What does "Permeability of Free Space" mean? I know the value, but what really is it?

Answer 1

Its an arbitrary number which creates a particular system of units.

Experimentally, ampere noticed that the force between two wires is proportional to the current in the first wire and proportional to the current in the second wire.

At the time, "force" was already defined, but current was not. Permeability of Free Space is just a conversion factor between the two, kind of like we have a conversion factor between calories and joules. The 4pi was chosen for geometric reasons and I suppose 10^-7 made the units more convienient.

It is possible to have a system of units without a permeability of space. With Gaussian units the only constant in Maxwells equations is the speed of light.

You could also invent a system where the speed of light is defined to be one, but this isn't the same type of arbitrarines.

Actually changing the speed of light would have real consequences, whereas changing the permeability of space simply changes the number of electrons in what we call a coulomb.

You could imagine a world where a particular number of electrons would create a larger force, but this change would be linked to a change in the speed of light, so you do not need a second fundamental constant (permeability)

Another way to think of it is to say that magnetic fields do not even exist. They are simply electrostatic fields distorted by a lorentz transformation (Relativity). So you cannot adjust the strengths of electrostatic and magnetic forces separately.

Answer 2

See the reference for a fuller explanation. Imagine you're 1000 feet above the earth's surface, where the earth's magnetic field is fairly uniform. Picture the magnetic flux lines passing you uniformly. Now bring a chunk of weakly ferromagnetic material and you'll see the flux lines concentrating somewhat through that material. Repeat with a strongly ferromagnetic material and they will concentrate even more. The relative measurement of how much the materials concentrate the flux is permeability. Make that measurement when nothing's there, and you have the permeability of free space.

As you'll see from the reference, it's not an arbritrary number.