Understanding Magnetic Permeability

Magnetic permeability measures how easily a material magnetizes in an external magnetic field and how much magnetic flux it can carry as a result. This page covers the definition, formula, units, material types and the factors that change permeability.

What is Magnetic Permeability?

Magnetic permeability is the ratio of the magnetic flux density (B) to the magnetizing force (H) within a material or medium:

Magnetic flux density (B) describes the strength and direction of the field inside a material, counted as field lines per unit area. Its unit is the tesla (T), also written as Weber per square meter (Wb/m^2).

Magnetizing force (H) sets the intensity and direction of the external field doing the magnetizing. A wire coil produces it in proportion to the electric current it carries, and its unit is ampere per meter (A/m).

A high permeability therefore means the material carries a large flux for a given magnetizing force, while a low permeability means it resists magnetization and lets little flux inside.

Unit of Magnetic Permeability

The unit depends on the measurement system in use. In the SI system, magnetic permeability is measured in Henry per meter (H/m), which equals Newton per square ampere (N/A^2). In the CGS system it is treated as a dimensionless quantity.

Permeability of Free Space

Free space, or vacuum, is a reference medium with no matter and no magnetization. The permeability of free space, denoted by μ0, has a constant value of approximately 4π×10−7 H/m.

That exact figure once anchored the SI definition of the ampere. Since the ampere was redefined in 2019, though, μ0​ is no longer exactly equal to 4π×10−7 H/m and must now be determined experimentally. The relative shift from the old value is tiny, about 5.5×10−10.

The permeability of free space, also known as the magnetic constant or vacuum permeability, is a fundamental physical constant. It appears in many electromagnetism equations, like Maxwell’s equations.

Relative Permeability and Magnetic Susceptibility

The permeability of a material or medium other than free space is denoted by μ. It varies depending on the type and composition of the material, as well as other factors such as temperature, frequency and position.

To compare different materials or media directly, engineers use a dimensionless quantity called relative permeability (μr​): the permeability of a material or medium divided by that of free space:

A relative permeability above 1 says the medium supports more magnetic flux than free space under the same magnetizing force.

A relative permeability below 1 says it supports less, which is the signature of diamagnetism.

Magnetic susceptibility (χm​) forms another dimensionless companion of relative permeability, defined as:

Magnetic susceptibility measures how strongly a material responds by becoming magnetized. A positive value means the internal magnetization points along the applied field; a negative value means it points against it.

Types of Materials Based on Permeability

Materials can be classified into three main types based on their relative permeability and magnetic susceptibility: diamagnetic, paramagnetic and ferromagnetic.

Diamagnetic Materials

Diamagnetic materials have a relative permeability slightly below 1 (μr​<1) and a negative susceptibility (χm​<0). They respond weakly against an applied field and push most of the magnetic flux out of their interior. Bismuth, copper, water and air are common examples.

Placed in an external field, a diamagnetic material develops a small opposing field inside itself. The net magnetic flux density drops, so these materials are gently repelled by magnets.

Paramagnetic Materials

Paramagnetic materials have a relative permeability slightly above 1 (μr​>1) and a positive susceptibility (χm​>0). They feel a weak pull toward an applied field and let extra flux pass through their interior. Platinum, aluminum, oxygen and manganese are typical examples.

When such a material sits in an external field, some of its atomic or molecular magnets line up with that field and produce a weak magnetization in the same direction. The net magnetic flux density rises slightly, so paramagnetic materials are attracted by magnets.

Ferromagnetic Materials

Ferromagnetic materials have a relative permeability far above 1 (μr​>>1) with very large positive susceptibility (χm​>>0). They respond strongly to an external field and can channel enormous amounts of flux through their interior. Iron, nickel, cobalt and steel are the classic examples.

When a ferromagnetic material enters an external field, most of its atomic magnets snap into alignment and the net magnetic flux density rises enormously, which is why these materials are strongly attracted by magnets.

Unlike diamagnetic and paramagnetic materials, ferromagnetic materials can retain their magnetization even after the external field is removed. This phenomenon is called hysteresis, and it depends on the history of the applied field. Ferromagnetic materials can also exhibit domains, which are regions where the atomic or molecular magnets are aligned in the same direction. The domains can be influenced by the external field, temperature, stress and impurities.

