In electromagnetism, permeability is the measure
of the ability of a material to support the
formation of a magnetic field within itself.
In other words, it is the degree of magnetization
that a material obtains in response to an
applied magnetic field. Magnetic permeability
is typically represented by the Greek letter
μ. The term was coined in September 1885
by Oliver Heaviside. The reciprocal of magnetic
permeability is magnetic reluctivity.
In SI units, permeability is measured in henries
per meter, or newtons per ampere squared.
The permeability constant, also known as the
magnetic constant or the permeability of free
space, is a measure of the amount of resistance
encountered when forming a magnetic field
in a classical vacuum. The magnetic constant
has the exact value µ0 = 4π×10−7 H·m−1≈
1.2566370614…×10−6 H·m−1 or N·A−2).
A 
closely related property of materials is magnetic
susceptibility, which is a measure of the
magnetization of a material in addition to
the magnetization of the space occupied by
the material.
Explanation
In electromagnetism, the auxiliary magnetic
field H represents how a magnetic field B
influences the organization of magnetic dipoles
in a given medium, including dipole migration
and magnetic dipole reorientation. Its relation
to permeability is
where the permeability, μ, is a scalar if
the medium is isotropic or a second rank tensor
for an anisotropic medium.
In general, permeability is not a constant,
as it can vary with the position in the medium,
the frequency of the field applied, humidity,
temperature, and other parameters. In a nonlinear
medium, the permeability can depend on the
strength of the magnetic field. Permeability
as a function of frequency can take on real
or complex values. In ferromagnetic materials,
the relationship between B and H exhibits
both non-linearity and hysteresis: B is not
a single-valued function of H, but depends
also on the history of the material. For these
materials it is sometimes useful to consider
the incremental permeability defined as
This definition is useful in local linearizations
of non-linear material behavior, for example
in a Newton–Raphson iterative solution scheme
that computes the changing saturation of a
magnetic circuit.
Permeability is the inductance per unit length.
In SI units, permeability is measured in henries
per metre = N A−2). The auxiliary magnetic
field H has dimensions current per unit length
and is measured in units of amperes per metre.
The product μH thus has dimensions inductance
times current per unit area. But inductance
is magnetic flux per unit current, so the
product has dimensions magnetic flux per unit
area. This is just the magnetic field B, which
is measured in webers per square-metre, or
teslas.
B is related to the Lorentz force on a moving
charge q:
The charge q is given in coulombs, the velocity
v in meters per second, so that the force
F is in newtons:
H is related to the magnetic dipole density.
A magnetic dipole is a closed circulation
of electric current. The dipole moment has
dimensions current times area, units ampere
square-metre, and magnitude equal to the current
around the loop times the area of the loop.
The H field at a distance from a dipole has
magnitude proportional to the dipole moment
divided by distance cubed, which has dimensions
current per unit length.
Relative permeability and magnetic susceptibility
Relative permeability, sometimes denoted by
the symbol μr, is the ratio of the permeability
of a specific medium to the permeability of
free space, μ0:
where μ0 = 4π × 10−7 N A−2. In
terms of relative permeability, the magnetic
susceptibility is
χm, a dimensionless quantity, is sometimes
called volumetric or bulk susceptibility,
to distinguish it from χp and χM.
Diamagnetism
Diamagnetism is the property of an object
which causes it to create a magnetic field
in opposition of an externally applied magnetic
field, thus causing a repulsive effect. Specifically,
an external magnetic field alters the orbital
velocity of electrons around their nuclei,
thus changing the magnetic dipole moment in
the direction opposing the external field.
Diamagnets are materials with a magnetic permeability
less than μ0.
Consequently, diamagnetism is a form of magnetism
that a substance exhibits only in the presence
of an externally applied magnetic field. It
is generally a quite weak effect in most materials,
although superconductors exhibit a strong
effect.
Paramagnetism
Paramagnetism is a form of magnetism which
occurs only in the presence of an externally
applied magnetic field. Paramagnetic materials
are attracted to magnetic fields, hence have
a relative magnetic permeability greater than
one. The magnetic moment induced by the applied
field is linear in the field strength and
rather weak. It typically requires a sensitive
analytical balance to detect the effect. Unlike
ferromagnets, paramagnets do not retain any
magnetization in the absence of an externally
applied magnetic field, because thermal motion
causes the spins to become randomly oriented
without it. Thus the total magnetization will
drop to zero when the applied field is removed.
Even in the presence of the field there is
only a small induced magnetization because
only a small fraction of the spins will be
oriented by the field. This fraction is proportional
to the field strength and this explains the
linear dependency. The attraction experienced
by ferromagnets is non-linear and much stronger,
so that it is easily observed, for instance,
in magnets on one's refrigerator.
Gyromagnetism
For gyromagnetic media the magnetic permeability
response to an alternating electromagnetic
field in the microwave frequency domain is
treated as a non-diagonal tensor expressed
by:
Values for some common materials
The following table should be used with caution
as the permeability of ferromagnetic materials
varies greatly with field strength. For example
4% Si steel has an initial relative permeability
of 2,000 and a maximum of 35,000 and, indeed,
the relative permeability of any material
at a sufficiently high field strength trends
toward 1.
A good magnetic core material must have high
permeability.
For passive magnetic levitation a relative
permeability below 1 is needed.
Permeability varies with 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 frequency is considered, the permeability
can be complex, corresponding to the in phase
and out of phase response.
Note that the magnetic constant μ0 has an
exact value in SI units because the definition
of the ampere fixes its value to 4π × 10−7 H/m
exactly.
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. These fields
can be written as phasors, such that
where is the phase delay of from . 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
so that the permeability becomes a complex
number. By Euler's formula, the complex permeability
can be translated from polar to rectangular
form,
The ratio of the imaginary to the real part
of the complex permeability is called the
loss tangent,
which provides a measure of how much power
is lost in a material versus how much is stored.
See also
Antiferromagnetism
Diamagnetism
Electromagnet
Ferromagnetism
Figure of merit
Magnetic reluctance
Paramagnetism
Permittivity
SI electromagnetism units
Notes
^ The permeability of Austenitic Stainless
Steel strongly depends on the history of mechanical
stress applied to it, such as cold working
References
External links
Electromagnetism - a chapter from an online
textbook
Relative Permeability
Soil Permeability Test
Magnetic Properties of Materials
