Meaning of permeability in free space

[TheCanadian]

Permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. And the inductance of an electric circuit is one henry (H) when an electric current that is changing at one ampere per second results in an electromotive force of one volt across the inductor. Although when we state that the permeability of free space is ##\mu_0 = 4\pi \times 10^{−7} H \cdot m^{−1}##, what does this actually mean? How exactly can a vacuum have an inductance? What acts as the inductor/material in this case upon which this measurement of ##\mu_0## is made? I realize electromagnetic waves do not require a medium to propagate, but what is the mechanism behind why this value of ##\mu_0## is not 0? Why is it precisely ##4\pi \times 10^{−7} H \cdot m^{−1}##?

 

[andrewkirk]

The meaning of the permeability of free space is perhaps more easily understood by looking at its role in Ampere's Law ##\nabla \times \mathbf B=\mu_0\mathbf J##, which relates the magnetic field around a conductor to the current through the conductor and the permeability of the volume surrounding the conductor. The value of the constant is determined by the relationship between the fundamental units used in Ampere's equation, which are seconds, metres, kg and Coulombs. Each of those four units is chosen independently, and different choices for the combinations of the units give different values for ##\mu_0##. It is analogous in a way to the gravitational constant, which is needed to make Newton's gravitational equation balance, and whose value is determined by our choice of the three units: seconds, metres and kg.

 

[Orodruin]

To add to that, like the speed of light, the permeability of vacuum is not something you measure. That is the reason we can say it is exactly that number. There are no error bars. Since also the speed of light is a defined value, an exact value for the permittivity of vacuum follows.

the fundamental units used in Ampere's equation, which are seconds, metres, kg and Coulombs

Just nitpicking, but Coulomb is not a fundamental SI unit, Ampere is (although I always thought the unit of charge should be the one to be considered fundamental instead of the unit of current).

 

[Dale]

what is the mechanism behind why this value of μ0μ0\mu_0 is not 0?

There is no physical mechanism. It's value is determined by the voting members of the BIPM, the committee that defines SI units. In other commonly used unit systems it is 1.

Why is it precisely 4π×10−7H⋅m−14π×10−7H⋅m−14\pi \times 10^{−7} H \cdot m^{−1}?

Because that is what the committee voted to define it as.

 

[TheCanadian]

The meaning of the permeability of free space is perhaps more easily understood by looking at its role in Ampere's Law ##\nabla \times \mathbf B=\mu_0\mathbf J##, which relates the magnetic field around a conductor to the current through the conductor and the permeability of the volume surrounding the conductor. The value of the constant is determined by the relationship between the fundamental units used in Ampere's equation, which are seconds, metres, kg and Coulombs. Each of those four units is chosen independently, and different choices for the combinations of the units give different values for ##\mu_0##. It is analogous in a way to the gravitational constant, which is needed to make Newton's gravitational equation balance, and whose value is determined by our choice of the three units: seconds, metres and kg.

There is no physical mechanism. It's value is determined by the voting members of the BIPM, the committee that defines SI units. In other commonly used unit systems it is 1.

Because that is what the committee voted to define it as.

To add to that, like the speed of light, the permeability of vacuum is not something you measure. That is the reason we can say it is exactly that number. There are no error bars. Since also the speed of light is a defined value, an exact value for the permittivity of vacuum follows.Just nitpicking, but Coulomb is not a fundamental SI unit, Ampere is (although I always thought the unit of charge should be the one to be considered fundamental instead of the unit of current).

I suppose I am wondering what in the vacuum field permits the propagation of the electromagnetic wave? For example, are quantum fluctuations in vacuum essential to the non-zero permeability of free space? Are there any other properties or phenomena in the vacuum field that permit inductance?

 

[Nugatory]

I suppose I am wondering what in the vacuum field permits the propagation of the electromagnetic wave?

The electric wave propagates because of the way that changes in the magnetic field and the electrical field are related, as described by Maxwell's equations. So this question comes down to asking how it is that these fields can exist in a vacuum. There may not be any answer more satisfying than "Because that's the way the universe we live in works".

For example, are quantum fluctuations in vacuum essential to the non-zero permeability of free space?

No - these are classical fields described by classical physics.

 

[TheCanadian]

The electric wave propagates because of the way that changes in the magnetic field and the electrical field are related, as described by Maxwell's equations. So this question comes down to asking how it is that these fields can exist in a vacuum. There may not be any answer more satisfying than "Because that's the way the universe we live in works".

No - these are classical fields described by classical physics.

Yes, the evolution of these fields is given by Maxwell's equations, but there is a corresponding analog when considering the quantized field, correct? Permeability is not a measure limited to classical physics, right? So when using formalism such as second quantization, I suppose I am trying to understand how the concept or permeability follows.

 

[Dale]

It is the result of a committee decision. It does not follow from classical physics nor quantum physics. It follows from a vote.

 

[Nugatory]

Permeability is not a measure limited to classical physics, right? So when using formalism such as second quantization, I suppose I am trying to understand how the concept or permeability follows.

We start with Maxwell's equations there too, so permeability is already part of the phenomenon that we're quantizing. You might try this overview of how the electromagnetic field is quantized: http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

 

[Vanadium 50]

If you're struggling with classical vacuum pemeability, jumping to quantum will certainly not clear anything up.

How exactly can a vacuum have an inductance?

It doesn't. It has a permeability. If the Henrys (Henries?) confuse you, I could rewrite it as N/A². In short, it's a measure of a force between two currents a distance apart. But that's how the Ampere is defined - which is why you've been told this was determined by committee. Essentially "the permeability of free space" and "how much is an Ampere" are the same quantity. You tell me what one is, and I'll tell you the other.

 

[DrDu]

The SI system is really a horrible choice to do theoretical physics. But it is probably excellent to do metrology which is what it is really designed for. In doubt, it is always easier to force a theoretical physicist to express his final results in ugly units than to measure the value of a beautiful unit which is difficult to access directly in an experiment.