In Chapter 11, we discussed the anisotropic properties of a magnetized plasma by modeling it as an anisotropic dielectric whose permittivity is a tensor. Ferrites in the presence of an external static magnetic field behave like an anisotropic magnetic material whose permeability is a tensor. Such material has many applications in microwave engineering. Here we briefly discuss the permeability tensor [1].

Ferrites are magnetic primarily because of the magnetic dipole moment $\overline{m}$ created due to the electron spin. In the presence of a magnetic field $\overline{H}$, torque $\overline{T}$ exerted on the electron is given by

The equation of angular motion of mechanics is given by

where $\overline{L}$ is the angular momentum.

The motion of the electron is gyroscopic around $\overline{H}$. The difference between a mechanical gyroscope and the spinning electron motion is that in the case of electrons the torque is of magnetic origin rather than mechanical. The dipole moment to be used in Equation 11B.1 is given by

where

m_e being the mass of the electron, γ_m the gyromagnetic ratio (=1.76 × 10¹¹ rad/sT).

From Equation 11B.1, Equation 11B.2 and Equation 11B.3, we obtain

Equation 11B.5 describes (Larmor) precession motion with the frequency ω₀ (Larmor frequency) given by

If there are N electrons per unit volume with each electron creating magnetic dipole moment $\overline{m}$ due to its spin, the magnetization vector $\overline{M}$ is given by

Equation 11B.5 can now be written as

For harmonic variation of $\overline{M}$ with frequency ω,

Equation 11B.9 contains the origin of the anisotropy of the permeability. Let us assume that the external magnetic field is

and the small-signal H₁ in the z-direction is neglected. From Equation 11B.9, one can obtain the relation between the small-signal value of $\overline{M}$ and the small-signal values of H_x and H_y as follows:

We can now write

where

The elements of the matrix are:

If μ_r of the ferrite material is known, then

Equation 11B.13 looks similar to the dielectric tensor $\overline{K}$ discussed in Chapter 11. One can thus expect phenomena of Faraday rotation. Sohoo [2] discusses a number of microwave applications.