The elementary charge has three numerical values in use today. In SI units, $e = 1.602 \times 10^{-19}$ coulombs. In the electromagnetic CGS system (emu), $e = 1.602 \times 10^{-20}$ abcoulombs. In the electrostatic CGS system (esu), $e = 4.803 \times 10^{-10}$ statcoulombs. Three numbers, three units, the same physical charge. The story of how this happened — and what Planck-unit decomposition reveals about it — is one of the cleanest examples of what can become visible when a formula is written in universal units.
Why there are three units at all
In the SI system, electric charge has its own fundamental dimension — the coulomb — independent of mass, length, and time. In the CGS systems (both emu and esu), charge does not have its own dimension. Charge is expressed entirely in mechanical units: grams, centimeters, and seconds. This was not a stylistic choice. Before Millikan’s 1909 oil-drop experiment, physicists did not have a reliable handle on the elementary charge, and there was no compelling reason to treat charge as a separate fundamental dimension. It had to be derived from the force laws that involved it.
There are two such force laws, and they give two different mechanical expressions for charge. The esu derivation starts from Coulomb’s law, and reads off the dimensions of charge from the electrostatic force between two stationary charges. The emu derivation starts from Ampère’s force law, and reads off the dimensions of current from the magnetic force between two parallel wires. Both are self-consistent. They disagree about the units because electrostatics and magnetostatics contain a factor of $c^{2}$ between them — the same factor of $c^{2}$ that appears in every relativistic rewriting of Maxwell’s equations.
The Planck-unit identity
The simplest way to see what the different systems are doing is to write the magnetic constant in Planck-unit form. Starting from the natural decomposition (EJP 2024 eq. 43):
The middle expression treats $\mu_{0}/(4\pi)$ as force per current squared; the right-hand form rewrites the current as charge per time, giving the combination of Planck units $l_{\mathrm{P}} m_{\mathrm{P}} / q_{\mathrm{P}}^{2}$. Numerically, $\mu_{0}/(4\pi)$ is exactly $10^{-7}\;\mathrm{N\,A^{-2}}$ in pre-2019 SI, essentially $10^{-7}$ in post-2019 SI, and whatever you set it to in CGS. This identity is the single sentence out of which the emu and esu systems fall.
The electromagnetic system (emu)
The emu choice is to set the magnetic constant equal to $1$, which immediately gives a definition of charge in mechanical units. Setting $\mu_{0}/(4\pi) = 1$ in the identity above yields:
In this system the abcoulomb has the dimensions $M^{1/2} L^{1/2}$, and the abampere has dimensions $M^{1/2} L^{1/2} T^{-1}$ — the square root of force. The Planck charge in abcoulombs is:
The conversion factor to SI coulombs is exactly $10$: $1\;\mathrm{abC} = 10\;\mathrm{C}$. This factor is not mysterious — it is the square root of the $10^{-2}\;\mathrm{cm\,g\,C^{-2}}$ value that the pre-SI CGS system had for $\mu_{0}/(4\pi)$, compared to the emu value of $1$. The two systems disagree on the numerical value of $\mu_{0}/(4\pi)$, and the unit of charge absorbs the difference.
The electrostatic system (esu)
The esu choice is to set the electric constant (rather than the magnetic one) equal to a fixed value. Starting from the Coulomb constant in Planck-unit form:
Setting $1/(4\pi\varepsilon_{0}) = 1$ gives a different definition of charge — this time with an extra factor of $c^{2}$ embedded in the mechanical units:
The statcoulomb has dimensions $M^{1/2} L^{3/2} T^{-1}$: the square root of energy times length. The numerical Planck charge in statcoulombs is the abcoulomb value multiplied by $c$ in cgs units:
The conversion factor to coulombs is $1\;\mathrm{C} = 2.998 \times 10^{9}\;\mathrm{statC}$ — that is $c$ with the conventional factor of $10$. The numerical coincidence that the emu-to-esu conversion factor is exactly the speed of light in cgs units is the same fact that $1/(4\pi\varepsilon_{0}) \cdot 4\pi = \mu_{0} c^{2}$. It is the single factor of $c^{2}$ that separates electrostatics from magnetostatics.
Three systems, one structure
| System | Dimensional choice | Charge definition | $q_{\mathrm{P}}$ numerical |
|---|---|---|---|
| SI (post-2019) | charge is fundamental | coulomb defined via fixed $e$ | $1.876\times 10^{-18}\;\mathrm{C}$ |
| emu (CGS) | $\mu_{0}/(4\pi) = 1$ | $[\mathrm{charge}]^{2} = [\mathrm{mass}][\mathrm{length}]$ | $1.876\times 10^{-19}\;\mathrm{abC}$ |
| esu (CGS, Gaussian) | $1/(4\pi\varepsilon_{0}) = 1$ | $[\mathrm{charge}]^{2} = [\mathrm{energy}][\mathrm{length}]$ | $5.623\times 10^{-9}\;\mathrm{statC}$ |
| Universal units | $q_{\mathrm{P}}$ as natural scale | $e/q_{\mathrm{P}} = \sqrt{\alpha}$ | $0.0854$ (dimensionless) |
Each row is a different way to dispose of the same Planck-charge degree of freedom. The bottom row is what the other three have in common: the ratio $e / q_{\mathrm{P}} = \sqrt{\alpha}$ is the same number in every system, independent of whether charge is expressed in coulombs, abcoulombs, or statcoulombs. That invariant dimensionless ratio is the physical content; the three unit systems are different conventions for packaging it.
Why the Planck-charge reading is the clean one
The historical CGS systems were trying to eliminate charge as a fundamental dimension because its natural scale was unknown. Once Millikan’s measurement established that an indivisible charge exists, the motivation for deriving charge from mechanical dimensions disappeared — but the systems remained in use because they were convenient for electrodynamics textbooks and because conversion tables were already familiar. The 2019 SI redefinition fixed $e$ exactly, making the coulomb a defined multiple of the elementary charge. The Planck-charge reading goes one step further and treats $q_{\mathrm{P}}$ as the reference scale, with $e/q_{\mathrm{P}} = \sqrt{\alpha} = 0.08542…$ as the unit-invariant coupling ratio.
Viewed through this lens, the emu and esu systems are not wrong; they are two different Planck-unit packagings of the same physics, each one having absorbed a factor of $\sqrt{l_{\mathrm{P}} m_{\mathrm{P}}}$ or $\sqrt{l_{\mathrm{P}} m_{\mathrm{P}}}\cdot c$ into its unit of charge. The emu charge unit is $\sqrt{l_{\mathrm{P}} m_{\mathrm{P}}}$ in universal form; the esu charge unit is $\sqrt{l_{\mathrm{P}} m_{\mathrm{P}} c^{2}}$. The three-unit history is an accident of which mechanical force law was given unit-one priority.
Further reading on this site: the elementary charge profile develops $e = \sqrt{\alpha}\, q_{\mathrm{P}}$ in detail, and the Planck charge profile covers $q_{\mathrm{P}}$ as the base Planck unit of charge. The published source is EJP 2024, “Understanding the natural units,” Section 4.4 and equations 43–51.