Value of Vacuum Permeability/Proof 1

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Theorem

The value of the vacuum permeability is calculated as:

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

with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$.


Proof

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 $\mathrm m \, \mathrm s^{-1}$


$e$ is defined precisely as:

$e = 1 \cdotp 60217 \, 6634 \times 10^{-19} \, \mathrm C$ (coulombs)

$\alpha$ has the value determined experimentally as:

$\alpha \approx 0 \cdotp 00729 \, 73525 \, 693 (11)$

$h$ is defined precisely as:

$h = 6 \cdotp 62607 \, 015 \times 10^{-34} \, \mathrm J \, \mathrm s$ (joule seconds)

$c$ is defined precisely as:

$c = 299 \, 792 \, 458 \, \mathrm m \, \mathrm s^{-1}$ (metres per second)


Hence $\mu_0$ can be calculated as:

\(\ds \mu_0\) \(=\) \(\ds \dfrac {2 \alpha h} {e^2 c}\) \(\ds \dfrac {\mathrm J \, s} {\mathrm C^2 \times \mathrm m \, \mathrm s^{-1} }\)
\(\ds \) \(=\) \(\ds \dfrac {0 \cdotp 00729 \, 73525 \, 693 (11) \times 6 \cdotp 62607 \, 015 \times 10^{-34} } {\paren {1 \cdotp 60217 \, 6634 \times 10^{-19} }^2 \times 299 \, 792 \, 458}\) \(\ds \dfrac {\paren {\frac {\mathrm {kg} \times \mathrm m^2} {\mathrm s} } } {\paren {\mathrm A \, \mathrm s}^2 \times \mathrm s \times \mathrm m \, \mathrm s^{-1} }\) Base Units of Coulomb, Base Units of Joule
\(\ds \) \(=\) \(\ds 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6}\) \(\ds \dfrac {\mathrm {kg} \times \mathrm m} {\mathrm A^2 \times \mathrm s^4}\)
\(\ds \) \(=\) \(\ds 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6}\) \(\ds \dfrac {\mathrm H} {\mathrm m}\) Base Units of Henry

$\blacksquare$