Introduction

The study of magnetism forms a critical field in the field of physics and engineering. The earthly objects and immediate materials have various responses to the magnetic fields, and the knowledge of these phenomena will help in the imagining of magnetic tools, transformative organs, electromechanical actuators, and even biomedical machinery such as magnetic resonance imaging (MRI).

The terrestrial environment is infested with a wide array of elements, metallic alloys, and chemical compounds. Some materials are strongly magnetic responsive and some attenuated or have weak magnetic responsiveness. To systematize the classification of these responses and their quantification, physicists use terms like magnetization, magnetic intensity, susceptibility, and permeability. These parameters allow a strict evaluation of the behavior of materials under the influence of external magnetic fields.

Magnetization (M)

1. Definition

Every circulating electron in an atom contributes a small magnetic moment. In bulk matter, these moments can add vectorially. If the sum is non-zero, the material exhibits a net magnetic effect.

To measure this effect, we define magnetization (M) of a sample as the net magnetic moment per unit volume:

M=m_net/V

where:

m_net = total magnetic moment of the sample
V = volume of the material

2. Nature of Magnetization

Magnetization is a vector quantity. Its SI unit is A·m⁻¹, and its dimensions are L⁻¹A. It represents how strongly a material becomes magnetized when subjected to an external magnetic field.

For example, a piece of iron placed in a magnetic field develops magnetization because the atomic magnetic moments within it tend to align with the applied field.

As an example, an iron sample in a magnetic field becomes magnetized as the magnetic moments of the atoms in the sample prefer to follow the field.

Magnetic Field inside a Solenoid

1. Solenoid without Magnetic Material

A solenoid is a helical coil that is designed such that it produces a magnetic field in the inside that, under the influence of an electric current, can be considered as being spatially homogeneous. A good example is a long solenoid that is a single turn long and it carries a current identified by I; thus, the field in that solenoid is:

B₀=μ₀nI

where μ₀ is the permeability of free space (4π×10⁻⁷ Hm⁻¹).

This B₀ is entirely due to the existing current and geometry of the solenoid (whatever material is inserted therein) which occurs regardless of any material inserted therein.

2. Solenoid with a Magnetic Material Core

The magnetic field inside the solenoid is then boosted when a material with non-zero magnetization is placed inside the solenoid. The modified field may be indicated as:

B=B₀+B_m

where:

B₀ = field due to solenoid current
B_m = additional field due to magnetization of the material core

This contribution B_m, is proportional to the magnetization M of the material:

B_m=μ₀M

Thus, the total field inside the solenoid becomes:

B=μ₀(nI+M)

Magnetic Intensity (H)

1. Definition of H

To distinguish between the contribution of the external source (the solenoid current) and the material itself, physicists define a new vector quantity called magnetic intensity (H). It is defined as:

H=B/μ₀−M

This can be rewritten as:

H=nI

The magnetic field, H, is only dependent on the current and the geometrical characteristics of the solenoid, but not on the properties of the materials.

2. Expression for Total Magnetic Field

The total magnetic field inside a material can now be expressed as:

B=μ₀(H+M)

Thus, the magnetic field is partitioned into two contributions:

External factors (represented by H) such as the current in a solenoid.
Material's own response (represented by M).

Magnetic Susceptibility (χ)

1. Definition

That is the proportionality between magnetization and the magnetic intensity is given by:

M=χH

In this term, χ is a non-dimensional constant often known as magnetic susceptibility; it is used to measure how well a substance responds to an applied magnetic field.

2. Types of Materials Based on χ

Paramagnetic Materials: In paramagnetic materials, the magnetic susceptibility χ is not only small but it is also positive, thus resulting in a magnetization that is parallel to the applied external field. Examples of such representatives are aluminum and platinum.
Diamagnetic Materials: In diamagnetic materials, the magnetic susceptibility χ is small and negative, resulting in antiparallel magnetization and antiparallel to the applied field. Examples of such are bismuth, copper, and gold.

