# Impedance of free space

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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 currently has an exactly defined value

A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

## Contents

${\displaystyle Z_{0}=(119.916~983~2)\pi ~\Omega =376.730~313~461~77\ldots ~\Omega .}$

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. However, with the redefinition of the SI base units which are going into force on May 20, 2019, this value is subject to experimental measurement.

In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

The physical constant μ0,, commonly called the vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant, is the magnetic permeability in a classical vacuum. Vacuum permeability is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field production in a vacuum.

The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second. It is exact because by international agreement a metre is defined to be the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most commonly associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

## 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:

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The medium has an intrinsic impedance, given by

Eta is the seventh letter of the Greek alphabet. Originally denoting a consonant /h/, its sound value in the classical Attic dialect of Ancient Greek was a long vowel, raised to [i] in hellenistic Greek, a process known as iotacism.

• 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,

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

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

where

μ0 is the magnetic constant,
ε0 is the electric constant,
c0 is the speed of light in free space. [8] [9]

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

## 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,

The ampere, often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.

The henry is the SI derived unit of electrical inductance. If a current of 1 ampere flowing through the coil produces flux linkage of 1 weber turn, the coil has a self inductance of 1 henry.‌ The unit is named after Joseph Henry (1797–1878), the American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday (1791–1867) in England.

The metre or meter is the base unit of length in the International System of Units (SI). The SI unit symbol is m. The metre is defined as the length of the path travelled by light in a vacuum in 1/299 792 458 second.

${\displaystyle Z_{0}=\mu _{0}c_{0}=119.916\,9832\,\pi ~\Omega }$exactly,

or

${\displaystyle Z_{0}=376.730\,313\,461\,77\ldots ~\Omega .}$

This chain of dependencies will change when the ampere is redefined on 20 May 2019. 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 Z0. This is equivalent to taking the speed of light c0 to be precisely 3×108 m/s in conjunction with the current definition of μ0. For example, Cheng 1989 states [2] that the radiation resistance of a Hertzian dipole is

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

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

## 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. 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.

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