My name is Alexander FufaeV and here I write about:

What is the Vacuum Permeability?

The vacuum permeability is a physical constant and is written as $ \mu_0 $ (pronounced: "Mu zero"). It has the following value:

$$ \begin{align} \mu_0 ~=~ 1.256 \, 637 \, 062 \, 12 ~\cdot~ 10^{-6} \, \frac{\mathrm{Vs}}{\mathrm{Am}} \end{align} $$

The unit of $\mu_0$ is, for example, volt-second per ammeter or Henry per meter:

$$ \begin{align} \frac{\mathrm{Vs}}{\mathrm{Am}} ~=~ \frac{\mathrm{H}}{\mathrm{m}} ~=~ \frac{\mathrm{N}}{\mathrm{A}^2} ~=~ \frac{\mathrm{kg}\,\mathrm{m}}{\mathrm{A}^2\,\mathrm{s}^2} \end{align} $$

The vacuum permeability occurs in the equations that have to do with magnetic fields. For example, in the Biot-Savart law or in the wave equation for electromagnetic waves.

The vacuum permeability, for example, determines how strongly two current-carrying wires attract or repel each other.

Two current-carrying wires attracting each other.

Two repulsive current-carrying wires.

The vacuum permeability, together with the vacuum permittivity $\varepsilon_0$, determines how large the speed of light $c$ should be in a vacuum:

$$ \begin{align} c ~=~ \sqrt{\frac{1}{\mu_0 \, \varepsilon_0}} \end{align} $$

Vacuum permeability describes how good the magnetic fields can penetrate the vacuum.

The magnetic field constant also determines how strongly the vacuum can be magnetized. If we want to know how strongly other substances can be magnetized, then we can calculate their permeability $\mu$:

$$ \begin{align} \mu ~=~ \mu_{\text r} \, \mu_0 \end{align} $$

Here $\mu_{\text r}$ is the relative permeability and varies from material to material. The relative permeability value describes whether the material is diamagnetic, paramagnetic or ferromagnetic. Iron, for example, has a value of $\mu_{\text r} = 300 $ to $\mu_{\text r}= 10\, 000$ depending on the temperature and is very easy to magnetize.

The vacuum permeability can be measured, for example, with the help of two long current-carrying wires which are at a distance $r$ from each other. For this purpose, the attractive force $\class{green}{F}$ of two wires is measured, through which (in each) the current $\class{blue}{I}$ flows:

One of the two wires.

$$ \begin{align} \mu_0 ~=~ \frac{2\pi \, r \, \class{green}{F}}{L \, \class{blue}{I}^2} \end{align} $$

A more precise way to determine the vacuum permeability experimentally is to measure the fine structure constant $\alpha$ and then use the following equation:

$$ \begin{align} \mu_0 ~=~ \frac{2\,h\,\alpha}{ c \, e^2 } \end{align} $$

Here $h$ is the Planck's constant, $e$ is the elementary charge, and $c$ is the speed of light.