Magnetisation and Magnetic Intensity

Magnetisation ($\mathbf{M}$)

In any bulk material, the magnetic moments of individual atoms (due to circulating electrons) add up as vectors. The net magnetic moment per unit volume of the sample is called its magnetisation, denoted by $\mathbf{M}$.

• $\mathbf{m}_{\text{net}}$ is the vector sum of all atomic magnetic moments in the sample.
• $V$ is the volume of the sample.
• $\mathbf{M}$ is a vector quantity. Its SI unit is $\text{A m}^{-1}$ (ampere per metre), and its dimensions are $[\text{L}^{-1}\text{A}]$.

Magnetic Field Inside a Solenoid with a Core

Consider a long solenoid with $n$ turns per unit length carrying a current $I$. In vacuum (or air), the magnetic field inside is:

• $\mu_0 = 4\pi \times 10^{-7} \, \text{N/A}^2$ is the permeability of free space.

If the solenoid is filled with a magnetic material (a core), the total magnetic field $\mathbf{B}$ inside becomes the sum of two contributions:

• $\mathbf{B}_0$ is the field due to the solenoid current alone.
• $\mathbf{B}_m$ is the additional field contributed by the magnetised core.

It is found experimentally that $\mathbf{B}_m$ is proportional to the magnetisation $\mathbf{M}$ of the material:

Thus, the total field inside the material is:

Magnetic Intensity ($\mathbf{H}$)

To separate the effect of the external current from the material's response, we define a new vector field $\mathbf{H}$, called magnetic intensity (or magnetic field strength). It is defined by:

• $\mathbf{H}$ has the same dimensions and SI unit as $\mathbf{M}$: $\text{A m}^{-1}$.
• $\mathbf{H}$ represents the contribution to the total field from external sources (like the current in the solenoid). For a solenoid, $H = nI$, independent of the core material.

Magnetic Susceptibility ($\chi$)

The response of a material to an external magnetic field is quantified by its magnetic susceptibility, $\chi$ (chi). It is defined as the ratio of magnetisation to magnetic intensity:

• $\chi$ is a dimensionless quantity.
• It tells us how easily a material gets magnetised.
  • Paramagnetic materials: $\chi$ is small and positive ($\mathbf{M}$ is parallel to $\mathbf{H}$).
  • Diamagnetic materials: $\chi$ is small and negative ($\mathbf{M}$ is opposite to $\mathbf{H}$).

Relative Permeability ($\mu_r$) and Absolute Permeability ($\mu$)

Using $\mathbf{M} = \chi \mathbf{H}$ in $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})$, we get:

$\mathbf{B} = \mu_0 (1 + \chi) \mathbf{H}$ …