Elementary Charge

e
Electromagnetic constant
$1.602\,176\,634 \times 10^{-19}$ C
Dimensions: Q  ·  Relative uncertainty: 0 (exact since the 2019 SI redefinition)
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At a glance
In Planck units, the elementary charge can be expressed as $e = \sqrt{\alpha}\, q_{\mathrm{P}}$ — the Planck charge reduced by the square root of the fine-structure constant. The single dimensionless ratio $e/q_{\mathrm{P}} = \sqrt{\alpha} \approx 0.0854$, the same in every system of units, encodes the strength of all electromagnetic interactions. In physical formulas, $e$ enters through powers of $\alpha$ multiplied by Planck-unit combinations: the charge itself supplies the electromagnetic coupling, while the Planck units supply the dimensional scaffolding. Every formula involving $e$ can be decomposed into a Planck-scale expression with $\sqrt{\alpha}$ as the only remnant of the charge’s specific value.

Universal form

The Planck-unit decomposition of $e$ is:

Elementary charge in Planck units
$$e = \sqrt{\alpha}\, q_{\mathrm{P}}$$

A single Planck charge multiplied by a single dimensionless coefficient $\sqrt{\alpha}$. The Planck charge $q_{\mathrm{P}} = \sqrt{4\pi \varepsilon_0 \hbar c}$ carries the full dimension of charge; the factor $\sqrt{\alpha}$ carries no dimensions — it is a pure number that encodes how much smaller the elementary charge is than the Planck charge.

Equivalently, since $\alpha = \tfrac{e^2}{4\pi \varepsilon_0 \hbar c}$:

Squared form (most common in formulas)
$$e^2 = \alpha \, q_{\mathrm{P}}^2 = 4\pi \alpha \, \varepsilon_0 \hbar c$$

This is the form that appears most often in physical formulas — the square of the charge, paired with the coupling constant $\alpha$.

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Dimensions
$e$ contributes Q — one dimension of electric charge. Every power of $e$ in a formula contributes one factor of $q_{\mathrm{P}}$ (dimensional) and one factor of $\sqrt{\alpha}$ (dimensionless).

Equivalent expressions

The elementary charge appears in several distinct groupings, each foregrounding a different physical role.

The charge ratio

$$e = \sqrt{\alpha}\, q_{\mathrm{P}}$$

The elementary charge as a fraction of the Planck charge. This is $e$ in its most fundamental form: the coupling constant is contained entirely in the dimensionless prefactor $\sqrt{\alpha}$.

From the fine-structure constant

$$e = \sqrt{4\pi \alpha \, \varepsilon_0 \hbar c}$$

Expanding $q_{\mathrm{P}}$ makes the dependence on $\hbar$, $c$, and $\varepsilon_0$ explicit. This expression is the bridge between the elementary charge and the mechanical constants whose dimensional structure contains the Planck units.

From the von Klitzing constant

$$e = \sqrt{\frac{h}{R_K}} = \sqrt{\frac{2\pi \, m_{\mathrm{P}} l_{\mathrm{P}}^2}{t_{\mathrm{P}} R_K}}$$

The von Klitzing constant $R_K = h/e^2$ sets the quantized Hall resistance. This relation connects $e$ directly to precision quantum electrical metrology.

From the magnetic flux quantum

$$e = \frac{h}{2\Phi_0} = \frac{\pi \, m_{\mathrm{P}} l_{\mathrm{P}}^2}{t_{\mathrm{P}} \Phi_0}$$

The magnetic flux quantum $\Phi_0 = h/(2e)$ sets the flux threading a superconducting loop. The factor of 2 reflects Cooper pairing.

From the Faraday constant

$$e = \frac{F}{N_A}$$

The Faraday constant $F = e N_A$ measures the total charge carried by one mole of electrons. This connection made $e$ accessible to electrochemistry long before individual electrons could be counted.

Dimensional verification

Multiple independent formulas involving $e$ from different areas of physics can be decomposed into Planck units to verify dimensional consistency.

