Electric Constant

ε0
Universal Constant
$8.8541878188(14) \times 10^{-12}$ F/m
Dimensions: M⁻¹ L⁻³ T² Q²  ·  Relative uncertainty: $1.6 \times 10^{-10}$
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At a glance
Expressed in Planck units, $\varepsilon_0$ can be written as $\varepsilon_0 = q_\mathrm{P}^2/(4\pi F_\mathrm{P} l_\mathrm{P}^2)$ — a Planck force, two Planck charges ($q_\mathrm{P}^2$), and two Planck lengths ($l_\mathrm{P}^2$), with a geometric $4\pi$. This composition matches the structure of Coulomb’s law: for two charges $q_1$, $q_2$ separated by a distance $L$, the electrostatic force is $F = F_\mathrm{P} (q_1/q_\mathrm{P})(q_2/q_\mathrm{P})(l_\mathrm{P}/L)^2$. The Planck force from $\varepsilon_0$ supplies the force scale; the Planck charges convert $q_1$ and $q_2$ into dimensionless charge quantities (Q); the Planck lengths convert the squared distance into a dimensionless length quantity (L). Equivalently, expanding $F_\mathrm{P}$ into its mechanical Planck units, $\varepsilon_0 = t_\mathrm{P}^2 q_\mathrm{P}^2 / (4\pi l_\mathrm{P}^3 m_\mathrm{P})$.

Universal form

The complete Planck-unit decomposition of $\varepsilon_0$ is:

Electric constant in Planck units
$$\varepsilon_0 = \frac{t_\mathrm{P}^2 \, q_\mathrm{P}^2}{4\pi \, l_\mathrm{P}^3 \, m_\mathrm{P}}$$

The factor of $4\pi$ is the geometric convention of rationalized units — the same $4\pi$ that appears in Coulomb’s law $F = q_1 q_2/(4\pi\varepsilon_0 r^2)$. Apart from this convention, $\varepsilon_0$ is a pure product of Planck-unit quantities: a Planck time squared, a Planck charge squared, divided by a Planck length cubed and a Planck mass.

Rearranging in terms of the compound constants $\hbar$ and $c$:

Compact form via compound constants
$$\varepsilon_0 = \frac{q_\mathrm{P}^2}{4\pi \, \hbar \, c}$$

which is simply the inverse of the relation $q_\mathrm{P}^2 = 4\pi \varepsilon_0 \hbar c$ used in the customary calculation of the Planck charge. The equivalence of the two forms follows from substituting $\hbar = m_\mathrm{P} l_\mathrm{P}^2/t_\mathrm{P}$ and $c = l_\mathrm{P}/t_\mathrm{P}$.

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Dimensions
The product $t_\mathrm{P}^2 \, q_\mathrm{P}^2$ contributes T² Q². The denominator $l_\mathrm{P}^3 \, m_\mathrm{P}$ contributes L³ M. Combined: M⁻¹ L⁻³ T² Q² — the dimensions of permittivity.

Equivalent expressions

The same dimensional content appears in several useful forms.

Planck charge squared per action-velocity

$$\varepsilon_0 = \frac{q_\mathrm{P}^2}{4\pi \, \hbar \, c}$$

The product $\hbar c$ has units of action × velocity, or equivalently energy × length, since $\hbar c = E_\mathrm{P} \, l_\mathrm{P}$. In this form, $\varepsilon_0$ normalizes the Planck charge squared against the quantum-relativistic energy-length scale.

Via the magnetic constant

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

Maxwell’s relation between permittivity, permeability, and the speed of light. This identity arises from the requirement that electromagnetic waves propagate at $c$, and holds independently of the Planck-unit interpretation.

Via the fine-structure constant

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

Solving the definition $\alpha = e^2/(4\pi\varepsilon_0 \hbar c)$ for $\varepsilon_0$. This form makes explicit that $\varepsilon_0$ and $\alpha$ together determine the electromagnetic regime: specifying $e$, $\alpha$, $\hbar$, and $c$ fixes $\varepsilon_0$.

