Permeability (electromagnetism)

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In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction ${\displaystyle B}$ to the magnetizing field ${\displaystyle H}$ as a function of the field ${\displaystyle H}$ in a material. The term was coined by

In

A closely related property of materials is magnetic susceptibility, which is a dimensionless proportionality factor that indicates the degree of magnetization of a material in response to an applied magnetic field.

Explanation

In the macroscopic formulation of electromagnetism, there appear two different kinds of magnetic field:

• the magnetizing field H which is generated around electric currents and displacement currents, and also emanates from the poles of magnets. The SI units of H are amperes/meter.
• the magnetic flux density B which acts back on the electrical domain, by curving the motion of charges and causing electromagnetic induction. The SI units of B are volt-seconds/square meter (teslas).

The concept of permeability arises since in many materials (and in vacuum), there is a simple relationship between H and B at any location or time, in that the two fields are precisely proportional to each other:^[2]

${\displaystyle \mathbf {B} =\mu \mathbf {H} }$,

where the proportionality factor μ is the permeability, which depends on the material. The

In the microscopic formulation of electromagnetism, where there is no concept of an H field, the vacuum permeability μ₀ appears directly (in the SI Maxwell's equations) as a factor that relates total electric currents and time-varying electric fields to the B field they generate. In order to represent the magnetic response of a linear material with permeability μ, this instead appears as a magnetization M that arises in response to the B field: ${\displaystyle \mathbf {M} =\left(\mu _{0}^{-1}-\mu ^{-1}\right)\mathbf {B} }$. The magnetization in turn is a contribution to the total electric current—the

Relative permeability and magnetic susceptibility

Relative permeability, denoted by the symbol ${\displaystyle \mu _{\mathrm {r} }}$, is the ratio of the permeability of a specific medium to the permeability of free space μ₀:

${\displaystyle \mu _{\mathrm {r} }={\frac {\mu }{\mu _{0}}},}$

where ${\displaystyle \mu _{0}\approx }$ 4π × 10⁻⁷ H/m is the magnetic permeability of free space.^[3] In terms of relative permeability, the magnetic susceptibility is

${\displaystyle \chi _{m}=\mu _{r}-1.}$

The number χ_m is a dimensionless quantity, sometimes called volumetric or bulk susceptibility, to distinguish it from χ_p (magnetic mass or specific susceptibility) and χ_M (molar or molar mass susceptibility).

Diamagnetism

Diamagnetism is the property of an object which causes it to create a

Consequently, diamagnetism is a form of

Paramagnetism

Paramagnetism is a form of

The following table should be used with caution as the permeability of ferromagnetic materials varies greatly with field strength and specific composition and fabrication. For example, 4% electrical steel has an initial relative permeability (at or near 0 T) of 2,000 and a maximum of 38,000 at T = 1 [5]^[6] and different range of values at different percent of Si and manufacturing process, and, indeed, the relative permeability of any material at a sufficiently high field strength trends toward 1 (at magnetic saturation).

A good magnetic core material must have high permeability.^[35]

For passive magnetic levitation a relative permeability below 1 is needed (corresponding to a negative susceptibility).

Permeability varies with a magnetic field. Values shown above are approximate and valid only at the magnetic fields shown. They are given for a zero frequency; in practice, the permeability is generally a function of the frequency. When the frequency is considered, the permeability can be complex, corresponding to the in-phase and out of phase response.

Complex permeability

A useful tool for dealing with high frequency magnetic effects is the complex permeability. While at low frequencies in a linear material the magnetic field and the auxiliary magnetic field are simply proportional to each other through some scalar permeability, at high frequencies these quantities will react to each other with some lag time.

${\displaystyle H=H_{0}e^{j\omega t}\qquad B=B_{0}e^{j\left(\omega t-\delta \right)}}$

where ${\displaystyle \delta }$ is the phase delay of ${\displaystyle B}$ from ${\displaystyle H}$.

Understanding permeability as the ratio of the magnetic flux density to the magnetic field, the ratio of the phasors can be written and simplified as

${\displaystyle \mu ={\frac {B}{H}}={\frac {B_{0}e^{j\left(\omega t-\delta \right)}}{H_{0}e^{j\omega t}}}={\frac {B_{0}}{H_{0}}}e^{-j\delta },}$

so that the permeability becomes a complex number.

By Euler's formula, the complex permeability can be translated from polar to rectangular form,

${\displaystyle \mu ={\frac {B_{0}}{H_{0}}}\cos(\delta )-j{\frac {B_{0}}{H_{0}}}\sin(\delta )=\mu '-j\mu ''.}$

The ratio of the imaginary to the real part of the complex permeability is called the

${\displaystyle \tan(\delta )={\frac {\mu ''}{\mu '}},}$

which provides a measure of how much power is lost in material versus how much is stored.

See also

• Antiferromagnetism
• Diamagnetism
• Electromagnet
• Ferromagnetism
• Magnetic reluctance
• Paramagnetism
• Permittivity
• SI electromagnetism units

Notes