From Coulomb’s law

$$F = \frac{e^2}{4\pi \varepsilon_0 r^2}$$
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Planck-unit reduction
$$\require{cancel} \frac{e^2}{4\pi \varepsilon_0 r^2} = \frac{\alpha \, q_{\mathrm{P}}^2}{4\pi \varepsilon_0 r^2} = \frac{\alpha}{4\pi} \cdot \frac{4\pi l_{\mathrm{P}}^3 m_{\mathrm{P}}}{t_{\mathrm{P}}^2 \cancel{q_{\mathrm{P}}^2}} \cdot \frac{\cancel{q_{\mathrm{P}}^2}}{r^2} = \alpha \cdot F_{\mathrm{P}} \cdot \left(\frac{l_{\mathrm{P}}}{r}\right)^2$$ The Coulomb force between two elementary charges is $\alpha$ times the Planck force, scaled by the square of the distance in Planck lengths. The charge $e$ has been absorbed entirely into the dimensionless coupling $\alpha$.

From the hydrogen ground-state energy

$$E_1 = -\frac{m_e e^4}{2(4\pi \varepsilon_0)^2 \hbar^2}$$
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Planck-unit reduction
$$\frac{m_e e^4}{2(4\pi \varepsilon_0)^2 \hbar^2} = \frac{\alpha^2}{2} \cdot \frac{m_e}{m_{\mathrm{P}}} \cdot E_{\mathrm{P}}$$ The hydrogen binding energy is the Planck energy, reduced by the mass ratio $m_e/m_{\mathrm{P}}$ and $\alpha^2/2$. The four powers of $e$ have collapsed into $\alpha^2$ — two from the Coulomb potential and two from the virial theorem.

Physical characterization

The elementary charge enters physics as the electromagnetic coupling unit. In Planck-decomposed formulas, $e$ never appears by name — it has been replaced by its dimensionless content $\sqrt{\alpha}$ and the Planck charge $q_{\mathrm{P}}$. What makes $e$ physically interesting is the range of distinct roles that the coupling $\alpha$ plays once $e$ has been absorbed.

Where the charge sets the force scale

In Coulomb’s law, the electromagnetic force between two elementary charges at separation $r$ is:

$$F = \alpha \cdot F_{\mathrm{P}} \cdot \left(\frac{l_{\mathrm{P}}}{r}\right)^2$$

Compare this with the gravitational force between two masses $m_1$ and $m_2$:

$$F_g = F_{\mathrm{P}} \cdot \frac{m_1}{m_{\mathrm{P}}} \cdot \frac{m_2}{m_{\mathrm{P}}} \cdot \left(\frac{l_{\mathrm{P}}}{r}\right)^2$$

Both are the Planck force multiplied by dimensionless ratios. For Coulomb’s law, the prefactor is $\alpha \approx 1/137$. For gravity between two electrons, the prefactor is $(m_e/m_{\mathrm{P}})^2 \approx 1.7 \times 10^{-45}$. The electromagnetic force is stronger not because of anything special about charge (both forces have the same Planck-scale structure) but because $\alpha$ is vastly larger than $(m_e/m_{\mathrm{P}})^2$.

Numerical verification — Coulomb force at Bohr radius
$r = a_0$5.292 × 10−11 m
$\alpha$7.297 × 10−3
$F_{\mathrm{P}}$1.210 × 1044 N
$(l_{\mathrm{P}}/a_0)^2$9.32 × 10−50
$F = \alpha \cdot F_{\mathrm{P}} \cdot (l_{\mathrm{P}}/a_0)^2$8.24 × 10−8 N

Matches the standard result $e^2/(4\pi\varepsilon_0 a_0^2) = 8.24 \times 10^{-8}$ N.

Where the charge sets the length hierarchy

The elementary charge, through $\alpha$, establishes the characteristic length scales of atomic physics. For the electron, three lengths are related by exact powers of $\alpha$:

$$a_0 = \frac{1}{\alpha} \cdot \frac{m_{\mathrm{P}}}{m_e} \cdot l_{\mathrm{P}} \qquad \bar{\lambda}_C = \frac{m_{\mathrm{P}}}{m_e} \cdot l_{\mathrm{P}} \qquad r_e = \alpha \cdot \frac{m_{\mathrm{P}}}{m_e} \cdot l_{\mathrm{P}}$$

The Bohr radius $a_0$ (orbital size), Compton wavelength $\bar{\lambda}_C$ (quantum wavelength), and classical electron radius $r_e$ (electromagnetic self-energy scale) share the common factor $(m_{\mathrm{P}}/m_e) \cdot l_{\mathrm{P}}$. They differ only by their power of $\alpha$ — or equivalently, by the number of factors of $e$ that entered their defining formulas.

