Английская Википедия:Impedance of free space
In electromagnetism, the impedance of free space, Шаблон:Math, is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, <math display=block>Z_0 = \frac{|\mathbf E|}{|\mathbf H|},</math> where Шаблон:Math is the electric field strength and Шаблон:Math is the magnetic field strength. Its presently accepted value isШаблон:Physconst
Where Ω is the ohm, the 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 Шаблон:Math and the speed of light in vacuum Шаблон:Math. 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 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 Шаблон:Math retains an exactly defined value.
Terminology
The analogous quantity for a plane wave travelling through a dielectric medium is called the intrinsic impedance of the medium, and designated Шаблон:Mvar (eta). Hence Шаблон:Math is sometimes referred to as the intrinsic impedance of free space,[1] and given the symbol Шаблон:Math.[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, <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
- Шаблон:Math is the magnetic constant, also known as the permeability of free space ≈ Шаблон:Val Henries/meter,
- Шаблон:Math is the electric constant, also known as the permittivity of free space ≈ Шаблон:Val Farads/meter,
- Шаблон:Math is the speed of light in free space.[8][9]
- Шаблон:Math is the elementary charge,
- Шаблон:Math is the fine structure constant, and
- Шаблон:Math is Planck's constant.
The reciprocal of Шаблон:Math is sometimes referred to as the admittance of free space and represented by the symbol Шаблон:Math.
Historical exact value
Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of Шаблон:Math to be exactly Шаблон:Nowrap. Similarly, since 1983 the SI metre has been defined relative to the second by choosing the value of Шаблон:Math to be Шаблон:Val. 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 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 120Шаблон:Pi ohms for Шаблон:Math. This is equivalent to taking the speed of light Шаблон:Math to be precisely Шаблон:Val in conjunction with the then-current definition of Шаблон:Math as Шаблон:Nowrap. For example, Cheng 1989 states[2] 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
- ↑ Шаблон:Cite book
- ↑ 2,0 2,1 Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ With ISO 31-5, NIST and the BIPM have adopted the notation Шаблон:Math for the speed of light in free space.
- ↑ "Current practice is to use Шаблон:Math to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol Шаблон:Math was used for this purpose." Quote from NIST Special Publication 330, Appendix 2, p. 45 Шаблон:Webarchive,
Further reading