Via the Planck length and charge

$$\varepsilon_0 = \frac{q_\mathrm{P}^2}{4\pi \, l_\mathrm{P} \, E_\mathrm{P}}$$

Using $\hbar c = l_\mathrm{P} E_\mathrm{P}$. The Planck charge squared per Planck-energy-times-Planck-length is, up to the $4\pi$ convention, the natural permittivity scale.

Dimensional verification

Several independent formulas for $\varepsilon_0$, drawn from different areas of physics, each reduce to the same Planck-unit expression.

From the Maxwell relation

$$\varepsilon_0 = \frac{1}{\mu_0 c^2}$$
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Planck-unit reduction
$$\require{cancel} \frac{1}{\mu_0 c^2} = \frac{1}{4\pi \, m_\mathrm{P} l_\mathrm{P}/q_\mathrm{P}^2} \cdot \frac{t_\mathrm{P}^2}{l_\mathrm{P}^2} = \frac{q_\mathrm{P}^2 \, t_\mathrm{P}^2}{4\pi \, m_\mathrm{P} \, l_\mathrm{P}^3}$$ Matching the universal form.

From the fine-structure constant definition

$$\varepsilon_0 = \frac{e^2}{4\pi \, \alpha \, \hbar \, c}$$
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Planck-unit reduction
Substituting $e = \sqrt{\alpha} \, q_\mathrm{P}$, $\hbar = m_\mathrm{P} l_\mathrm{P}^2/t_\mathrm{P}$, $c = l_\mathrm{P}/t_\mathrm{P}$: $$\frac{(\sqrt{\alpha} \, q_\mathrm{P})^2}{4\pi \, \alpha \, (m_\mathrm{P} l_\mathrm{P}^2/t_\mathrm{P}) \, (l_\mathrm{P}/t_\mathrm{P})} = \frac{\cancel{\alpha} \, q_\mathrm{P}^2 \, t_\mathrm{P}^2}{4\pi \, \cancel{\alpha} \, m_\mathrm{P} \, l_\mathrm{P}^3} = \frac{q_\mathrm{P}^2 \, t_\mathrm{P}^2}{4\pi \, m_\mathrm{P} \, l_\mathrm{P}^3}$$ Matching the universal form. The $\alpha$ in the numerator (from $e^2$) and the $\alpha$ in the denominator cancel.

Two independent routes — one from the electromagnetic wave equation, the other from the atomic-scale coupling — produce the same Planck-unit composition. The Planck-unit content of $\varepsilon_0$ is a property of the constant itself, not an artifact of how it is measured.

Physical characterization

Because $\varepsilon_0$ carries Planck-unit quantities in its dimensions, it offers an independent route to the Planck-unit values — a route that does not pass through $\hbar$. This demonstrates that the standard calculation path through $\hbar$, $c$, and $G$ is convenient, but not uniquely fundamental: the electromagnetic constants contain the same Planck-scale information, and can recover the same Planck-unit values, with the fine-structure constant $\alpha$ serving as the bridge between the two paths.

Planck units calculated from the electromagnetic constants

The definition of the fine-structure constant,

$$\alpha = \frac{e^2}{4\pi \varepsilon_0 \hbar c}$$

rearranges to give $\hbar$ in terms of the electromagnetic constants:

$$\hbar = \frac{e^2}{4\pi \varepsilon_0 \alpha c}$$

Substituting this expression for $\hbar$ into the standard Planck-unit formulas eliminates $\hbar$ entirely:

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Planck units via the electromagnetic route
$$l_\mathrm{P} = \sqrt{\frac{\hbar G}{c^3}} = \frac{e}{c^2} \sqrt{\frac{G}{4\pi \varepsilon_0 \, \alpha}}$$ $$m_\mathrm{P} = \sqrt{\frac{\hbar c}{G}} = \frac{e}{\sqrt{4\pi \varepsilon_0 \, \alpha \, G}}$$ $$t_\mathrm{P} = \sqrt{\frac{\hbar G}{c^5}} = \frac{e}{c^3} \sqrt{\frac{G}{4\pi \varepsilon_0 \, \alpha}}$$

Each expression recovers the same Planck-unit value as the standard $\hbar$-based formula. The electromagnetic constants $\varepsilon_0$ and $e$ — together with $G$, $c$, and $\alpha$ — contain sufficient Planck-unit content to pin down the Planck length, Planck mass, and Planck time without invoking the Planck constant.