Each step from $a_0$ down to $r_e$ compresses the length by a factor of $\alpha = (e/q_{\mathrm{P}})^2 \approx 1/137$. The Bohr radius is inflated by weak coupling ($1/\alpha$). The classical radius is compressed by strong-field effects ($\alpha$). The Compton wavelength sits at the boundary — independent of the charge altogether.

Where the charge determines energy scales

The electron volt itself is defined by the elementary charge: $1\;\mathrm{eV} = e \times 1\;\mathrm{V}$. In Planck units:

$$1\;\mathrm{eV} = 8.19 \times 10^{-29} \, E_{\mathrm{P}}$$

The hydrogen binding energy illustrates how $e$ controls atomic energy scales. The ground-state energy:

$$E_1 = \frac{\alpha^2}{2} \cdot \frac{m_e}{m_{\mathrm{P}}} \cdot E_{\mathrm{P}} = 13.6\;\mathrm{eV}$$

Two powers of $\alpha$ (equivalently, four powers of $e/q_{\mathrm{P}}$) reduce the Planck energy to the atomic scale. This is why chemistry operates at electron-volt energies rather than Planck energies: the electromagnetic coupling $\alpha$ is small, and it enters twice.

Numerical verification — hydrogen ground-state energy
$\alpha^2/2$2.663 × 10−5
$m_e/m_{\mathrm{P}}$4.185 × 10−23
$E_{\mathrm{P}}$1.956 × 109 J
Product2.179 × 10−18 J = 13.6 eV

The constant in context

The following formulas show $e$ at work across different physical settings.

Coulomb’s law

$$F = \frac{e^2}{4\pi \varepsilon_0 r^2} = \alpha \cdot F_{\mathrm{P}} \cdot \left(\frac{l_{\mathrm{P}}}{r}\right)^2$$

The force between two elementary charges. The charge content $e^2$ has become the dimensionless coupling $\alpha$, and the distance is measured in Planck lengths.

Bohr radius

$$a_0 = \frac{\hbar}{\alpha \, m_e c} = \frac{1}{\alpha} \cdot \frac{m_{\mathrm{P}}}{m_e} \cdot l_{\mathrm{P}}$$

The characteristic size of hydrogen. The charge enters through $\alpha$ in the denominator — weaker coupling ($\alpha \to 0$) would push $a_0 \to \infty$, making atoms infinitely large. The Planck length, amplified by $m_{\mathrm{P}}/m_e$ and $1/\alpha$, gives the atomic scale directly.

Hydrogen energy levels

$$E_n = -\frac{\alpha^2 m_e c^2}{2n^2} = -\frac{\alpha^2}{2n^2} \cdot \frac{m_e}{m_{\mathrm{P}}} \cdot E_{\mathrm{P}}$$

The binding energies of hydrogen. Two powers of $e$ enter through the Coulomb potential, and two more through the virial theorem. All four collapse into $\alpha^2$.

Quantized Hall resistance

$$R_K = \frac{h}{e^2} = \frac{2\pi}{\alpha} \cdot \frac{m_{\mathrm{P}} l_{\mathrm{P}}^2}{t_{\mathrm{P}} q_{\mathrm{P}}^2}$$

The von Klitzing constant, measured to extraordinary precision in the quantum Hall effect. In Planck units, the $e^2$ in the denominator supplies the factor $\alpha$ — the resistance quantum is a Planck-scale impedance divided by $\alpha$.

Magnetic flux quantum

$$\Phi_0 = \frac{h}{2e} = \frac{\pi}{\sqrt{\alpha}} \cdot \frac{m_{\mathrm{P}} l_{\mathrm{P}}^2}{t_{\mathrm{P}} q_{\mathrm{P}}}$$

The flux quantum of superconductivity. A single factor of $e$ in the denominator contributes $\sqrt{\alpha}$ to the Planck decomposition. The factor of 2 reflects Cooper pairing — each flux quantum is threaded by a pair of charges.

