Derive the relation between magnetic field and magnetisatiin for a material placed in a magnetizing field

To derive the relationship between the magnetic field and magnetization for a material placed in a magnetizing field, we need to delve into some fundamental concepts of magnetism. This relationship is crucial in understanding how materials respond to external magnetic fields and how they become magnetized.

Understanding Key Concepts

First, let's clarify some terms:

• Magnetic Field (B): This is the field around a magnet where magnetic forces can be detected. It is measured in teslas (T).
• Magnetization (M): This refers to the magnetic moment per unit volume of a material, indicating how much the material is magnetized. It is measured in amperes per meter (A/m).
• Magnetizing Field (H): This is the external magnetic field applied to a material, also measured in amperes per meter (A/m).

The Relationship Between B, H, and M

The relationship between the magnetic field (B), the magnetizing field (H), and the magnetization (M) can be expressed through the following equation:

B = μ₀(H + M)

Here, μ₀ is the permeability of free space, which is a constant that describes how a magnetic field interacts with a vacuum. The value of μ₀ is approximately 4π × 10⁻⁷ T·m/A.

Breaking Down the Equation

Let’s analyze this equation step by step:

• Magnetizing Field (H): This is the external field applied to the material. It creates a magnetic influence that can align the magnetic dipoles within the material.
• Magnetization (M): When the external field is applied, the material's internal magnetic dipoles tend to align with the field, resulting in an increase in magnetization. This is particularly significant in ferromagnetic materials, where the alignment can be substantial.
• Magnetic Field (B): The total magnetic field inside the material is a combination of the external magnetizing field and the contribution from the material's own magnetization.

Example of Magnetization

Consider a ferromagnetic material like iron. When you place it in a strong magnetic field, the domains within the iron, which are small regions where the magnetic moments are aligned, start to align with the external field. As a result, the magnetization (M) increases significantly. This leads to a much stronger magnetic field (B) inside the material than the applied magnetizing field (H) alone would suggest.

Understanding Permeability

The relationship can also be expressed in terms of permeability:

B = μH

Where μ is the permeability of the material, which can be expressed as:

μ = μ₀(1 + χ)

Here, χ is the magnetic susceptibility of the material, a dimensionless quantity that indicates how susceptible a material is to becoming magnetized in an external magnetic field. The higher the susceptibility, the more the material will become magnetized.

Conclusion

In summary, the relationship between the magnetic field (B), magnetizing field (H), and magnetization (M) is foundational in magnetism. The equation B = μ₀(H + M) encapsulates how external fields interact with materials, leading to magnetization. Understanding this relationship is essential for applications in electromagnetism, materials science, and various technologies that rely on magnetic properties.