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| magnetic field strength. It has an exact irrational value, given approximately as 376.73031... ohms.^[1]

The impedance of free space (more correctly, the wave-impedance of a plane wave in free space) equals the product of the vacuum permeability or magnetic constant μ₀ and the speed of light in vacuum c₀. Since the numerical values of the magnetic constant and of the speed of light are fixed by the definitions of the ampere and the metre respectively, the exact value of the impedance of free space is likewise fixed by definition relative to the S.I. base units.

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,^[2] and given the symbol η₀.^[3] It has numerous other synonyms, including:

• wave impedance of free space,^[4]
• the vacuum impedance,^[5]
• intrinsic impedance of vacuum,^[6]
• characteristic impedance of vacuum,^[7]
• wave resistance of free space.^[8]

Relation to other constants

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

$Z_0 = \frac{E}{H} = \mu_0 c_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \frac{1}{\varepsilon_0 c_0}$

where

$\mu_0 =$ the magnetic constant

$\varepsilon_0 =$ the electric constant and

$c_0 =$ the speed of light in free space.^[9][10]

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

Exact value

Since 1948, the SI unit ampere has been 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 by choosing the value of c₀ to be 299 792 458 m/s. Consequently

$Z_{0} = \mu_{0} c_0 = 119.9169832 \; \pi \ \Omega$ exactly,

or

$Z_{0} \approx 376.730\ 313\ 461\ 77 \ldots \Omega$. This situation may change if the ampere is redefined in 2015. See New SI definitions.

120π-approximation

It is very common in textbooks and learned papers written before about 1990 to substitute the approximate value $120 \pi$ for $Z_{0}$. This is equivalent to taking the speed of light to be 3×10⁸ m/s. For example, Cheng 1989 states^[3] that the radiation resistance of a Hertzian dipole is

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

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

See also

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

References and notes

1. "Characteristic impedance of vacuum, Z₀". The NIST reference on constants, units, and uncertainty: Fundamental physical constants. NIST. Retrieved 2011-11-28.
2. 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.
3. David K Cheng (1989). Field and wave electromagnetics (Second ed.). New York: Addison-Wesley. ISBN 0-201-12819-5.
4. 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.
5. Clemmow, P. C. (1973). An introduction to electromagnetic theory. University Press. p. 183. ISBN 978-0-521-09815-1.
6. Kraus, John Daniel (1984). Electromagnetics. McGraw-Hill series in electrical engineering. McGraw-Hill. p. 396. ISBN 978-0-07-035423-4.
7. Cardarelli, François (2003). Encyclopaedia of scientific units, weights, and measures: their SI equivalences and origins. Springer. p. 49. ISBN 978-1-85233-682-0.
8. Ishii, Thomas Koryu (1995). Handbook of Microwave Technology: Applications. Academic Press. p. 315. ISBN 978-0-12-374697-9.
9. With ISO 31-5, NIST and the BIPM have adopted the notation c₀ for the speed of light in free space.
10. "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

Further reading

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