Numerical verification — Planck length from $\varepsilon_0$
$e$1.602 × 10−19 C
$\varepsilon_0$8.854 × 10−12 F/m
$G$6.674 × 10−11 m³ kg⁻¹ s⁻²
$c$2.998 × 108 m/s
$\alpha$7.297 × 10−3
$l_\mathrm{P} = \frac{e}{c^2} \sqrt{\frac{G}{4\pi \varepsilon_0 \alpha}}$1.616 × 10−35 m

Matching the standard CODATA value of the Planck length to four significant figures.

The electromagnetic–quantum bridge

Why does $\alpha$ have to appear in these alternate formulas? The dimensional count provides the answer. From the five dimensional constants $\{\varepsilon_0, c, G, \hbar, e\}$, exactly one dimensionless combination can be formed — and that combination is $\alpha$ itself. Any expression that substitutes the electromagnetic constants in place of $\hbar$ must therefore carry $\alpha$ as its trade-in factor. The fine-structure constant is what relates the electromagnetic regime (set by $e$ and $\varepsilon_0$) to the quantum regime (set by $\hbar$).

Without $\hbar$ at all

If $\hbar$ is dropped from the input set entirely and the question becomes: what length can be assembled from $\{e, \varepsilon_0, c, G\}$ alone? Dimensional analysis forces a unique answer:

$$\ell = \frac{e}{c^2} \sqrt{\frac{G}{\varepsilon_0}} = \sqrt{4\pi \alpha} \cdot l_\mathrm{P}$$

The four constants $\{e, \varepsilon_0, c, G\}$ determine a length — but it is not exactly the Planck length. It is $\sqrt{4\pi\alpha} \approx 0.3028$ times the Planck length. The Planck length itself cannot be recovered precisely from the purely classical-electromagnetic set without reintroducing $\alpha$ by hand. This makes the structural role of $\alpha$ concrete: $\alpha$ is what distinguishes the elementary charge from the Planck charge, and nothing in $\{e, \varepsilon_0, c, G\}$ encodes that distinction.

Numerical verification — length from $\{e, \varepsilon_0, c, G\}$
$\ell = \frac{e}{c^2} \sqrt{\frac{G}{\varepsilon_0}}$4.894 × 10−36 m
$\sqrt{4\pi\alpha} \cdot l_\mathrm{P}$0.3028 × 1.616 × 10−35 = 4.895 × 10−36 m
Agreementwithin rounding

The pattern

The ability to recover Planck-unit values from multiple compound-constant combinations makes visible a structural fact: the Planck units are the single-dimensional components contained in every universal constant, and which compound constants are used to calculate their values is a matter of experimental convenience. The standard route through $\hbar$, $c$, and $G$ is the most compact, but the electromagnetic route through $\varepsilon_0$, $c$, $G$, and $e$ is equally valid — and the two routes are bridged by $\alpha$.

The constant in context

The following formulas show $\varepsilon_0$ at work across different physical settings.

Coulomb’s law

$$F = \frac{q_1 q_2}{4\pi \varepsilon_0 \, r^2} = \alpha \cdot \left(\frac{q_1}{e}\right)\left(\frac{q_2}{e}\right) \cdot F_\mathrm{P} \cdot \left(\frac{l_\mathrm{P}}{r}\right)^2$$

The electrostatic force separates into a Planck force scaled by the charge ratios (expressed as multiples of $e$), the squared distance ratio, and a single factor of $\alpha$ — the dimensionless coupling of electromagnetism.

Gauss’s law

$$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\text{enc}}{\varepsilon_0}$$

The electric flux through a closed surface equals the enclosed charge divided by $\varepsilon_0$. The permittivity here functions as the conversion factor between charge (the source) and flux (the field’s geometric expression).

Electric field energy density

$$u = \frac{1}{2} \varepsilon_0 \, E^2$$

The energy stored per unit volume in an electric field. In universal form, the Planck-unit content of $\varepsilon_0$ combines with $E^2$ to yield an energy density — Planck energy per Planck volume, reduced by the dimensionless field strength ratio.

