# What will be the uncertainty in $\mu_0$ under the new SI scheme?

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 But I also note that all of the answers there are based on the soon to be superseded current system in which there is no uncertainty in $$\mu_0$$ or $$\epsilon_0$$ and are therefore about to become incorrect!

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.

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