Impedance of free space

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For the Canadian IDM and electronica group, see Impedance of Free Space.

The impedance of free space, $Z 0$ is a universal constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space.

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

where

E =

electric field strength

H =

magnetic field strength

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

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 $Z 0$ is sometimes referred to as the intrinsic impedance of free space, and given the symbol $η 0$ . It has numerous other synonyms, including:

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

1 Relation to other constants
2 Exact value
3 120π-approximation
4 Further reading

[edit] Relation to other constants

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

$Z_{0} = \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} = \mu_{0} c = \frac{1}{\varepsilon_{0} c}$

where

μ 0 =

permeability of free space

$\varepsilon_{0} =$ permittivity of free space

c =

speed of light in vacuo

The reciprocal of $Z 0$ is sometimes referred to as the admittance of free space, and represented by the symbol $Y 0$ .

[edit] Exact value

Since 1948, the SI Ampere has been defined by choosing $μ 0$ to be $4 \pi \times 10^{- 7}$ H/m. Similarly, since 1983 the SI metre has been defined by choosing the speed of light $c$ to be 299792458 m/s. Consequently

Z 0 = μ 0 c = 119.9169832πΩ

exactly, or

Z 0 = 376.73031346177...Ω

approximately. This situation may change if the Ampere is redefined in 2011.

[edit] 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 $3 \times 10^8$ m/s. For example, Cheng 1998 states that the radiation resistance of a Hertzian dipole is

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

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