# Impedance of free space

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, Z0 = | E | / | H |, where | E | is the electric field strength and | H | is the magnetic field strength. It has an exactly defined value

${\displaystyle Z_{0}=119.9169832\;\pi \ \Omega \approx 376.730\ 313\ 461\ 77\ldots \Omega }$.

The impedance of free space (more correctly, the wave-impedance of a plane wave in free space) equals the product of the vacuum permeability μ0 and the speed of light in vacuum c0. Since the values of these constants are exact (they are given in the definitions of the ampere and the metre respectively), the value of the impedance of free space is likewise exact.

## 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,[1] and given the symbol η0.[2] It has numerous other synonyms, including:

• wave impedance of free space,[3]
• the vacuum impedance,[4]
• intrinsic impedance of vacuum,[5]
• characteristic impedance of vacuum,[6]
• wave resistance of free space.[7]

## Relation to other constants

From the above definition, and the plane wave solution to Maxwell's equations,

${\displaystyle Z_{0}={\frac {E}{H}}=\mu _{0}c_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}={\frac {1}{\varepsilon _{0}c_{0}}}}$

where

${\displaystyle \mu _{0}=}$ the magnetic constant
${\displaystyle \varepsilon _{0}=}$ the electric constant and
${\displaystyle c_{0}=}$ the speed of light in free space.[8][9]

The reciprocal of ${\displaystyle Z_{0}\ }$ is sometimes referred to as the admittance of free space, and represented by the symbol ${\displaystyle Y_{0}\ }$.

## Exact value

Since 1948, the definition of the SI unit ampere has relied upon 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,

${\displaystyle Z_{0}=\mu _{0}c_{0}=119.9169832\;\pi \ \Omega }$ exactly,

or

${\displaystyle Z_{0}\approx 376.730\ 313\ 461\ 77\ldots \Omega .}$

This chain of dependencies will change if the ampere is redefined in 2018. See New SI definitions.

## Approximation as 120π ohms

It is very common in textbooks and papers written before about 1990 to substitute the approximate value 120π ohms for ${\displaystyle Z_{0}}$. This is equivalent to taking the speed of light ${\displaystyle c_{0}}$ to be 3×108 m/s in conjunction with the current definition of ${\displaystyle \mu _{0}}$. For example, Cheng 1989 states[2] that the radiation resistance of a Hertzian dipole is

${\displaystyle R_{r}\approx 80\pi ^{2}\left({\frac {\ell }{\lambda }}\right)^{2}.}$ [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

${\displaystyle R_{r}={\frac {2\pi }{3}}Z_{0}\left({\frac {\ell }{\lambda }}\right)^{2}.}$

## References and notes

1. ^ 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.
2. ^ a b David K Cheng (1989). Field and wave electromagnetics (Second ed.). New York: Addison-Wesley. ISBN 0-201-12819-5.
3. ^ 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.
4. ^ Clemmow, P. C. (1973). An introduction to electromagnetic theory. University Press. p. 183. ISBN 978-0-521-09815-1.
5. ^ Kraus, John Daniel (1984). Electromagnetics. McGraw-Hill series in electrical engineering. McGraw-Hill. p. 396. ISBN 978-0-07-035423-4.
6. ^ Cardarelli, François (2003). Encyclopaedia of scientific units, weights, and measures: their SI equivalences and origins. Springer. p. 49. ISBN 978-1-85233-682-0.
7. ^ Ishii, Thomas Koryu (1995). Handbook of Microwave Technology: Applications. Academic Press. p. 315. ISBN 978-0-12-374697-9.
8. ^ With ISO 31-5, NIST and the BIPM have adopted the notation c0 for the speed of light in free space.
9. ^ "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