Both fields, the external and the self, overlap, giving a resulting field.Al, regardless of the material, external field is called the magnetic field strength H , while the superposition of the external field plus the material is induced in magnetic induction B .
Figure 1. Solenoid with a core of μ magnetic permeability material. Source: Wikimedia Commons.
When dealing with homogeneous and isotropic materials, the H and B fields are proportional. And the proportionality constant (scalar and positive) is the magnetic permeability, denoted by the Greek letter μ:
B = μ H
In the International System SI, magnetic induction B is measured in Tesla (T), while magnetic field strength H is measured in Ampere on the meter (A / m).
As μ must guarantee dimensional homogeneity in the equation, the unit of μ in the SI system is:
[μ] = (Tesla) / Ampere = (T ⋅ m) / A
Vacuum magnetic permeability
Let’s see how magnetic fields are produced, whose absolute values we denote by B and H , in a coil or solenoid. From there, the concept of magnetic vacuum permeability will be introduced.
The solenoid consists of a coiled conductor. Each turn of the spiral is called a spiral . If current i is passed by the solenoid then it has an electromagnet that produces a magnetic field B .
Furthermore, the value of magnetic induction B is greater as current i increases. And also when the density of turns n increases ( N number 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 μ magnetic permeability of the internal material. Finally, the magnitude of this field is:
B = µ. i. n = µ. i. (AT)
As stated in the previous section, the strength of the magnetic field H is:
H = i. (AT)
This field of magnitude H, which depends only on the circulating current and the solenoid winding density, “permeates” the material with magnetic permeability μ , causing it to magnetize.
Then a total field of magnitude B is produced , which depends on the material inside the solenoid .
Likewise, if the material inside the solenoid is a vacuum, the H field “permeates” the vacuum producing a B field. The ratio of the B field in the vacuum to 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. From that date, a revision of the International System was made, which leads to μ or can be experimentally measured.
However, measurements taken so far indicate that this value is extremely accurate.
Magnetic permeability table
The materials have a characteristic magnetic permeability. It is now possible to find magnetic permeability with other units. For example, let’s take the unit of inductance, which is the henry (H):
1H = 1 (t ⋅ m 2 ) / A .
Comparing this unit with the one that occurred at the beginning, one can see a similarity, although the difference is the square meter that Henry has. For this reason, magnetic permeability is considered an inductance per unit length:
[µ] = H / m .
The magnetic permeability μ is closely related to another physical property of materials, called magnetic susceptibility χ , which is defined as:
μ = μ o (1 + χ)
In the above expression μ o, is the magnetic permeability of vacuum .
The magnetic susceptibility χ is proportionality between the external field H and the magnetization of the material M .
It is very common to express magnetic permeability in relation to vacuum permeability. It is known as relative permeability and is just the ratio of the permeability of the material to that of a vacuum.
According to this definition, relative permeability has no units. But it’s a useful concept for classifying materials.
For example, materials are ferromagnetic as long as their relative permeability is much greater than unity.
Likewise, paramagnetic substances have a relative permeability just above 1.
And finally, diamagnetic materials have relative permeabilities just below unity. The reason is that they magnetize in such a way that they produce a field that opposes the external magnetic field.
It should be mentioned that ferromagnetic materials have a phenomenon known as “hysteresis”, in which they keep the memory of previously applied fields. Under this feature, they can form a permanent magnet.
Due to the magnetic memory of ferromagnetic materials, the memories of the original digital computers were small ferrite toroids pierced by conductors. There they stored, extracted, or deleted the contents (1 or 0) from memory.
Materials and their permeability
Here are some materials, with their magnetic permeability in H / m and in parentheses their relative permeability:
Iron: 6.3 x 10 -3 (5000)
Cobalt iron : 2.3 x 10 -2 (18000)
Nickel-iron: 1.25 x 10 -1 (100,000)
Zinc-manganese: 2.5 x 10 -2 (2000)
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 (1.001001)
Dry wood 1,256 x 10 -6 ( 1.0000003)
Covers 1.27 x10 -6 (0.999)
Pure water 1.26 x 10 -6 (0.999992)
Superconductor: 0 (0)
Observing the values in this table, it can be seen that there is a first group with magnetic permeability in relation to vacuum with high values. They are ferromagnetic materials, very suitable for the manufacture of electromagnets for the production of large magnetic fields.
Figure 3. B vs. H for ferromagnetic, paramagnetic and diamagnetic materials. Source: Wikimedia Commons.
So we have a second group of materials, with relative magnetic permeability just above 1. These are paramagnetic materials.
So materials with relative magnetic permeability can be seen just below the unit. These are diamagnetic materials like pure water and copper.
Finally, we have a superconductor. Superconductors have zero magnetic permeability because they completely exclude the internal magnetic field. Superconductors are not intended to be used in the core of an electromagnet.
However, superconducting electromagnets are generally constructed, but the superconductor is used in the winding to establish very high electrical currents that produce high magnetic fields.