# What's the physical interpretation of $\mu_0$ and $\varepsilon_0$?

1. Jul 20, 2014

### HakimPhilo

What are the physical interpretations of $\mu_0$ and $\varepsilon_0$, the magnetic permeability and electric permittivity of vacuum? Can these be directly measured? How?

2. Jul 20, 2014

### Staff: Mentor

They are basically artifacts of the units that we use for the magnetic field and the electric field. In Gaussian units, in vacuum, they don't exist at all. Look at Maxwell's equations in Gaussian units.

3. Jul 20, 2014

### HallsofIvy

Rather than "they don't exist at all", you mean they are "1" don't you?

4. Jul 20, 2014

### Staff: Mentor

That is a dimensionless 1, which can always be factored out or in. So I don't think there much of a difference between being a dimensionless factor 1 and not existing.

In SI units for Newtons 2nd law (f=ma) would you say the conversion factor between N and kg m/s^2 doesn't exist or is 1? I think either way is equivalent.

5. Jul 20, 2014

### HallsofIvy

I would say it was 1. To say a conversion factor "doesn't exist", to me, would imply that you can't convert one to the other.

6. Jul 20, 2014

### AlephZero

The key difference between CGS units and SI units is that SI invented an extra independent base unit (the ampere) which is logically unnecessary, and CGS units did not. Since the physics doesn't depend on the units, there is necessarily an extra non-dimensionless constant in SI units to mop up the extra unit.

So in SI units you have two "arbitrary units conversion factors" $\mu_0$ and $\epsilon_0$, and an equation giving the speed of light in a vacuum in terms of the two factors.

In CGS you only have one "arbitrary constant" which depends on the units of length and time, namely the speed of light.

EDIT: this crossed with Hall's previous post, but IMO the point is that in CGS units there is nothing to convert. As an example, the units of electrical resistivity are just seconds-1, not something derived from amperes. Similarly the units of electrical capacitance in CGS are just centimeters.

Last edited: Jul 20, 2014
7. Jul 20, 2014

### Staff: Mentor

Fair enough. As long as you are consistent between Newton's 2nd law in SI units and Maxwells equations in Gaussian I think it is fine.

8. Jul 21, 2014

### PhilDSP

Where SI starts showing a potential logical advantage over Gaussian units is that $\mu_0$ and $\varepsilon_0$ become no longer constants but vectors or tensors when dealing with media. Outside of a vacuum, magnetic permeability and electric permittivity vary continuously. They are the basis for most optical equations.

This means too that the possibilities for solutions to the Maxwell equations is increased. You may choose any combination of E to D or B to H variables using $\mu$ and $\varepsilon$ as conversion factors. Maxwell determined that there is a fundamental difference between a force intensity and a flux amount (the difference between E and D and subsequently B and H) that is defined in geometric terms.

$B = \mu H$
$D = \varepsilon E$

Last edited: Jul 21, 2014