Permeability

Permeability $\mu$ is the material parameter linking the magnetic flux density $\mathbf{B}$ to the magnetic field intensity $\mathbf{H}$:

$\mathbf{B}=\mu \mathbf{H}.$

Units: henrys per meter (H/m). Typically split as

$\mu ={\mu }_r{\mu }_0$

with ${\mu }_0=4\pi \times 10^{-7}$ H/m the permeability of free space and ${\mu }_r$ the relative permeability (dimensionless).

For most materials (vacuum, air, water, copper, aluminum, dielectrics), ${\mu }_r$ is within parts per thousand of 1, and you can use $\mu \approx {\mu }_0$ without loss. Exceptions are the ferromagnetic materials (iron, cobalt, nickel, and their alloys), where ${\mu }_r$ can be hundreds to hundreds of thousands. Electromagnetics does not cover ferromagnetism — assume $\mu ={\mu }_0$ unless stated otherwise.

Why ${\mu }_0$ has the value it does

${\mu }_0=4\pi \times 10^{-7}$ H/m looks oddly arbitrary, but it’s actually a definition. Before the 2019 SI redefinition, the ampere was defined so that two parallel wires 1 m apart carrying 1 A each experience a force of $2\times 10^{-7}$ N/m. From the Ampère’s-law derivation for parallel wires, this forces ${\mu }_0=4\pi \times 10^{-7}$ exactly.

Post-2019, ${\mu }_0$ is now measured rather than defined, but it remains very close to the historical value.

Connection to speed of light

The free-space permeability and Permittivity together set the speed of light:

$c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}=3\times 10^8\text{ m/s}.$

This is one of EM’s deepest results: a constant from magnetostatics (${\mu }_0$) and one from electrostatics (${ϵ}_0$) combine to give the speed of EM waves — which equals the speed of light. Maxwell’s prediction in 1865 that light is an EM wave hinged on plugging ${\mu }_0,{ϵ}_0$ in and recognizing the result.

How magnetic materials respond

Microscopically: atoms have intrinsic magnetic dipole moments from electron spins and orbital motions. In most materials these dipoles are randomly oriented and average to zero (no net magnetization). Applying an external $\mathbf{H}$ can:

• Diamagnetic (${\mu }_r<1$, very slightly): induced dipoles oppose $\mathbf{H}$, weakening the field. Almost all non-magnetic materials. The effect is tiny.
• Paramagnetic (${\mu }_r>1$, very slightly): existing dipoles align with $\mathbf{H}$, strengthening the field. Also tiny — aluminum, oxygen, etc.
• Ferromagnetic (${\mu }_r\gg 1$): strong nonlinear response, hysteresis, domain structure. The realm of iron, nickel, cobalt, electrical steel, ferrites. Outside Electromagnetics scope.

The magnetization $\mathbf{M}$ (dipole moment per unit volume) parallels the polarization story in electrostatics:

$\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M}),\mathbf{M}={\chi }_m\mathbf{H},$

with ${\chi }_m$ the magnetic susceptibility. This gives ${\mu }_r=1+{\chi }_m$.

In wave propagation

Phase velocity of an EM wave in a medium with $ϵ,\mu$:

$u_p=\frac{1}{\sqrt{\mu ϵ}}=\frac{c}{\sqrt{{\mu }_r{ϵ}_r}}.$

In a non-magnetic dielectric (${\mu }_r=1$), this simplifies to $c/\sqrt{{ϵ}_r}$, the formula used throughout transmission line analysis. In a ferromagnetic or ferrite material, ${\mu }_r$ would also slow the wave.

The intrinsic impedance of a medium is

$\eta =\sqrt{\mu /ϵ}.$

In free space, ${\eta }_0=\sqrt{{\mu }_0/{ϵ}_0}\approx 377\Omega$ — the impedance of free space, the ratio $E/H$ in a free-space plane wave.