Fine-structure constant

$$\alpha = \frac{e^2}{4\pi \varepsilon_0 \hbar c} = \left(\frac{e}{q_{\mathrm{P}}}\right)^2$$

The defining relationship. All Planck units cancel, leaving only the charge ratio squared. This is the purest expression of the elementary charge’s physical content — a single dimensionless number.

Why this value?

Since 2019, the SI defines $e = 1.602\,176\,634 \times 10^{-19}$ C exactly. The ampere, previously defined through the force between parallel wires, is now defined by fixing the numerical value of $e$. The coulomb follows as $1\;\mathrm{C} = e / (1.602\,176\,634 \times 10^{-19})$. Current is a count of elementary charges per second.

In Planck units, $e/q_{\mathrm{P}} = \sqrt{\alpha} \approx 0.0854$ — a charge quantity (Q). The elementary charge is about 8.5% of the Planck charge. This single ratio (equivalently, $\alpha \approx 1/137$) determines the entire electromagnetic energy and length scale hierarchy. Why $\alpha$ takes the particular value $\approx 1/137.036$ is one of the most persistent open questions in physics, and the Standard Model takes it as an empirical input rather than a derived quantity.

The experimental determination of $e$ has a long history. Millikan’s oil-drop experiment (1909–1913) established that charge is quantized and measured $e$ to within 1%. Modern measurements rely on quantum electrical standards: the quantum Hall effect (which measures $h/e^2$) and the Josephson effect (which measures $h/(2e)$) together determine $e$ in terms of $h$. The 2019 SI redefinition fixed both $h$ and $e$ to exact values, so these quantum effects now serve as primary standards for the volt, ohm, and ampere.

That electric charge is quantized (all observed free charges are integer multiples of $e$) is a deep fact. The Standard Model accommodates charge quantization but does not require it from first principles. Grand unified theories predict it as a consequence of magnetic monopoles (via the Dirac quantization condition), but no monopole has been observed. The quantization of charge remains an empirical law without a universally accepted theoretical origin.

Connections

Fine-structure constant: $\alpha = (e/q_{\mathrm{P}})^2$ — the square of the charge ratio. This is the master coupling constant of electromagnetism, and $e$ is its physical carrier.

Permittivity of free space: $\varepsilon_0 = \tfrac{t_{\mathrm{P}}^2 q_{\mathrm{P}}^2}{4\pi l_{\mathrm{P}}^3 m_{\mathrm{P}}}$ — the vacuum’s capacitive property, expressed in Planck units. Together with $e$, it sets the scale of Coulomb’s law.

Magnetic constant: $\mu_0 = \tfrac{4\pi m_{\mathrm{P}} l_{\mathrm{P}}}{q_{\mathrm{P}}^2}$ — the vacuum’s inductive property. The product $\mu_0 \varepsilon_0 = 1/c^2$ involves no charge; only in formulas that include $e$ (or $\alpha$) does the specific electromagnetic coupling appear.

Impedance of free space: $Z_0 = \mu_0 c = \tfrac{4\pi m_{\mathrm{P}} l_{\mathrm{P}}^2}{t_{\mathrm{P}} q_{\mathrm{P}}^2}$ — a Planck-scale impedance independent of $\alpha$. The elementary charge does not appear in the impedance of the vacuum itself — only in the coupling of charges to the electromagnetic field.

Planck charge: $q_{\mathrm{P}} = e/\sqrt{\alpha} = \sqrt{4\pi \varepsilon_0 \hbar c} \approx 1.876 \times 10^{-18}$ C — the Planck unit of charge, customarily calculated from $\hbar$, $c$, and $\varepsilon_0$. It is about 11.7 times larger than $e$, and the ratio $e/q_{\mathrm{P}} = \sqrt{\alpha}$ is the fundamental electromagnetic coupling.

Planck’s constant: The product $\hbar c$ provides the quantum-relativistic scale against which $e^2/(4\pi \varepsilon_0)$ is measured. In Planck units, $\hbar c = E_{\mathrm{P}} \, l_{\mathrm{P}}$ — an energy times a length — and $\alpha$ is the ratio $\tfrac{e^2}{4\pi \varepsilon_0 \hbar c}$.