{P}}}\;\frac{q_{\mathrm{P}}^2}{l_{\mathrm{P}}^2}} \quad \text{C}^2\text{/(N·m}^2\text{)}
\end{aligned}$$

$K_{\mathrm{J}}$ — Josephson constant

SI formula

$$K_{\mathrm{J}} = \frac{2e}{h}$$

$$\begin{aligned}
\frac{2e}{h} &= 2\left(\sqrt{\alpha}\, q_{\mathrm{P}}\right)\left(\frac{t_{\mathrm{P}}}{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\sqrt{\alpha}}{\pi}\;\frac{q_{\mathrm{P}}}{E_{\mathrm{P}}}\;\frac{1}{t_{\mathrm{P}}}\;\frac{\cancel{2}}{\cancel{2}} \\[1.25em]
&= \;\boxed{\frac{\sqrt{\alpha}}{\pi}\;\frac{1}{V_{\mathrm{P}}\, t_{\mathrm{P}}}} \quad \text{Hz/V}
\end{aligned}$$

$\Phi_0$ — Magnetic flux quantum

SI formula

$$\Phi_0 = \frac{h}{2e}$$

$$\begin{aligned}
\frac{h}{2e} &= \frac{1}{2}\left(\frac{2\pi\, l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right)\left(\frac{1}{\sqrt{\alpha}\, q_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\pi}{\sqrt{\alpha}}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2\, q_{\mathrm{P}}}\;t_{\mathrm{P}}\;\frac{\cancel{2}}{\cancel{2}} \\[1.25em]
&= \;\boxed{\frac{\pi}{\sqrt{\alpha}}\;V_{\mathrm{P}}\, t_{\mathrm{P}}} \quad \text{V·s}
\end{aligned}$$

$G_0$ — Conductance quantum

SI formula

$$G_0 = \frac{2e^2}{2\pi\hbar}$$

$$\begin{aligned}
\frac{2e^2}{2\pi\hbar} &= \frac{2}{2\pi}\left(\alpha\, q_{\mathrm{P}}^2\right)\left(\frac{t_{\mathrm{P}}}{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\alpha}{\pi}\;\frac{q_{\mathrm{P}}}{E_{\mathrm{P}}}\;\frac{q_{\mathrm{P}}}{t_{\mathrm{P}}}\;\frac{\cancel{2}}{\cancel{2}} \\[1.25em]
&= \;\boxed{\frac{\alpha}{\pi}\;\frac{I_{\mathrm{P}}}{V_{\mathrm{P}}}} \quad \text{A/V}
\end{aligned}$$

$\mu_{\mathrm{B}}$ — Bohr magneton

SI formula

$$\mu_{\mathrm{B}} = \frac{e\hbar}{2m_{\mathrm{e}}}$$

$$\begin{aligned}
\frac{e\hbar}{2m_{\mathrm{e}}} &= \frac{1}{2m_{\mathrm{e}}}\left(\sqrt{\alpha}\, q_{\mathrm{P}}\right)\left(\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}}\right) \\[1.25em]
&= \frac{\sqrt{\alpha}}{2}\;\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\;\frac{l_{\mathrm{P}}^2\, m_{\mathrm{P}}}{t_{\mathrm{P}}^2}\;\frac{t_{\mathrm{P}}^2}{m_{\mathrm{P}}}\;\frac{q_{\mathrm{P}}}{t_{\mathrm{P}}} \\[1.25em]
&= \;\boxed{\frac{\sqrt{\alpha}}{2}\left(\frac{m_{\mathrm{P}}}{m_{\mathrm{e}}}\right)E_{\mathrm{P}}\;\frac{t_{\mathrm{P}}^2}{m_{\mathrm{P}}}\;I_{\mathrm{P}}} \quad \text{J/T}
\end{aligned}$$

Summary

Fifteen constants, drawn from electromagnetism, quantum mechanics, atomic physics, and precision metrology — and one short list of ingredients. Every universal form in the right-hand column below is built from nothing but the four Planck units, the coupling $\alpha$, the electron mass ratio $m_{\mathrm{e}}/m_{\mathrm{P}}$, and a geometric factor of $2\pi$ or $4\pi$. Read this way, the constants are not fifteen independent facts about nature; they can be expressed as one small Planck-unit kit, packaged fifteen different ways for the convenience of fifteen different measurements.

