MAGNETIC PERMEABILITY: CONSTANT AND TABLE - PHYSICAL

The magnetic permeability is the physical quantity of the property of matter to generate its own magnetic field, when it is permeated by an outer magnetic field.

Both fields: the external and the own, are superimposed giving a resulting field. A the, independent of the material, external field is called magnetic field strength H, while overlapping the outer field plus the material is induced in the magnetic induction B.

Figure 1. Solenoid with a μ magnetic permeability material core. Source: Wikimedia Commons.

When it comes to homogeneous and isotropic materials, the H and B fields are proportional. And the constant of proportionality (scalar and positive) is the magnetic permeability, denoted by the Greek letter μ:

B = μ H

In the SI International System the magnetic induction B is measured in Tesla (T), while the magnetic field intensity H is measured in Ampere over meter (A / m).

Since μ must guarantee dimensional homogeneity in the equation, the unit of μ in the SI system is:

= (Tesla ⋅ meter) / Ampere = (T ⋅ m) / A

Magnetic permeability of vacuum

Let's see how magnetic fields, whose absolute values we denote by B and H, are produced in a coil or solenoid. From there, the concept of magnetic permeability of the vacuum will be introduced.

The solenoid consists of a spirally wound conductor. Each turn of the spiral is called a turn. If current is passed through the solenoid i, then we have an electromagnet that produces a magnetic field B.

Furthermore, the value of the magnetic induction B is greater, as the current i is increased. And also when the density of turns n increases (number N of turns between the length d of the solenoid).

The other factor that affects the value of the magnetic field produced by a solenoid is the magnetic permeability μ of the material that is inside it. Finally, the magnitude of said field is:

B = μ. i.n = μ. i. (N / a)

As stated in the previous section, the magnetic field intensity H is:

H = i. (N / d)

This field of magnitude H, which only depends on the circulating current and the density of turns of the solenoid, "permeates" the material of magnetic permeability μ, causing it to become magnetized.

Then a total field of magnitude B is produced, which does depend on the material that is inside the solenoid.

Solenoid in vacuum

Similarly, if the material inside the solenoid is vacuum, then the H field "permeates" the vacuum producing a resultant field B. The quotient between the B field in vacuum and the H produced by the solenoid defines the permeability of the vacuum., whose value is:

μ o = 4π x 10 ^-7 (T⋅m) / A

It turns out that the previous value was an exact definition until May 20, 2019. As of that date, a revision of the International System was made, which leads to μ or being measured experimentally.

However, measurements made so far indicate that this value is extremely accurate.

Magnetic permeability table

Materials have a characteristic magnetic permeability. Now, it is possible to find the magnetic permeability with other units. For example, let's take the unit of inductance, which is Henry (H):

1H = 1 (T * m ²) / A.

Comparing this unit with the one that was given at the beginning, it is seen that there is a similarity, although the difference is the square meter that Henry owns. For this reason, magnetic permeability is considered an inductance per unit length:

= H / m.

The magnetic permeability μ is closely related to another physical property of materials, called the magnetic susceptibility χ, which is defined as:

μ = μ or (1 + χ)

In the previous expression μ o, is the magnetic permeability of the vacuum.

The magnetic susceptibility χ is the proportionality between the external field H and the magnetization of the material M.

Relative permeability

It is very common to express the magnetic permeability in relation to the permeability of the vacuum. It is known as relative permeability and it is nothing more than the ratio of the permeability of the material to that of the vacuum.

According to this definition, relative permeability is unitless. But it is a useful concept for classifying materials.

For example, materials are ferromagnetic as long as their relative permeability is much greater than unity.

In the same way, paramagnetic substances have relative permeability just above 1.

And finally, diamagnetic materials have relative permeabilities just below unity. The reason is that they become magnetized in such a way that they produce a field that opposes the external magnetic field.

It is worth mentioning that ferromagnetic materials present a phenomenon known as "hysteresis", in which they keep memory of the fields previously applied. By virtue of this characteristic they can form a permanent magnet.

Figure 2. Ferrite magnetic memories. Source: Wikimedia Commons

Due to the magnetic memory of ferromagnetic materials, the memories of early digital computers were small ferrite toroids traversed by conductors. There they saved, extracted or erased the content (1 or 0) of the memory.

Materials and their permeability

Here are some materials, with their magnetic permeability in H / m and their relative permeability in parentheses:

Iron: 6.3 x 10 ^-3 (5000)

Cobalt-iron: 2.3 x 10 ^-2 (18000)

Nickel-iron: 1.25 x 10 ^-1 (100000)

Manganese-zinc: 2.5 x 10 ^-2 (20000)

Carbon Steel: 1.26 x 10 ^-4 (100)

Neodymium magnet: 1.32 x 10 ^-5 (1.05)

Platinum: 1.26 x 10 ^-6 1.0003

Aluminum: 1.26 x 10 ^-6 1.00002

Air 1.256 x 10 ^-6 (1.0000004)

Teflon 1.256 x 10 ^-6 (1.00001)

Dry wood 1.256 x 10 ^-6 (1.0000003)

Copper 1.27 x10 ^-6 (0.999)

Pure water 1.26 x 10 ^-6 (0.999992)

Superconductor: 0 (0)

Table analysis

Looking at the values in this table, it can be seen that there is a first group with magnetic permeability relative to that of vacuum with high values. These are ferromagnetic materials, very suitable for the manufacture of electromagnets for the production of large magnetic fields.

Figure 3. Curves B vs. H for ferromagnetic, paramagnetic and diamagnetic materials. Source: Wikimedia Commons.

Then we have a second group of materials, with relative magnetic permeability just above 1. These are the paramagnetic materials.

Then you can see materials with relative magnetic permeability just below unity. These are diamagnetic materials such as pure water and copper.

Finally we have a superconductor. Superconductors have zero magnetic permeability because it completely excludes the magnetic field inside them. Superconductors are useless to be used in the core of an electromagnet.

However, superconducting electromagnets are often built, but the superconductor is used in the winding to establish very high electrical currents that produce high magnetic fields.

References

Dialnet. Simple experiments to find magnetic permeability. Recovered from: dialnet.unirioja.es
Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 6. Electromagnetism. Edited by Douglas Figueroa (USB). 215-221.
Giancoli, D. 2006. Physics: Principles with Applications. 6th Ed Prentice Hall. 560-562.
Kirkpatrick, L. 2007. Physics: A Look at the World. 6th abridged edition. Cengage Learning. 233.
Youtube. Magnetism 5 - Permeability. Recovered from: youtube.com
Wikipedia. Magnetic field. Recovered from: es.wikipedia.com
Wikipedia. Permeability (Electromagnetism). Recovered from: en.wikipedia.com

MAGNETIC PERMEABILITY: CONSTANT AND TABLE - PHYSICAL - 2024