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As you may be aware, a new SI system is likely to be adopted in November 2018 (see https://www.nist.gov/si-redefinition/kilogram-introduction).

Whilst the speed of light remains a fixed quantity and hence $\epsilon_0 \mu_0$ remains fixed, the definition of the kg and Ampere will change such that $\mu_0$ will have an experimental uncertainty.

What is the current level of that uncertainty?

I note Uncertainty of permittivity of vacuum, which is clearly linked. But I also note that all of the answers there are based on the soon to be superseded current system in which there is no uncertainty in $\mu_0$ or $\epsilon_0$ and are therefore about to become incorrect!

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In the new system of units the electron charge, speed of light and Planck's constant all assume defined values.

The permeability of vacuum or "magnetic constant", $\mu_0$ can be written as $$ \mu_0 = \frac{2h}{e^2 c}\alpha,$$ where $\alpha$ is the fine structure constant.

According to NIST, the fine structure constant has a value of $$ \alpha = (7.2973525664 \pm 0.0000000017) \times 10^{-3}$$ corresponding to a relative precision of $2.3\times 10^{-10}$.

I assume then, that this will be the relative precision with which $\mu_0$ is known after the re-definition.

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  • $\begingroup$ Yes, this is right. +1 $\endgroup$ – Dale Nov 10 '18 at 18:00
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    $\begingroup$ Yes, this is correct. The full roster is given in this Wikipedia table. $\endgroup$ – Emilio Pisanty Nov 13 '18 at 14:31
  • $\begingroup$ ... and, even better, this is mentioned explicitly in the changeover resolution (as linked here). $\endgroup$ – Emilio Pisanty Nov 18 '18 at 17:25

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