Magnetization and Magnetic Intensity

Magnetisation ($\vec{M}$), Magnetic Intensity ($\vec{H}$), Magnetic Susceptibility ($\chi_m$), and Permeability ($ \mu $)

When a material is placed in an external magnetic field, it becomes magnetised to some extent. The response of a material to an external magnetic field is described by concepts like Magnetization, Magnetic Intensity, Magnetic Susceptibility, and Permeability. These quantities help us understand the magnetic properties of different materials and how they modify the magnetic field.

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Magnetisation ($\vec{M}$)

When a material is subjected to a magnetic field, the microscopic magnetic dipoles within the material (due to electron motion and spin) tend to align with the field. This alignment results in the material itself producing a net magnetic dipole moment.

Magnetisation ($\vec{M}$) is defined as the net magnetic dipole moment per unit volume of the material.

$ \vec{M} = \frac{\text{Net magnetic dipole moment}}{\text{Volume}} $

Magnetisation is a vector quantity. Its direction is the direction of the net magnetic dipole moment in the material. Its SI unit is Ampere per meter ($A/m$).

The magnetisation $\vec{M}$ represents the extent to which the material has become magnetised when placed in a magnetic field.

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Magnetic Intensity ($\vec{H}$)

The magnetic field $\vec{B}$ inside a material placed in an external magnetic field is a result of two contributions:

1. The magnetic field produced by the external sources (like current-carrying wires or external magnets) in the absence of the material.
2. The additional magnetic field produced by the magnetisation of the material itself.

To distinguish the field produced by external sources from the total field inside the material, a new vector field called Magnetic Intensity ($\vec{H}$) is introduced. Magnetic Intensity is a measure of the ability of an external magnetic field (or current) to magnetise a material or create a magnetic field. It is defined such that it depends only on the external current distribution, not on the magnetic properties of the medium.

The total magnetic field $\vec{B}$ inside a material is related to the magnetic intensity $\vec{H}$ and the magnetisation $\vec{M}$ by the relation:

$ \vec{B} = \mu_0 (\vec{H} + \vec{M}) $

Where $\mu_0$ is the permeability of free space.

From this equation, $\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}$.

The SI unit of magnetic intensity ($\vec{H}$) is also Ampere per meter ($A/m$), the same as Magnetisation.

Consider a solenoid with $n$ turns per unit length carrying current $I$. The magnetic field inside it in vacuum is $B_0 = \mu_0 n I$. If a material core is inserted, the field inside becomes $B$. The external source (the current in the solenoid) creates a magnetic intensity $H$. For a long solenoid, the magnetic intensity inside is related to the current per unit length: $H = nI$. Then $\vec{B} = \mu_0 (\vec{H} + \vec{M})$.

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Magnetic Susceptibility ($\chi_m$)

In many materials, especially for relatively weak magnetic fields, the magnetisation $\vec{M}$ induced in the material is directly proportional to the magnetic intensity $\vec{H}$ applied to it.

$ \vec{M} = \chi_m \vec{H} $

The constant of proportionality $\chi_m$ (chi-em) is called the magnetic susceptibility of the material.

Magnetic susceptibility is a dimensionless scalar quantity that indicates how easily a material can be magnetised by an external magnetic field.

• For materials that are easily magnetised (like ferromagnetic materials), $\chi_m$ is large and positive.
• For materials that are weakly magnetised in the direction of the field (paramagnetic materials), $\chi_m$ is small and positive.
• For materials that are weakly magnetised in the opposite direction of the field (diamagnetic materials), $\chi_m$ is small and negative.

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Permeability ($\mu$)

The magnetic field $\vec{B}$ inside a material is often related directly to the magnetic intensity $\vec{H}$ through the concept of permeability ($\mu$).

Substitute the relation $\vec{M} = \chi_m \vec{H}$ into the equation $\vec{B} = \mu_0 (\vec{H} + \vec{M})$:

$ \vec{B} = \mu_0 (\vec{H} + \chi_m \vec{H}) $

$ \vec{B} = \mu_0 (1 + \chi_m) \vec{H} $

The term $\mu_0 (1 + \chi_m)$ is defined as the permeability ($\mu$) of the material:

$ \mu = \mu_0 (1 + \chi_m) $

The permeability $\mu$ is a measure of how well a material supports the formation of a magnetic field within itself. It indicates how "permeable" the material is to magnetic field lines.

Using the permeability, the relation between $\vec{B}$ and $\vec{H}$ inside the material becomes:

$ \vec{B} = \mu \vec{H} $

The SI unit of permeability is the same as $\mu_0$, which is Tesla meter per Ampere ($T \cdot m/A$) or Henry per meter ($H/m$).

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Relative Permeability ($\mu_r$)

It is often convenient to express the permeability of a material relative to the permeability of free space. The relative permeability ($\mu_r$) is defined as the ratio of the permeability of the material to the permeability of free space:

$ \mu_r = \frac{\mu}{\mu_0} $

From the definition of $\mu = \mu_0 (1 + \chi_m)$, we can relate relative permeability and magnetic susceptibility:

$ \mu_r = \frac{\mu_0 (1 + \chi_m)}{\mu_0} = 1 + \chi_m $

$ \chi_m = \mu_r - 1 $

Relative permeability ($\mu_r$) is a dimensionless scalar quantity, just like magnetic susceptibility ($\chi_m$).

• For diamagnetic materials, $\chi_m$ is small and negative (e.g., $-10^{-5}$ to $-10^{-6}$), so $\mu_r < 1$. Diamagnetic materials slightly weaken the magnetic field inside them.
• For paramagnetic materials, $\chi_m$ is small and positive (e.g., $10^{-3}$ to $10^{-5}$), so $\mu_r > 1$. Paramagnetic materials slightly strengthen the magnetic field inside them.
• For ferromagnetic materials, $\chi_m$ is large and positive (e.g., hundreds or thousands), so $\mu_r \gg 1$. Ferromagnetic materials greatly strengthen the magnetic field inside them.
• For vacuum, there is no material to be magnetised, so $\vec{M}=0$. Thus $\vec{B} = \mu_0 \vec{H}$. This implies $\chi_m = 0$ and $\mu_r = 1$ for vacuum.

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Summary of Relationships

The key relationships between these magnetic vectors and properties are:

$ \vec{B} = \mu_0 (\vec{H} + \vec{M}) $ (Definition relating $\vec{B}, \vec{H}, \vec{M}$)

$ \vec{M} = \chi_m \vec{H} $ (Relation between Magnetisation and Magnetic Intensity for many materials)

$ \vec{B} = \mu \vec{H} $ (Relation between $\vec{B}$ and $\vec{H}$ using permeability)

$ \mu = \mu_0 (1 + \chi_m) $ (Relation between permeability and susceptibility)

$ \mu_r = \frac{\mu}{\mu_0} = 1 + \chi_m $ (Relation between relative permeability and susceptibility)

These concepts are essential for understanding how different materials behave in magnetic fields and are used in designing devices like electromagnets, transformers, and magnetic storage media.

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