# Why is permeability of free space (in calculation of magnetic field) a non zero number?

I've been reading in Khan academy about the formula for calculating magnetic field lines (Ampere's law). Some materials have the ability to concentrate magnetic fields, which is described by those materials having higher permeability. The permeability of vacuum is $$4\pi *10^-7$$ , which means it can concentrate magnetic field lines. But how can nothing—no matter, energy, or force—influence or manipulate the waves passing through it (practically through nothingness). I mean why is the permeability not 0? (This just comes out of the rigid intuition that vacuum does nothing to anything because it is nothing—ignoring the particle-antiparticle pairs or similar very distant stuff.)

As can be seen from the fuzziness of the question, I'm learning physics (not an expert) and any theoretical answer is okay but answers involving complex math like calculus or differentials are unwelcome but if it needs to be used let me be informed.

• If it was zero, there would be no magnetic field at all. – my2cts Apr 8 '20 at 19:37

The "ability to concentrate fields" is, at best, a very loose description of limited validity. Loose descriptions of limited validity always lead to paradoxes when taken too literally.

The magnetic permeability is a proportionality factor in the relationship between field and current. In free space, the relationship is $$\nabla\times\mathbf{B} = \mu_0 \mathbf{J}$$ Don't be distracted by the derivative $$\nabla$$; the calculus isn't important here. The important thing is that a non-zero current produces a non-zero field, even in free space, so the permeability of free space is non-zero.

Inside a material where some of the current is due to bound charges, the current associated with free charges is the curl of the quantity $$\mathbf{H} = \frac{1}{\mu_0}\mathbf{B} - \mathbf{M}$$ where the magnetization $$\mathbf{M}$$ accounts for the currents due to bound charges. The magnetic permeability $$\mu$$ of the material is defined by the relationship $$\mathbf{H} = \frac{1}{\mu}\mathbf{B}.$$ This is meant to make the equations involving the free current look like the equation involving the total current, with $$\mathbf{H}$$ in place of $$\mathbf{B}$$. In free space, where the magnitization $$\mathbf{M}$$ is zero, the preceding equations imply $$\mu=\mu_0$$.

• Thanks for the derivation which I was looking for. Where does the value of that constant come from 4*pi*10^-7 ?? Is it defined or found out by experimental calculations like G( universal gravitational constant)? – The Infinity Apr 9 '20 at 1:56
• @Theinfinity The value used to be defined as $4\pi\cdot 10^{-7} \text{H/m}$. Now, it is experimentally determined to be within uncertainty of the original value. Also, note that permeability has units; it is not dimensionless. – Sandejo Apr 9 '20 at 2:17
• Tesla metres per ampere . Right. Thank you – The Infinity Apr 9 '20 at 6:30

If the vacuum permeability were zero, then there would be no magnetic fields, as $$B \propto \mu_0$$, so it must be non-zero in order for there to be magnetic fields. The condition of zero permeability actually describes superconductors, where the magnetic field is zero inside. Since you inquired about something being zero in the vacuum, it is worth noting that there is a quantity related to permeability, called magnetic susceptibility, which is zero for the vacuum. This is used to describe how linear media compare to the vacuum, by the relation $$\mu = \mu_0 (1 + \chi_m)$$, where $$\mu$$ is the permeability of some medium with magnetic susceptibility $$\chi_m$$.

• I understood that but my doubt is why is it called permeability of vacuum. Is it just a historical conventional practice or does it have any practical implications? – The Infinity Apr 9 '20 at 1:41

The permeability of vacuum arises in classical field theory from the idea that space is not a nothing, but is a substantive something which supports the notion of a field. It appears differently in quantum electrodynamics, in which electromagnetic forces can be seen as arising from the exchange of photons between charged particles. Although it is still called the permeability of the vacuum, Standards Organizations have recently moved to using magnetic constant as the preferred name for $$μ_0$$,

While $$\mu_0$$ has the name of permeability of vacuum, its origin is related to the forces between conductors:

$$\frac{F}{\Delta L}=\frac{\mu_0 I_1I_2}{2\pi d}$$

What the experiments shows is that the force is proportional to the currents and to the inverse of the distance between wires. It is possible to set the constant of proportionality ($$\mu_0$$) as $$1$$. But in this case the unit of charge has to be modified, because $$I = Q/t$$.

At first, that experiment seems a pure electrical stuff, but conducting wires also deflects a compass needle, so the force is described as mediated by a magnetic field produced by the currents. And it changes depending on the material between the wires.

That is the reason for the name magnetic permeability, and why $$\mu_0\ne 1$$ for the vacuum.