Maxwell’s equations (displacement current)

$$\nabla \times \mathbf{B} = \mu_0 \left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$$

The Ampère–Maxwell law with Maxwell’s displacement current $\varepsilon_0 \partial_t \mathbf{E}$. This term is what allows electromagnetic waves to propagate in vacuum, and its presence is tied directly to the finite value of $\varepsilon_0$.

Fine-structure constant

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

The dimensionless coupling of electromagnetism. $\varepsilon_0$ appears in the denominator alongside $\hbar c$; both together set the scale against which $e^2$ is measured. Equivalently, $\alpha = (e/q_\mathrm{P})^2$ — the elementary charge squared, measured in Planck charges.

Why this value?

Before 2019, the electric constant had an exact value fixed by the SI definition of the ampere. With $\mu_0$ defined exactly as $4\pi \times 10^{-7}$ H/m and $c$ defined exactly as $299\,792\,458$ m/s, the Maxwell relation $\varepsilon_0 = 1/(\mu_0 c^2)$ fixed $\varepsilon_0$ exactly at $8.854\,187\,817\ldots \times 10^{-12}$ F/m.

The 2019 SI redefinition changed this. The ampere was redefined in terms of the elementary charge $e$, which fixed $e$ exactly rather than $\mu_0$. Both $\mu_0$ and $\varepsilon_0$ became measured quantities, tied to the fine-structure constant $\alpha$ through the relation $\alpha = e^2/(4\pi\varepsilon_0 \hbar c)$. The 2022 CODATA value $\varepsilon_0 = 8.854\,187\,8188(14) \times 10^{-12}$ F/m carries the same relative uncertainty as $\alpha$ — about $1.6 \times 10^{-10}$.

In Planck units, $\varepsilon_0 = q_\mathrm{P}^2/(4\pi\hbar c)$. The factor of $4\pi$ is the convention of rationalized units — it arises from writing Coulomb’s law with a $4\pi$ in the denominator rather than absorbing it into the permittivity itself. In unrationalized (Gaussian) units, $\varepsilon_0$ is not present as a separate quantity; the factor of $4\pi$ is absent from Coulomb’s law and present in Gauss’s law instead. The Planck-unit content of $\varepsilon_0$ is the same in either convention; only the $4\pi$ moves. For the full account of how the SI, emu, and esu systems each package the same charge differently, see Why are there so many electromagnetic unit systems?

The numerical smallness of $\varepsilon_0$ in SI units ($\sim 10^{-11}$) reflects the SI choice of units — a large electrical unit (the coulomb) and conventional mechanical units. In Planck units, $\varepsilon_0$ has a natural magnitude set by $q_\mathrm{P}^2/(\hbar c)$. The physical content is invariant; only the numerical expression changes with the unit system.

Connections

$\varepsilon_0$ sits at the center of the electromagnetic constants, all of which are interrelated through the Planck charge and the fine-structure constant.

Planck charge: $\displaystyle q_\mathrm{P} = \sqrt{4\pi \varepsilon_0 \, \hbar c}$ — the Planck charge is customarily calculated from $\varepsilon_0$, $\hbar$, and $c$.

Magnetic constant: $\displaystyle \mu_0 = \frac{1}{\varepsilon_0 c^2}$ — the electric and magnetic constants are bound by the speed of light.

Elementary charge: $\displaystyle e = \sqrt{\alpha} \cdot q_\mathrm{P} = \sqrt{4\pi\alpha\,\varepsilon_0 \hbar c}$ — the elementary charge is $\sqrt{\alpha}$ times the Planck charge.

Fine-structure constant: $\displaystyle \alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c}$ — the dimensionless coupling, measured as the ratio $(e/q_\mathrm{P})^2$.

Vacuum impedance: $\displaystyle Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \mu_0 c = \frac{1}{\varepsilon_0 c}$ — the characteristic impedance of free space, expressible through $\varepsilon_0$ and $c$ alone.

Speed of light: $\displaystyle c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ — Maxwell’s identification of light as electromagnetic radiation.