3. Magnetic Susceptibility Values (Table 5.2)

Different materials have a broad spectrum of susceptibility at temperatures of 300K.

Diamagnetic substances (negative χ):

Bismuth	−1.66 × 10⁻⁵
Copper	−9.8 × 10⁻⁶
Gold	−3.6 × 10⁻⁵
Diamond	−2.2 × 10⁻⁵

Paramagnetic substances (positive χ):

Aluminium	2.3 × 10⁻⁵
Calcium	1.9 × 10⁻⁵
Chromium	2.7 × 10⁻⁴
Platinum	2.9 × 10⁻⁴

This table defines the wide difference between weak and relatively strong diamagnetism and paramagnetism.

Magnetic Permeability (μ)

1. Relation between χ, μ_r, and μ

Another important parameter is the magnetic permeability parameter expressed as μ, which is a measure of the ease through which the magnetic field lines penetrate a material.

The relative permeability (μ_r) of a substance is given by:

μ_r=1+χ

The absolute permeability is:

μ=μ₀μ_r=μ₀(1+χ)

2. Significance of μ

Magnetic permeability operates similarly to the dielectric constant in the realm of electrostatics. Materials with higher permeability will condense the magnetic flux density, thus making them especially useful in applications involving electromagnets and transformers.

Example Problem

Problem Statement:

The solenoid in question has the winding density of 1000 turns per meter, the current of 2 amperes (A) added to it, and the core with the relative permeability of μ_r = 400. Calculate:

(a) Magnetic intensity H

(b) Magnetic field B

(c) Magnetization M

(d) Magnetizing current I_m

Step-by-Step Solution:

(a) Magnetic Intensity (H):

H=nI=1000×2=2.0×10³ A/m

(b) Magnetic Field (B):

B=μ_rμ₀H

B=400×4π×10⁻⁷×2×10³

B=1.0 T

(c) Magnetization (M):

M=(μ_r−1)H=399×H

M≈8.0×10⁵ A/m

(d) Magnetizing Current (I_m):

The additional current required is:

I_m=B/μ₀n−I

Substituting values:

I_m≈794 A

Therefore, a more magnetizing current of 794 A on the solenoid without the presence of the core is necessary to generate the same magnetic field.

Interrelationship between χ, μ, and μ_r

Of the three parameters that include the susceptibility (χ), relative permeability (μ_r), and absolute permeability (μ), only one of these is the independent variable.

If χ is known, we can compute μ_r and μ.
If μ_r is given, χ = μ_r − 1.
If μ is given, dividing by μ₀ gives μ_r, and χ follows.

This interdependence makes these parameters versatile in both experimental and theoretical studies of magnetism.

Applications of Magnetization and Magnetic Intensity

Electromagnets: Highly magnetic permeable material concentrates the magnetic flux, hence increasing its use in the handling of heavy loads.
Transformers: The working efficiency of transformer items depends upon the magnetic properties of the core material.
Magnetic Storage: Hard disk drives use digital information written in a magnetically oriented medium.
Medical Imaging: The magnetic resonance imaging (MRI) devices are based on the interaction of the nuclear spin populations in the high-strength magnetic field.
Material Classification: Magnetic susceptibility ( χ ) and permeability ( μ ) can be used to classify materials as paramagnetic, diamagnetic or ferromagnetic.

Conclusion

Magnetization and magnetic intensity are powerful analytical tools of quantification and categorization of magnetic materials. Dividing the external field component (H) and the inherent material response (M) allows to gain a more pronounced understanding and proper prediction of the behavior of various substances.

Magnetic susceptibility (χ) and permeability (μ) are then introduced, which allows materials to be classically categorized into diamagnetic, paramagnetic, and strongly ferromagnetic. Table 5.2. shows that these parameters are highly varied among various elements.

The practical applications of these quantities, in the processes of a real-world application, between the example of electromagnets and the medical applications, are explained through theoretical derivations and illustrative examples, including the example. The study of these concepts is hence essential to scholars working in the fields of electromagnetism, condensed-matter physics or electrical engineering.

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