Vacuum impedance

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For Impedance of Free Space (also known as Z0), a Canadian IDM and electronica group, see Impedance of Free Space.

The impedance of free space, $Z 0$ is a physical constant with a defined (not measured) value that relates the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is,

$Z_{0} = \frac{|E|}{|H|}$

where

$|E| = \$ electric field strength

$|H| = \$ magnetic field strength.

The impedance of free space has units of ohms (Ω).

1 Terminology
2 Relation to other constants
3 Exact value
4 120π-approximation
5 See also
6 References and notes
7 Further reading

Terminology

NIST uses the term characteristic impedance of vacuum for Z₀ and posts its defined value in SI units at NIST Z₀.

There are numerous other synonyms, including:

intrinsic impedance of vacuum,
intrinsic impedance of free space,
the vacuum impedance,
wave resistance 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 η.

Relation to other constants

From the above definition, and the plane wave solution to Maxwell's equations,^[1]

$Z_{0} \overset{\underset{\mathrm{def}}{}}{=}\ \mu_{0} c_0 = \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} = \frac{1}{\varepsilon_{0} c_0}$

where

$\mu_0 \ \overset{\underset{\mathrm{def}}{}}{=}\ 4 \pi \times 10^{-7}\ \approx 1.256\ 637\ 061\ 4 \ldots \times 10^{-6}$ H/m magnetic constant^[2]

$\varepsilon_0 \ \overset{\underset{\mathrm{def}}{}}{=}\ \frac {1}{\mu_0 {c_0}^2} \approx 8.854\ 187\ 817 \ldots \times 10^{-12}$ F/m electric constant^[3]

$c_0 \ \overset{\underset{\mathrm{def}}{}}{=}\ 299,792,458 \ \mathrm {m/s} \ ,$ speed of light in free space^[4]^[5]

(These values are taken from NIST μ₀, NIST ε₀, and NIST c₀.)

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 μ₀ $\ \stackrel{\mathrm{def}}{=}\$ 4π × 10^-7 H/m. Similarly, since 1983 the SI metre has been defined by c₀ $\ \stackrel{\mathrm{def}}{=}\$ 299 792 458 m/s. Consequently

$Z_{0} \ \stackrel{\mathrm{def}}{=}\ \mu_{0} c_0 = 119.9169832 \pi \ \Omega$

exactly, or

$Z_{0} \approx 376.730\ 313\ 461\ 77 \ldots \Omega$

approximately.^[6] This situation may change if the ampere is redefined in 2011.

120π-approximation

It is very common in textbooks and learned papers to substitute the approximate value $120π$ for $Z 0$ . This is equivalent to taking the speed of light to be 3×10⁸ m/s. For example, Cheng 1998 states that the radiation resistance of a Hertzian dipole is

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

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

References and notes

^ With ISO 31-5, NIST and the BIPM have adopted the notation c₀ for the speed of light in free space.
^ "Magnetic constant". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.
^ "Electric constant". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.
^ "Speed of light". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.
^ "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
^ NIST Z₀

Vacuum impedance

Contents

Terminology

Relation to other constants

Exact value

120π-approximation

See also

References and notes

Further reading