View source for Impedance of free space

{{short description|Physical constant; ratio of electric to magnetic field strength in a vacuum}}

In [[electromagnetism]], the '''impedance of free space''', {{math|''Z''<sub>0</sub>}}, is a [[physical constant]] relating the magnitudes of the [[electric field|electric]] and [[magnetic field]]s of [[electromagnetic radiation]] travelling through [[Vacuum#In electromagnetism|free space]]. That is, 
<math display=block>Z_0 = \frac{|\mathbf E|}{|\mathbf H|},</math> 
where {{math|{{abs|'''E'''}}}} is the [[electric field strength]] and {{math|{{abs|'''H'''}}}} is the [[magnetic field strength]]. Its presently accepted  value is{{physconst|Z0|ref=only}}

:{{math|{{physconst|Z0|symbol=yes|ref=no}}}}.

Where Ω is the [[ohm]], the [[international system of units|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]] {{math|''μ''<sub>0</sub>}} and the [[speed of light|speed of light in vacuum]] {{math|''c''<sub>0</sub>}}. 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 [[2019 redefinition of the SI base units|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 {{math|''c''<sub>0</sub>}} retains an exactly defined value.

==Terminology==
The analogous quantity for a [[plane wave]] travelling through a [[dielectric]] [[optical medium|medium]] is called the ''[[intrinsic impedance]]'' of the medium, and designated {{mvar|η}} ([[eta]]). Hence {{math|''Z''<sub>0</sub>}} is sometimes referred to as the ''intrinsic impedance of [[Vacuum|free space]]'',<!-- 1210 Google Books hits --><ref>{{cite book|last=Haslett|first=Christopher J.|title=Essentials of radio wave propagation|url=https://books.google.com/books?id=rx-1YAmPZycC|series=The Cambridge wireless essentials series|year=2008|publisher=Cambridge University Press|isbn=978-0-521-87565-3|page=29}}</ref> and given the symbol {{math|''η''<sub>0</sub>}}.<ref name="cheng" /> It has numerous other synonyms, including:
<!-- sorted by decreasing Google Books hits-->
* ''wave impedance of free space'',<!-- 464 Google Books hits --><ref>{{cite book|last1=Guran|first1=Ardéshir|last2=Mittra|first2=Raj|last3=Moser|first3=Philip J.|title=Electromagnetic wave interactions|series=Series on stability, vibration, and control of systems|url=https://books.google.com/books?id=xY3ed--JZFAC|year=1996|publisher=World Scientific|isbn=978-981-02-2629-9|page=41}}</ref>
* ''the vacuum impedance'',<ref><!-- 422 Google Books hits -->{{cite book|last=Clemmow|first=P. C. |title=An introduction to electromagnetic theory|url=https://books.google.com/books?id=ahQ7AAAAIAAJ|year=1973|publisher=University Press|isbn=978-0-521-09815-1|page=183}}</ref>
* ''intrinsic impedance of vacuum'',<!-- 72 Google Books hits --><ref>{{cite book|last=Kraus|first=John Daniel|title=Electromagnetics|url=https://archive.org/details/electromagnetics00krau|url-access=registration|series=McGraw-Hill series in electrical engineering|year=1984|publisher=McGraw-Hill|isbn=978-0-07-035423-4|page=[https://archive.org/details/electromagnetics00krau/page/396 396]}}</ref>
* ''characteristic impedance of vacuum'',<!-- 59 Google Books hits --><ref>{{cite book|last=Cardarelli|first=François|title=Encyclopaedia of scientific units, weights, and measures: their SI equivalences and origins|url=https://archive.org/details/encyclopaediaofs0000card|url-access=registration|year=2003|publisher=Springer|isbn=978-1-85233-682-0|page=[https://archive.org/details/encyclopaediaofs0000card/page/49 49]}}</ref>
* ''wave resistance of free space''.<!-- 33 Google Books hits --><ref>{{cite book|last=Ishii|first=Thomas Koryu|title=Handbook of Microwave Technology: Applications|url=https://books.google.com/books?id=p8EO7N8qDNcC|year=1995|publisher=Academic Press|isbn=978-0-12-374697-9|page=315}}</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
*{{math|''μ''<sub>0</sub>}} is the [[magnetic constant]], also known as the permeability of free space ≈ {{val|12.566|e=-7}} Henries/meter,
*{{math|''ε''<sub>0</sub>}} is the [[electric constant]], also known as the permittivity of free space ≈ {{val|8.854|e=-12}} Farads/meter,
*{{math|''c''}} is the [[speed of light]] in free space.<ref>With [[ISO 31-5]], [[NIST]] and the [[BIPM]] have adopted the notation {{math|''c''<sub>0</sub>}} for the speed of light in [[Vacuum|free space]].</ref><ref>"Current practice is to use {{math|''c''<sub>0</sub>}} to denote the speed of light in vacuum according to [[ISO 31]]. In the original Recommendation of 1983, the symbol {{math|''c''}} was used for this purpose." Quote from [http://physics.nist.gov/Pubs/SP330/sp330.pdf NIST ''Special Publication 330'', Appendix 2, p. 45] {{Webarchive|url=https://web.archive.org/web/20160603215953/http://physics.nist.gov/Pubs/SP330/sp330.pdf |date=2016-06-03 }},</ref>
*{{math|''e''}} is the [[elementary charge]],
*{{math|''α''}} is the [[fine structure constant]], and
*{{math|''h''}} is  [[Planck's constant]].

The reciprocal of {{math|''Z''<sub>0</sub>}} is sometimes referred to as the ''admittance of free space'' and represented by the symbol {{math|''Y''<sub>0</sub>}}.

==Historical exact value==

Between 1948 and 2019, the [[SI]] unit the [[ampere]] was defined by ''choosing'' the numerical value of {{math|''μ''<sub>0</sub>}} to be exactly {{nowrap|4{{pi}} × {{val|e=-7|u=[[Henry (unit)|H]]/m}}}}. Similarly, since 1983 the SI [[metre]] has been defined relative to the second  by ''choosing'' the value of {{math|''c''<sub>0</sub>}} to be {{val|299792458|u=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 [[2019 redefinition of the SI base units#Ampere|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 120{{pi}}&nbsp;ohms for {{math|''Z''<sub>0</sub>}}. This is equivalent to taking the speed of light {{math|''c''}} to be precisely {{val|3|e=8|u=m/s}} in conjunction with the then-current definition of {{math|''μ''<sub>0</sub>}} as {{nowrap|4{{pi}} × {{val|e=-7|u=[[Henry (unit)|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">{{cite book
|author=David K Cheng
|title=Field and wave electromagnetics
|url=https://books.google.com/books?as_isbn=0-201-12819-5
|publisher=Addison-Wesley
|location=New York
|year=1989
|edition=Second
|isbn=0-201-12819-5}}</ref>
</references>

==Further reading==
* {{cite book
|author=John David Jackson
|title=Classical electrodynamics
|edition=Third
|publisher= Wiley
|location=New York
|year=1998
|isbn=0-471-30932-X
|url=http://worldcat.org/isbn/047130932X}}

[[Category:Electromagnetism]]
[[Category:Physical constants]]