 x Permeability (electromagnetism) Encyclopedia
In electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, permeability is the measure of the ability of a material to support the formation of a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

within itself. In other words, it is the degree of magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

that a material obtains in response to an applied magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

. Magnetic permeability is typically represented by the Greek letter μ
Mu (letter)
Carlos Alberto Vives Restrepo is a Grammy Award and three-time Latin Grammy Award winning-Colombian singer, composer and actor.-Biography:...

. The term was coined in September, 1885 by Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...

. The reciprocal of magnetic permeability is magnetic reluctivity.

We can simplify it by saying, the more conductive a material is to a magnetic field, the higher its permeability.

In SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, permeability is measured in the henry-meters (H·m−1), or newton-ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

squared (N·A−2). The permeability constant (μ0), 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
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

. The magnetic constant has the exact (defined) value µ0 = 4π×10−7 ≈ 1.2566370614…×10−6 H·m−1 or N·A−2).

## Explanation

In electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, 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
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

reorientation. Its relation to permeability is where the permeability μ is a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

if the medium is isotropic or a second rank tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

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
Humidity
Humidity is a term for the amount of water vapor in the air, and can refer to any one of several measurements of humidity. Formally, humid air is not "moist air" but a mixture of water vapor and other constituents of air, and humidity is defined in terms of the water content of this mixture,...

, temperature, and other parameters. In a nonlinear medium
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

, 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
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

materials, the relationship between B and H exhibits both non-linearity
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

and hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

: 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
Saturation (magnetic)
Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetizing field H cannot increase the magnetization of the material further, so the total magnetic field B levels off...

of a magnetic circuit.

Permeability is the inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

per unit length. In SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, permeability is measured in henries per metre (H·m−1 = J/(A2·m) = N A−2). The auxiliary magnetic field H has dimensions current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

per unit length and is measured in units of ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

s per metre (A m−1). The product μH thus has dimensions inductance times current per unit area (H·A/m2). But inductance is magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

per unit current, so the product has dimensions magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

per unit area. This is just the magnetic field B, which is measured in webers
Weber (unit)
In physics, the weber is the SI unit of magnetic flux. A flux density of one Wb/m2 is one tesla.The weber is named for the German physicist Wilhelm Eduard Weber .- Definition :...

(volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

-second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

s) per square-metre (V·s/m2), or teslas
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

(T).

B is related to the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

on a moving charge q: .

The charge q is given in coulombs (C), the velocity v in m/s, so that the force F is in newtons (N): 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 (A·m2), 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

Relative permeability, sometimes denoted by the symbol μr, is the ratio of the permeability of a specific medium to the permeability of free space given by the magnetic constant : .

In terms of relative permeability, the magnetic susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...

is: .

χm, a dimensionless quantity, is sometimes called volumetric or bulk susceptibility, to distinguish it from χp (magnetic mass or specific susceptibility) and χM (molar or molar mass susceptibility).

## Diamagnetism

Diamagnetism is the property of an object which causes it to create a magnetic field in opposition of an externally applied magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

, 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 (a relative permeability less than 1).

Consequently, diamagnetism is a form of magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

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
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

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 (or, equivalently, a positive magnetic susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...

).
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
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

, 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.

## 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 (at or near 0T) of 2,000 and a maximum of 35,000 and, indeed, the relative permeability of any material at a sufficiently high field strength tends to 1.
Magnetic susceptibility and permeability data for selected materials
Medium Susceptibility χm
(volumetric SI)
Permeability μ [H/m] Relative Permeability μ/μ0 Magnetic field Frequency max.
Metglas
Metglas
Metglas is a thin amorphous metal alloy ribbon produced by using rapid solidification process of approx: 1,000,000°C/s. This rapid solidification creates unique ferromagnetic properties that allows the ribbon to be magnetized and de-magnetized quickly and effectively with very low core losses of...

at 0.5 T 100kHz
Nanoperm at 0.5 T 10kHz
Mu-metal
Mu-metal
Mu-metal is a nickel-iron alloy that is notable for its high magnetic permeability. The high permeability makes mu-metal very effective at screening static or low-frequency magnetic fields, which cannot be attenuated by other methods. The name came from the Greek letter mu which represents...

at 0.002 T
Mu-metal
Mu-metal
Mu-metal is a nickel-iron alloy that is notable for its high magnetic permeability. The high permeability makes mu-metal very effective at screening static or low-frequency magnetic fields, which cannot be attenuated by other methods. The name came from the Greek letter mu which represents...

