Impedance of free space

In electromagnetism, the impedance of free space, Z₀, is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, <math display=block>Z_0 = \frac{|\mathbf E|}{|\mathbf H|},</math> where |E| is the electric field strength and |H| is the magnetic field strength. Its presently accepted value is<ref name="physconst-Z0">"2018 CODATA Value: characteristic impedance of vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-10-31.</ref>

Z₀ = 376.730313668(57) Ω.

Where Ω is the ohm, the SI unit of electrical resistance. The impedance of free space (that is the wave impedance of a plane wave in free space) is equal to the product of the vacuum permeability μ₀ and the speed of light in vacuum c₀. Before 2019, the values of both these constants were taken to be exact (they were given in the definitions of the ampere and the metre respectively), and the value of the impedance of free space was therefore likewise taken to be exact. However, with the redefinition of the SI base units that came into force on 20 May 2019, the impedance of free space is subject to experimental measurement because only the speed of light in vacuum c₀ retains an exactly defined value.

Terminology

The analogous quantity for a plane wave travelling through a dielectric medium is called the intrinsic impedance of the medium, and designated η (eta). Hence Z₀ is sometimes referred to as the intrinsic impedance of free space,<ref>Haslett, Christopher J. (2008). Essentials of radio wave propagation. The Cambridge wireless essentials series. Cambridge University Press. p. 29. ISBN 978-0-521-87565-3.</ref> and given the symbol η₀.<ref name="cheng" /> It has numerous other synonyms, including:

• wave impedance of free space,<ref>Guran, Ardéshir; Mittra, Raj; Moser, Philip J. (1996). Electromagnetic wave interactions. Series on stability, vibration, and control of systems. World Scientific. p. 41. ISBN 978-981-02-2629-9.</ref>
• the vacuum impedance,<ref>Clemmow, P. C. (1973). An introduction to electromagnetic theory. University Press. p. 183. ISBN 978-0-521-09815-1.</ref>
• intrinsic impedance of vacuum,<ref>Kraus, John Daniel (1984). Electromagnetics. McGraw-Hill series in electrical engineering. McGraw-Hill. p. 396. ISBN 978-0-07-035423-4.</ref>
• characteristic impedance of vacuum,<ref>Cardarelli, François (2003). Encyclopaedia of scientific units, weights, and measures: their SI equivalences and origins. Springer. p. 49. ISBN 978-1-85233-682-0.</ref>
• wave resistance of free space.<ref>Ishii, Thomas Koryu (1995). Handbook of Microwave Technology: Applications. Academic Press. p. 315. ISBN 978-0-12-374697-9.</ref>

Relation to other constants

From the above definition, and the plane wave solution to Maxwell's equations, <math display="block">Z_0 = \frac{|\mathbf E|}{|\mathbf H|} = \mu_0 c = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{1}{\varepsilon_0 c} = \frac{2 \alpha h}{e^2}</math> where

• μ₀ is the magnetic constant, also known as the permeability of free space ≈ 12.566×10⁻⁷ Henries/meter,
• ε₀ is the electric constant, also known as the permittivity of free space ≈ 8.854×10⁻¹² Farads/meter,
• c is the speed of light in free space.<ref>With ISO 31-5, NIST and the BIPM have adopted the notation c₀ for the speed of light in free space.</ref><ref>"Current practice is to use c₀ to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose." Quote from NIST Special Publication 330, Appendix 2, p. 45 Archived 2016-06-03 at the Wayback Machine,</ref>
• e is the elementary charge,
• α is the fine structure constant, and
• h is Planck's constant.

The reciprocal of Z₀ is sometimes referred to as the admittance of free space and represented by the symbol Y₀.

Historical exact value

Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of μ₀ to be exactly 4π × 10⁻⁷ H/m. Similarly, since 1983 the SI metre has been defined relative to the second by choosing the value of c₀ to be 299792458 m/s. Consequently, until the 2019 redefinition,

<math>Z_0 = \mu_0 c = 4\pi \times 29.979\,2458~\Omega</math> exactly,

or

<math>Z_0 = \mu_0 c = \pi \times 119.916\,9832~\Omega</math> exactly,

or

<math>Z_0 = 376.730\,313\,461\,77\ldots~\Omega.</math>

This chain of dependencies changed when the ampere was redefined on 20 May 2019.

Approximation as 120π ohms

It is very common in textbooks and papers written before about 1990 to substitute the approximate value 120π ohms for Z₀. This is equivalent to taking the speed of light c to be precisely 3×10⁸ m/s in conjunction with the then-current definition of μ₀ as 4π × 10⁻⁷ H/m. For example, Cheng 1989 states<ref name="cheng" /> that the radiation resistance of a Hertzian dipole is

<math>R_r \approx 80 \pi^2 \left( \frac{l}{\lambda}\right)^2</math> (result in ohms; not exact).

This practice may be recognized from the resulting discrepancy in the units of the given formula. Consideration of the units, or more formally dimensional analysis, may be used to restore the formula to a more exact form, in this case to

<math>R_r = \frac{2 \pi}{3} Z_0 \left( \frac{l}{\lambda}\right)^2.</math>

See also

• Electromagnetic wave equation
• Mathematical descriptions of the electromagnetic field
• Near and far field
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Space cloth
• Vacuum
• Wave impedance

References and notes

<references> <ref name="cheng">David K Cheng (1989). Field and wave electromagnetics (Second ed.). New York: Addison-Wesley. ISBN 0-201-12819-5.</ref> </references>

Further reading

• John David Jackson (1998). Classical electrodynamics (Third ed.). New York: Wiley. ISBN 0-471-30932-X.