The table below collects all fifteen constants and their common Planck-unit forms. Every formula listed above — regardless of which SI constants it uses — reduces to the expression in the right-hand column.

Constant SI formulas Common universal form Unit
$\alpha$ 3 formulas $\alpha$ —
$\mu_0$ 3 formulas $4\pi\, F_{\mathrm{P}} / I_{\mathrm{P}}^2$ F/Q²
$Z_0$ 3 formulas $4\pi\, V_{\mathrm{P}} / I_{\mathrm{P}}$ E/Q
$c\,R_\infty$ 4 formulas $(\alpha^2/4\pi)(m_{\mathrm{e}}/m_{\mathrm{P}})\, t_{\mathrm{P}}^{-1}$ T
$hc\,R_\infty$ 2 formulas $\tfrac{1}{2}\alpha^2(m_{\mathrm{e}}/m_{\mathrm{P}})\, E_{\mathrm{P}}$ E
$R_\infty$ 2 formulas $(\alpha^2/4\pi)(m_{\mathrm{e}}/m_{\mathrm{P}})\, l_{\mathrm{P}}^{-1}$ L
$a_0$ 2 formulas $\bar{\lambda}_{\mathrm{C}} / \alpha$ L
$R_{\mathrm{K}}$ 2 formulas $(2\pi/\alpha)\, V_{\mathrm{P}} / I_{\mathrm{P}}$ E/Q
$E_{\mathrm{h}}$ 4 formulas $\alpha^2(m_{\mathrm{e}}/m_{\mathrm{P}})\, E_{\mathrm{P}}$ E
$\kappa$ 2 formulas $\pi(m_{\mathrm{P}}/m_{\mathrm{e}})\, l_{\mathrm{P}}^2/t_{\mathrm{P}}$ Lv
$\varepsilon_0$ 1 formula $q_{\mathrm{P}}^2/(4\pi\, F_{\mathrm{P}}\, l_{\mathrm{P}}^2)$ Q²/FL
$K_{\mathrm{J}}$ 1 formula $(\sqrt{\alpha}/\pi)/(V_{\mathrm{P}}\, t_{\mathrm{P}})$ T/E
$\Phi_0$ 1 formula $(\pi/\sqrt{\alpha})\, V_{\mathrm{P}}\, t_{\mathrm{P}}$ ET
$G_0$ 1 formula $(\alpha/\pi)\, I_{\mathrm{P}} / V_{\mathrm{P}}$ Q/E
$\mu_{\mathrm{B}}$ 1 formula $(\sqrt{\alpha}/2)(m_{\mathrm{P}}/m_{\mathrm{e}})\, E_{\mathrm{P}}\, t_{\mathrm{P}}^2\, I_{\mathrm{P}}/m_{\mathrm{P}}$ E/M

!
What the cancellations reveal

The compound constants of physics — $\hbar$, $\mu_0$, $\varepsilon_0$, $h$, and all the rest — can be expressed as products and quotients of four Planck units ($l_{\mathrm{P}}$, $m_{\mathrm{P}}$, $t_{\mathrm{P}}$, $q_{\mathrm{P}}$) and two dimensionless numbers ($\alpha$ and $m_{\mathrm{e}}/m_{\mathrm{P}}$). When these constants appear in a formula, they bring Planck factors along with them. Different formulas combine different constants — but the Planck factors they carry are not independent. They share a common dimensional substrate. When you write everything out, the shared factors cancel, and what remains is the actual physical content: which Planck-scale quantity is being measured, and which dimensionless ratios scale it.

Related: 
Electromagnetic & Quantum Constants in Natural Form  | 
Natural Formulas Index  | 
Planck Units — Complete Reference

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