The impedance of free space, $Z_{0} \approx 376.730\,\Omega$, fixes the ratio of electric to magnetic field amplitude in every electromagnetic wave in vacuum. It can be expressed as $Z_{0} = 4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2})$ — exactly $4\pi$ times the Planck impedance $Z_{\mathrm{P}} = m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2}) \approx 29.98\,\Omega$. Equivalently $Z_{0} = 2\alpha R_{K}$, where $R_{K} = h/e^{2}$ is the von Klitzing constant. These two expressions connect the vacuum’s electromagnetic structure to precision quantum electrical metrology.
Planck-unit form
This follows from $\mu_{0} = 4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}/q_{\mathrm{P}}^{2}$ and $c = l_{\mathrm{P}}/t_{\mathrm{P}}$. Multiplying gives the combination above. The $4\pi$ is the geometric factor from Coulomb’s law in SI form; the remaining structure is the Planck impedance $Z_{\mathrm{P}}$, the unit of impedance at the Planck scale.
| Factor | Role | Source |
|---|---|---|
| $4\pi$ | geometric coefficient | spherical symmetry of Coulomb’s law (SI form) |
| $m_{\mathrm{P}} l_{\mathrm{P}}^{2}/t_{\mathrm{P}}$ | mechanical energy per time | from $\mu_{0} \cdot c$ rearranged |
| $1/q_{\mathrm{P}}^{2}$ | per squared Planck charge | fixes electromagnetic units |
Equivalent form through the von Klitzing constant
The same constant can be expressed through the von Klitzing constant $R_{K} = h/e^{2}$ and the fine-structure constant $\alpha = (e/q_{\mathrm{P}})^{2}$:
Using $\alpha = (e/q_{\mathrm{P}})^{2}$ and $h = 2\pi\, m_{\mathrm{P}} l_{\mathrm{P}}^{2}/t_{\mathrm{P}}$, the right side expands to $4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2})$ — the same Planck-unit combination as before. The $\alpha$ in the numerator and the $(e/q_{\mathrm{P}})^{2}$ from $e^{2}$ cancel exactly.
Both routes agree to four significant figures. The identity $4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2}) = 2\alpha h/e^{2}$ is structural, not coincidental: it follows from the definitions of $q_{\mathrm{P}}$ and $\alpha$.
Physical meaning: field ratio in a plane wave
In any electromagnetic plane wave in vacuum, the electric and magnetic field amplitudes are locked in a fixed ratio. With $H = B/\mu_{0}$ the magnetic field intensity,
This ratio is set by the vacuum itself; it does not depend on the wave’s frequency, wavelength, or direction. The time-averaged power per unit area (the Poynting flux) for a wave of electric-field amplitude $E_{0}$ is $\langle S \rangle = E_{0}^{2}/(2Z_{0})$.
Connection to the quantized Hall resistance
Rearranging $Z_{0} = 2\alpha R_{K}$ gives $R_{K} = Z_{0}/(2\alpha)$. In Planck-unit components,
$R_{K}$ is measured in quantum Hall experiments to a relative uncertainty near $10^{-10}$. The factor $2\alpha$ separating $R_{K}$ from $Z_{0}$ is the electromagnetic coupling: $Z_{0}$ is a vacuum property, while $R_{K}$ involves the specific coupling of electrons to the electromagnetic field. After the 2019 SI redefinition, $h$ and $e$ are exact, so the uncertainty in $Z_{0}$ is entirely the uncertainty in $\alpha$, roughly $1.6\times 10^{-10}$. This makes $Z_{0}$ more precisely known than any individual Planck-unit component (whose precision is limited by $G$).
Reflection and impedance matching
When an electromagnetic wave meets a material of impedance $Z \neq Z_{0}$, the reflection coefficient is $r = (Z – Z_{0})/(Z + Z_{0})$. A perfect conductor has $Z = 0$ and $r = -1$ (complete reflection with phase reversal); a matched medium has $Z = Z_{0}$ and $r = 0$ (no reflection). Practical radio-frequency impedances sit in the same territory: a half-wave dipole antenna radiates at about $73\,\Omega$, within a factor of a few of the Planck impedance $Z_{\mathrm{P}} \approx 30\,\Omega$. Radio engineering operates on impedance scales set by the vacuum itself.
Connections
Magnetic constant. $\mu_{0} = 4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}/q_{\mathrm{P}}^{2}$, and $Z_{0} = \mu_{0} c$. The factor $c = l_{\mathrm{P}}/t_{\mathrm{P}}$ converts $\mu_{0}$’s units to ohms.
Fine-structure constant. $\alpha = (e/q_{\mathrm{P}})^{2} \approx 1/137.036$. After 2019, $\alpha$ is the only measured input needed to fix $Z_{0}$.
Von Klitzing constant. $R_{K} = h/e^{2} \approx 25\,813\,\Omega$. The vacuum impedance and the quantized Hall resistance are separated by the electromagnetic coupling: $Z_{0} = 2\alpha R_{K}$.
Planck impedance. $Z_{\mathrm{P}} = m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2}) \approx 29.98\,\Omega$. The unit of impedance at the Planck scale; $Z_{0} = 4\pi Z_{\mathrm{P}}$.