Permalloy
Permalloy
Permalloy is a nickel-iron magnetic alloy, with about 20% iron and 80% nickel content. It is notable for its very high magnetic permeability, which makes it useful as a magnetic core material in electrical and electronic equipment, and also in magnetic shielding to block magnetic fields...

at 0.002 T
Electrical steel
Electrical steel
Electrical steel, also called lamination steel, silicon electrical steel, silicon steel, relay steel or transformer steel, is specialty steel tailored to produce certain magnetic properties, such as a small hysteresis area and high permeability.The material is usually manufactured in the form of...

at 0.002 T
Ferrite
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...

(nickel zinc)
16–640 100 kHz ~ 1 MHz
Ferrite (manganese zinc) > 640 (or more) 100 kHz ~ 1 MHz
Steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

100 at 0.002 T
Nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...

100 – 600 at 0.002 T
Neodymium magnet
Neodymium magnet
A neodymium magnet , the most widely-used type of rare-earth magnet, is a permanent magnet made from an alloy of neodymium, iron, and boron to form the Nd2Fe14B tetragonal crystalline structure. Developed in 1982 by General Motors and Sumitomo Special Metals, neodymium magnets are the strongest...

1.05
Platinum
Platinum
Platinum is a chemical element with the chemical symbol Pt and an atomic number of 78. Its name is derived from the Spanish term platina del Pinto, which is literally translated into "little silver of the Pinto River." It is a dense, malleable, ductile, precious, gray-white transition metal...

Aluminum
Wood
Wood
Wood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...

Air
Concrete
Concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...

1
Vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

0 0) 1
Hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

Teflon
Sapphire
Sapphire
Sapphire is a gemstone variety of the mineral corundum, an aluminium oxide , when it is a color other than red or dark pink; in which case the gem would instead be called a ruby, considered to be a different gemstone. Trace amounts of other elements such as iron, titanium, or chromium can give...

Copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

or
Water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

Bismuth
Bismuth
Bismuth is a chemical element with symbol Bi and atomic number 83. Bismuth, a trivalent poor metal, chemically resembles arsenic and antimony. Elemental bismuth may occur naturally uncombined, although its sulfide and oxide form important commercial ores. The free element is 86% as dense as lead...

Superconductors −1 0 0

A good magnetic core material must have high permeability.

For magnetic levitation
Magnetic levitation
Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields...

a relative permeability below 1 is needed (corresponding to a negative susceptibility).

Permeability varies with magnetic field. Values shown above are approximate and valid only at the magnetic fields shown. Moreover, 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
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, corresponding to the in phase and out of phase response.

Note that the magnetic constant has an exact value in SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units (that is, there is no uncertainty in its value),
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
Phasor (electronics)
In physics and engineering, a phase vector, or phasor, is a representation of a sine wave whose amplitude and angular frequency are time-invariant. It is a subset of a more general concept called analytic representation. Phasors decompose the behavior of a sinusoid into three independent...

, such that where is the phase delay of from . Understanding permeability as the ratio of the magnetic field to the auxiliary 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
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

, 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
Loss tangent
The loss tangent is a parameter of a dielectric material that quantifies its inherent dissipation of electromagnetic energy. The term refers to the tangent of the angle in a complex plane between the resistive component of an electromagnetic field and its reactive component.-Electromagnetic...

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

• Antiferromagnetism
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...

• Diamagnetism
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to 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...

• Electromagnet
Electromagnet
An electromagnet is a type of magnet in which the magnetic field is produced by the flow of electric current. The magnetic field disappears when the current is turned off...

• Ferromagnetism
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

• Figure of merit
Figure of merit
A figure of merit is a quantity used to characterize the performance of a device, system or method, relative to its alternatives. In engineering, figures of merit are often defined for particular materials or devices in order to determine their relative utility for an application...

• Magnetic reluctance
Magnetic reluctance
Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is analogous to resistance in an electrical circuit, but rather than dissipating magnetic energy it stores magnetic energy...

• Paramagnetism
Paramagnetism
Paramagnetism is a form of magnetism whereby the paramagnetic material is only attracted when in the presence of an externally applied magnetic field. In contrast with this, diamagnetic materials are repulsive when placed in a magnetic field...

• Permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...

• SI electromagnetism units 