impedance of free space

The analogous quantity for a plane wave travelling through a dielectric medium is called the intrinsic impedance of the medium and designated {{mvar|η}} (eta). Hence {{math|Z₀}} is sometimes referred to as the intrinsic impedance of free space,{{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}} and given the symbol {{math|η₀}}. It has numerous other synonyms, including:

• wave impedance of free space,{{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}}
• the vacuum impedance,{{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}}
• intrinsic impedance of vacuum,{{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]}}
• characteristic impedance of vacuum,{{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]}}
• wave resistance of free space.{{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}}

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

$$Z\_0\; =\; \backslash frac$$

where

: {{math|μ₀ ≈ {{val|12.566|e=-7}}}} H/m is the magnetic constant, also known as the permeability of free space,

: {{math|ε₀ ≈ {{val|8.854|e=-12}}}} F/m is the electric constant, also known as the permittivity of free space,

: {{math|c}} is the speed of light in free space,With ISO 31-5, NIST and the BIPM have adopted the notation {{math|c₀}} for the speed of light in free space."Current practice is to use {{math|c₀}} 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 }}.

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

Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of {{math|μ₀}} to be exactly {{nowrap|4{{pi}} × {{val|e=-7|u=H/m}}}}. Similarly, since 1983 the SI metre has been defined relative to the second by choosing the value of {{math|c₀}} to be {{val|299792458|u=m/s}}. Consequently, until the 2019 revision,

: $Z\_0\; =\; \backslash mu\_0\; c\; =\; 4\backslash pi\; \backslash times\; 29.979\backslash ,2458~\backslash Omega$ exactly,

or

: $Z\_0\; =\; \backslash mu\_0\; c\; =\; \backslash pi\; \backslash times\; 119.916\backslash ,9832~\backslash Omega$ exactly,

or

: $Z\_0\; =\; 376.730\backslash ,313\backslash ,461\backslash ,77\backslash ldots~\backslash Omega.$

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

It is very common in textbooks and papers written before about 1990 to substitute the approximate value 120{{pi}} ohms for {{math|Z₀}}. 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|μ₀}} as {{nowrap|4{{pi}} × {{val|e=-7|u=H/m}}}}. For example, Cheng 1989 states that the radiation resistance of a Hertzian dipole is

: $R\_r\; \backslash approx\; 80\; \backslash pi^2\; \backslash left(\; \backslash frac\{l\}\{\backslash lambda\}\backslash right)^2$ (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

: $R\_r\; =\; \backslash frac\{2\; \backslash pi\}\{3\}\; Z\_0\; \backslash left(\; \backslash frac\{l\}\{\backslash lambda\}\backslash right)^2.$

• 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

{{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

}}

• {{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