Impedance of free space

Relation to other constants

From the above definition, and the plane wave solution to Maxwell's equations, $${\displaystyle Z_0=\frac{|\mathbf{E}|}{|\mathbf{H}|}={\mu }_{0}c=\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}=\frac{1}{{\epsilon }_{0}c},}$$ where

μ₀ ≈ 12.566×10⁻⁷ H/m is the magnetic constant, also known as the permeability of free space,

ε₀ ≈ 8.854×10⁻¹² F/m is the electric constant, also known as the permittivity of free space,

c is the speed of light in free space,

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

Historical exact value

Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of μ₀ to be exactly 4π × 10⁻⁷ H/m. Similarly, since 1983 the SI metre has been defined relative to the second by choosing the value of c₀ to be 299792458 m/s. Consequently, until the 2019 revision,

${\displaystyle Z_0={\mu }_{0}c=4\pi \times 29.9792458\mathrm{\Omega }}$ exactly,

or

${\displaystyle Z_0={\mu }_{0}c=\pi \times 119.9169832\mathrm{\Omega }}$ exactly,

or

${\displaystyle Z_0=376.73031346177\dots \mathrm{\Omega }.}$

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 120π ohms for Z₀. This is equivalent to taking the speed of light c to be precisely 3×10⁸ m/s in conjunction with the then-current definition of μ₀ as 4π × 10⁻⁷ H/m. For example, Cheng 1989 states that the radiation resistance of a Hertzian dipole is

${\displaystyle R_r\approx 80{\pi }^2{\left(\frac{l}{\lambda }\right)}^2}$ (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

${\displaystyle R_r=\frac{2\pi }{3}{Z}_0{\left(\frac{l}{\lambda }\right)}^2.}$