# Understanding Magnetic Permeability

Contents

Magnetic permeability is a fundamental 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. In this article, we will explain the definition, formula, units, types, and factors affecting magnetic permeability.

## What is Magnetic Permeability?

Magnetic permeability is defined as the ratio of the magnetic flux density (B) to the magnetizing force (H) in a material or medium. Mathematically, it is expressed as:

Magnetic flux density (B) is a measure of the strength and direction of the magnetic field within a material. It is proportional to the number of magnetic field lines per unit area. The unit of B is the tesla (T) or Weber per square meter (Wb/m^2).

Magnetizing force (H) is a measure of the intensity and direction of the external magnetic field that causes magnetization in a material. It is proportional to the electric current flowing in a coil of wire that produces the magnetic field. The unit of H is ampere per meter (A/m).

Magnetic permeability indicates how easily a material can be magnetized by an external magnetic field. A high permeability means that the material can support a large amount of magnetic flux within itself for a given magnetizing force. A low permeability means that the material resists being magnetized and expels most of the magnetic flux.

## Unit of Magnetic Permeability

The unit of magnetic permeability depends on the system of units used. In the SI system, the unit of magnetic permeability is Henry per meter (H/m) or Newton per square ampere (N/A^2). In the CGS system, the unit of magnetic permeability is dimensionless and equal to 1.

## Permeability of Free Space

Free space, or vacuum, is a reference medium that has no matter and no magnetization. The permeability of free space is denoted by μ0​ and has a constant value of:

This value was defined so that the SI unit of electric current, the ampere, could be consistent with the practical unit. However, with the redefinition of the ampere in 2019, μ0​ is no longer exactly equal to 4π×10−7 H/m and must be determined experimentally. The relative difference between the old and new values is very small (5.5×10−10).

The permeability of free space is also known as the magnetic constant or the vacuum permeability. It is one of the fundamental physical constants that appears in many equations of electromagnetism, such as 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 the permeability of different materials or media, it is useful to define a dimensionless quantity called relative permeability (μr​). It is the ratio of the permeability of a material or medium to that of free space:

Relative permeability indicates how much more or less permeable a material or medium is than free space. A relative permeability greater than 1 means that the material or medium can support more magnetic flux than free space for a given magnetizing force.

A relative permeability of less than 1 means that the material or medium can support less magnetic flux than free space for a given magnetizing force.

Another dimensionless quantity related to relative permeability is magnetic susceptibility (χm​). It is defined as:

Magnetic susceptibility indicates how much a material or medium responds to an external magnetic field by becoming magnetized. A positive susceptibility means that the material or medium aligns with the external field and becomes weakly or strongly magnetized. A negative susceptibility means that the material or medium opposes the external field and becomes weakly demagnetized.

## 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 less than 1 (μr​<1) and a negative susceptibility (χm​<0). This means that they have a weak opposition to an external magnetic field and tend to expel most of the magnetic flux from their interior. Examples of diamagnetic materials are bismuth, copper, water, and air.

When diamagnetic material is placed in an external magnetic field, it induces a small opposite magnetic field within itself that cancels out some of the external fields. This reduces the net magnetic flux density inside the material. As a result, diamagnetic materials are repelled by magnets.

### Paramagnetic Materials

Paramagnetic materials have a relative permeability slightly greater than 1 (μr​>1) and a positive susceptibility (χm​>0). This means that they have a weak attraction to an external magnetic field and tend to increase the magnetic flux within their interior. Examples of paramagnetic materials are platinum, aluminum, oxygen, and iron oxide.

When a paramagnetic material is placed in an external magnetic field, it aligns some of its atomic or molecular magnets with the external field and becomes weakly magnetized in the same direction. This increases the net magnetic flux density inside the material. As a result, paramagnetic materials are attracted by magnets.

### Ferromagnetic Materials

Ferromagnetic materials have a relative permeability much greater than 1 (μr​>>1) and a very high positive susceptibility (χm​>>0). This means that they have a strong attraction to an external magnetic field and tend to support a large amount of magnetic flux within their interior. Examples of ferromagnetic materials are iron, nickel, cobalt, and steel.

When a ferromagnetic material is placed in an external magnetic field, it aligns most of its atomic or molecular magnets with the external field and becomes strongly magnetized in the same direction. This greatly increases the net magnetic flux density inside the material. As a result, ferromagnetic 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: The presence of water molecules can affect the alignment of the atomic or molecular magnets and change the permeability.
• Temperature: The thermal motion of the atoms or molecules can disrupt the alignment of the magnets and reduce the permeability.
• Position in the medium: The permeability may vary depending on where in the medium the measurement is taken due to variations in density, composition, structure, etc.
• Frequency of the applied field: The time-varying nature of an alternating current (AC) field can cause eddy currents, skin effects, losses, and other phenomena that affect 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

Magnetic permeability is an important parameter in many applications that involve magnetic fields and materials. 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. In the SI system, it is henry per meter (H/m) or Newton per square ampere (N/A^2). In the CGS system, it is dimensionless and equal to 1.

The permeability of free space is a constant value that is used as a reference for other materials or media. It is denoted by μ0​ and has a value of 4π×10−7 H/m.

Relative permeability is a dimensionless quantity that compares the permeability of a material or medium to that of free space. It is denoted by μr​ and is 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 is equal to (μr​−1)/μr​.

Materials can be classified into three main types based on their relative permeability and magnetic susceptibility: diamagnetic, paramagnetic, and ferromagnetic. Diamagnetic materials have a weak opposition to an external field and expel most of the flux. Paramagnetic materials have a weak attraction to an external field and increase the flux. Ferromagnetic materials have a strong attraction to an external field and support a large amount of 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.

Magnetic permeability is an important parameter in many applications that involve magnetic fields and materials, such as magnetic recording and storage devices, magnetic sensors and transducers, magnetic separation and filtration, magnetic levitation and propulsion, and magnetic shielding and protection.

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