# Units of permeability

## Main Question or Discussion Point

In Hartle's book on General Relativity, page 47 footnote 1, it says: "You might be used to thinking that quantities called $\epsilon_0$ and $\mu_0$ are the basic parameters in Maxwell's equations, but $\mu_0 \equiv 4\pi \times 10^{-7}$ is a pure number, and $\epsilon_0 = 1/(c^2 \mu_0)$."

But as far as I know permeability of free space does have units, for example:
http://scienceworld.wolfram.com/physics/PermeabilityofFreeSpace.html

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Simon Bridge
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I don't understand how this link explains the seemingly unitless permeability the textbook suggests

vanhees71
Gold Member
2019 Award
Of course these "artificial quantities" have non-unit dimension in the SI of units. The reason is that in this system of units, for electromagnetism one introduces a fourth base unit for charge (in fact in the SI it's the Ampere for the current). The magnetic permeability of the vacuum, $\mu_0$, is set arbitrarily to $\mu_0=4 \pi \dot 10^{-7} \; \text{N}/\text{A}^2$. The dielectrical constant of the vacuum is then fixed by the relation, $c^2=1/(\mu_0 \epsilon_0)$, where the vacuum-speed of light is fixed within the SI by relating the unit of length (metre) to the unit of time (second) by setting its value to $c=2.99792458 \cdot 10^8 \; \text{m}/\text{s}.$

From a physical point of view SI units are not very natural since electric and magnetic field components are just components of the Faraday tensor field in four-dimensional spacetime (Minkowski space in special pseudo-Riemannian space in general relativity).

The "electric" part of the SI system isn't perfect indeed but not worse than others.
Physicists have always tried to choose a "simple" system of units. This way, we have seen several variants of the CGS approach, where every law had its units that lead to a unitary, "simple" constant (forget the 4π that plagged those equations). Later on, people sought systems with unit speed of light, unit Planck's constant and so on.
However elegant those systems are, when time comes to measure (still an important subject in Physics) they're very complicated to live with.
I think Nature doesn't pay attention to our wishes and we will have to deal with constants (I hope not with Imperial constants)

vanhees71
Gold Member
2019 Award
Right. Of course the SI is made for everyday use and to provide well-defined units for measurement. However, when it comes to the logical structure of natural laws, other systems are better. At least $\vec{E}$ and $\vec{B}$ should have the same units and not some strange constants like $\epsilon_0$ and $\mu_0$, which are just chosen in a way to make everyday numbers in electrical engineering convenient, should appear in the equations, but the velocity of light. To also get rid of the factors $4 \pi$, one reationalizes the Gaussian CGS units and uses Heaviside-Lorentz units.

Of course, one can also use natural units by setting other fundamental constants to 1 (like $\hbar$ and/or $c$) to further simplify the equations.

clem
In the early part of the last century, Georgi (an electrical engineer) came to the mistaken notion that charge was a new physical unit on a par with mass, length, and time. This led eventually to the MKSA system of units, later voted into existence (in a close vote) as SI. With the purportedly 'fundamental' unit 'ampere', all SI quantities have complicated units.
If the definition of the ampere given in post #2 is used to replace 'ampere' as a unit, the SI units simplify.

clem
I misspelled Giorgi.

In Hartle's book on General Relativity, page 47 footnote 1, it says: "You might be used to thinking that quantities called $\epsilon_0$ and $\mu_0$ are the basic parameters in Maxwell's equations, but $\mu_0 \equiv 4\pi \times 10^{-7}$ is a pure number, and $\epsilon_0 = 1/(c^2 \mu_0)$."

But as far as I know permeability of free space does have units, for example:
http://scienceworld.wolfram.com/physics/PermeabilityofFreeSpace.html
Your book means that it is an exact number, not that it is unit-less. It is only exact because we choose it that way, with nothing physically significant about it being exact. The number π is exact and unitless in a fundamental way independent of choice of unit system. The permeability of free space is not.

Simon Bridge