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Mathematics for Physics

69 Magnetic Materials

A substance whose atomic magnetic dipoles respond to an external magnetic field that is used to modify the net field inside matter, where a magnetic dipole is a localized circulating current with a magnetic moment.

Magnetization. Magnetization is the magnetic dipole moment per unit volume. This principle is used to describe how the material responds to an applied field.

The magnetization of a linear medium is

\[ \mathbf{M} = \chi_{m}\mathbf{H} \]

where

  • \(\mathbf{M}\) is the magnetization.
  • \(\chi_{m}\) is the magnetic susceptibility.
  • \(\mathbf{H}\) is the auxiliary magnetic field.

The auxiliary field \(\mathbf{H}\). The auxiliary field \(\mathbf{H}\) subtracts the magnetization from \(\mathbf{B}\). This principle is used to write Ampère’s law in terms of free currents alone.

The definition of \(\mathbf{H}\) is

\[ \mathbf{H} = \dfrac{1}{\mu_{0}}\mathbf{B} - \mathbf{M} \]

where

  • \(\mathbf{H}\) is the auxiliary magnetic field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mathbf{M}\) is the magnetization.
  • \(\mu_{0}\) is the permeability of free space.

The classification of linear magnetic media. Linear media are classified by the sign of \(\chi_{m}\). Diamagnetism is a weak opposing magnetization. Paramagnetism is a weak aligning magnetization. Ferromagnetism is a strong spontaneous magnetization of domains. This principle is used to distinguish the three standard classes of magnetic materials.

The permeability of a linear medium is

\[ \mu = \mu_{0}\bigl(1+\chi_{m}\bigr) \]

where

  • \(\mu\) is the permeability of the material.
  • \(\mu_{0}\) is the permeability of free space.
  • \(\chi_{m}\) is the magnetic susceptibility.