Zero uncertainty constant and a unit change

Question

So, we know the speed of light with zero uncertainty. We also know that values of $\epsilon_0$ (electric constant) and $\mu_0$ (magnetic constant) are known with zero uncertainty.

My questions are quite simple, but I seek a sanity check.

1. Why are $\mu_0$ and $\epsilon_0$ known with zero uncertainty?
2. What might happen to the uncertainty of the present zero uncertainty constants if the definition of the units is changed in the future SI ?

My guess is :

1. $\Large \frac{1}{\sqrt{ \mu_0 \epsilon_0}}$

and

2. Nothing.

But I would appreciate hearing other's thoughts. Cheers.

Answer 1

1. We are free to define our units however we wish. These are examples where the units are essentially determined by the magnitudes of the constants from certain historically important equations. The simplest example here is the speed of light in vacuum. Its value is defined as $299\,792\,458$ meters per second. Now, the second is already defined with respect to something else, and the speed of light in vacuum is an invariant feature of nature, so we can equivalently think of this as defining the unit of meter as simply the distance that light goes in $1/(299\,792\,458)$ seconds. Similarly, the permeability $\mu_0$ is defined as $4\pi \times 10^{-7} \mathrm{N} / \mathrm{A}^2$, which essentially defines the unit Ampere (symbol $\mathrm{A}$). Similarly, the permittivity $\epsilon_0$ is defined as $1/(c_0^2\, \mu_0)$, as you suggest. This quantity is measured in units of Coulombs per volt per meter, which defines the unit Coulomb.

2. Anything could happen at all. You could multiply the numerical value of the speed of light by two, and create a new unit of distance might be called the "new meter". The new meter would just be half of what we currently call a meter. Much like new coke, this probably wouldn't get much traction, because the definition of units are just for convenient standards. But we're free to redefine them any way we please.