Definition:Vacuum Permeability

Physical Constant

The vacuum permeability is the physical constant denoted $\mu_0$ defined as:

$\mu_0:= \dfrac {2 \alpha h} {e^2 c}$

where:

$e$ is the elementary charge

$\alpha$ is the fine-structure constant

$h$ is Planck's constant

$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$

Of the above, only the fine-structure constant $\alpha$ is a measured value; the others are defined.

It can be defined as the capability of a magnetic field to permeate a vacuum.

From Value of Vacuum Permeability, it has the value:

$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6} \, \mathrm H \, \mathrm m^{-1}$ (henries per metre)

It can equivalently be defined as:

$\mu_0 := \dfrac 1 {\varepsilon_0 c^2}$

where:

$\varepsilon_0 $ is the vacuum permittivity defined in $\mathrm F \, \mathrm m^{-1}$ (farads per metre)

$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$

Dimension

The vacuum permeability has the dimension $\mathsf {M L T^{-2} I^{-2} }$.

This arises from its definition as inductance per unit length:

$\dfrac {\mathsf {M L^2 T^{-2} I^{-2} } } {\mathsf L}$

Also known as

The vacuum permeability is also known by the terms:

vacuum magnetic permeability

permeability of free space

permeability of vacuum

magnetic constant.

Some sources denote the vacuum permeability with the symbol $\Gamma_m$.

Also see

• Interconnection between Vacuum Permittivity and Vacuum Permeability
• Value of Vacuum Permeability
• Definition:Vacuum Permittivity

• Results about vacuum permeability can be found here.

Historical Note

Before the redefinition of the SI base units on $20$ May $2019$, the vacuum permeability was defined as exactly $4 \pi \times 10^{-7} \, \mathrm H \, \mathrm m^{-1}$ (henries per metre).

As a consequence, the vacuum permittivity was defined as

$\varepsilon_0 = 8 \cdotp 85418 \, 78176 \, 2039 \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre)

which was derived from the equation:

$\varepsilon_0 := \dfrac 1 {\mu_0 c^2}$

where $c$ is the speed of light defined as exactly $299 \, 792 \, 458 \, \mathrm m \, \mathrm s^{-1}$.

However, since $20$ May $2019$, the vacuum permeability has been redefined to be dependent upon the newly redefined electric charge on the electron, as follows:

$\mu_0 = \dfrac {2 \alpha} {e^2} \dfrac h c$

where:

$\alpha$ is the fine-structure constant

$e$ is the elementary charge

$h$ is Planck's constant

$c$ is the speed of light.

As a consequence, $\mu_0$ is now dependent upon the measured quantity $\alpha$, and its value is approximately:

$\mu_0 \approx 4 \pi \times 1 \cdotp 00000 \, 00005 \, 5 (15) \times 10^{-7} \, \mathrm H \, \mathrm m^{-1}$

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Appendix $\text A$: Units