Factors Affecting Permeability

The permeability of a material or medium depends on several factors, such as:

• Humidity: absorbed moisture measurably shifts the permeability of porous magnetic ceramics such as ferrites, though dense metals barely respond.
• Temperature: thermal motion of the atoms or molecules disturbs the magnetic alignment and lowers the permeability.
• Position in the medium: local variations in density, composition or structure can make the measured permeability differ from place to place.
• Frequency of the applied field: the time-varying nature of an alternating current (AC) field brings eddy currents, skin effects and other loss mechanisms into play, all of which change the effective permeability.

Complex Permeability

Complex permeability is a concept that accounts for high-frequency effects on magnetic fields. When an AC field is applied to a material or medium, there may be a phase lag between B and H due to losses such as eddy currents, hysteresis, etc. This phase lag can be represented by using complex numbers for B and H, such as:

where B0​ and H0​ are the amplitudes, ω is the angular frequency, t is time, and ϕB​ and ϕH​ are phase angles.

The complex permeability μ∗ can then be defined as:

The complex permeability can be written in polar form as:

where ∣μ∗∣ is called magnitude permeability and θ is called the phase angle.

Applications of Magnetic Permeability

Many technologies work only because designers can choose materials with the right permeability. Some examples are:

• Magnetic recording and storage devices: Magnetic permeability affects the performance and quality of magnetic tapes, disks, cards and memories. High permeability materials are used to store information by creating magnetic domains on the surface. Low permeability materials are used to shield the devices from external interference and noise.
• Magnetic sensors and transducers: Magnetic permeability affects the sensitivity and accuracy of devices that measure or convert magnetic fields into electrical signals or vice versa. Examples are Hall effect sensors, magnetometers, magnetic resonance imaging (MRI), transformers and inductors.
• Magnetic separation and filtration: Magnetic permeability affects the efficiency and selectivity of processes that separate or filter materials based on their magnetic properties. Examples are low-intensity magnetic separation (LIMS), high-intensity magnetic separation (HIMS) and high-gradient magnetic separation (HGMS).
• Magnetic levitation and propulsion: Magnetic permeability affects the stability and speed of systems that use magnetic forces to levitate or propel objects. Examples are maglev trains, linear motors and magnetic bearings.
• Magnetic shielding and protection: Magnetic permeability affects the ability of materials to block or redirect magnetic fields from unwanted sources. Examples are mu-metal, ferrites and superconductors.

Summary

Magnetic permeability is a property of materials that describes how easily they can be magnetized by an external magnetic field. It also determines how much magnetic flux they can support within themselves. It is defined as the ratio of the magnetic flux density to the magnetizing force in a material or medium.

The unit of magnetic permeability depends on the system of units used: henry per meter (H/m), equal to Newton per square ampere (N/A^2), within the SI system; the CGS system treats the quantity as dimensionless.

The permeability of free space serves as the reference for other materials or media. Denoted μ0​, this constant takes a value of approximately 4π×10−7 H/m.

Relative permeability (μr​) compares the permeability of any material or medium with that of free space as a dimensionless ratio, equal to μ/μ0​.

Magnetic susceptibility is another dimensionless quantity that indicates how much a material or medium responds to an external magnetic field by becoming magnetized. It is denoted by χm​ and equals μr​−1.

Materials fall into three main classes by relative permeability and susceptibility: diamagnetic, paramagnetic and ferromagnetic. Diamagnets weakly oppose an external field and expel most of the flux. Paramagnets weakly attract it and add flux. Ferromagnets respond powerfully and carry enormous flux.

The permeability of a material or medium depends on several factors, such as humidity, temperature, position in the medium and frequency of the applied field.

Complex permeability is a concept that accounts for high-frequency effects on magnetic fields. It represents the phase lag between B and H due to losses such as eddy currents, hysteresis, etc.

Recording and storage devices, sensors and transducers, separation and filtration systems, levitation and propulsion, magnetic shielding: each depends on choosing materials whose permeability suits the job.