2.20: Magnetization of Moving Media

It is natural to use polarization charge to represent the effect of macroscopic media on the macroscopic electric field. Actually, this is one of two alternatives for representing polarization. That such a choice has been made becomes clear when the analogous question is asked for magnetization. In the absence of magnetization, the free current density is the source of the magnetic field, and it is therefore natural to represent the macroscopic effects of magnetizable media on ý through an equivalent magnetization current density. Indeed, this viewpoint is often used and supported by the contention that what is modeled at the atomic level is really a system of currents (the electrons in their orbits). It is important to understand that the use of equivalent currents, or of equivalent magnetic charge as used here, if carried out self-consistently, results in the same predictions of physical processes. The choice of models in no way hinges on the microscopic processes accounting for the magnetization. Moreover, the magnetization is often dominated by dynamical processes that have more to do with the behavior of domains than with individual atoms, and these are most realistically pictured as small magnets (dipoles). With the Chu formulation postulated in Sec. 2.2, the dipole model for representing magnetization has been adopted.

An advantage of the Chu formulation is that magnetization is developed in analogy to polarization. But rather than starting with a magnetic charge density, and deducing its relation to the polarization density, think of the magnetic material as influencing the macroscopic fields through an intrinsic flux density \(\mu_o \overrightarrow{M}\) that might be given, or might be itself induced by the macroscopic \(\overrightarrow{H}\). For lack of evidence to support the existence of "free" magnetic monopoles, the total flux density due to all macroscopic fields must be solenoidal. Hence, the intrinsic flux density \(\mu_o \overrightarrow{M}\), added to the flux density in free space \(\mu_o \overrightarrow{M}\), must have no divergence:

\[ \nabla \cdot \mu_o (\overrightarrow{H} + \overrightarrow{M}) = 0 \label{1} \]

This is Equation 2.3.24b. It is profitable to think of \(-\nabla \cdot \mu_o \overrightarrow{M}\) as a source of \(\overrightarrow{H}\). That is, Equation \ref{1} can be written to make it look like Gauss' law for the electric field:

\[ \nabla \cdot \mu_o\overrightarrow{H} = \rho_m; \quad \rho_m = -\nabla \cdot \mu_o\overrightarrow{M} \label{2} \]

The magnetic charge density \(\rho_m\) is in this sense the source of the magnetic field intensity.

Faraday's law of induction must be revised if magnetization is present. If \(\mu_o \overrightarrow{M}\) is a magnetic flux density, then, through magnetic induction, its rate of change is capable of producing an induced electric field intensity. Also, if Faraday's law of induction were to remain valid without alteration, then its divergence must be consistent with Equation \ref{1}; obviously, it is not.

To generalize the law of induction to include magnetization, it is stated in integral form for a contour C enclosing a surface S fixed to the material in which the magnetized entities are imbedded. Then, because \(\mu_o (\overrightarrow{H} + \overrightarrow{M})\) is the total flux density,

\[ \int_{C} \overrightarrow{E^{'}} \cdot \overrightarrow{d}l = -\frac{d}{dt} \int_{S} \mu_o (\overrightarrow{H} + \overrightarrow{M}) \cdot \overrightarrow{n} da \label{3} \]

The electric field \(\overrightarrow{E^{'}}\) is evaluated in the frame of reference of the moving contour. With the time derivative taken inside the temporally varying surface integrals (Equation 2.6.4) and because of Equation \ref{1},

\[ \int_{C} \overrightarrow{E^{'}} \cdot \overrightarrow{d}l = -\int_{S} \frac{\partial}{\partial{t}} \big[ \mu_o (\overrightarrow{H} + \overrightarrow{M}) \big] \cdot \overrightarrow{n} da + \int_{S} \nabla \times \big[ \overrightarrow{v} \times \mu_o (\overrightarrow{H} + \overrightarrow{M}) \big] \cdot \overrightarrow{n} da \label{4} \]

The transformation law for \(\overrightarrow{E}\) (Equation 2.5.12b with \(\overrightarrow{u} = \overrightarrow{v}\)) is now used to evaluate \(\overrightarrow{E^{'}}\), and Stokes's theorem, Equation 2.6.3, used to convert the line integral to a surface integral. Because \(S\) is arbitrary, it then follows that the integrand must vanish:

\[ \nabla \times \overrightarrow{E} = \frac{\partial}{\partial{t}} \big[ \mu_o (\overrightarrow{H} + \overrightarrow{M}) \big] + \nabla \times (\overrightarrow{v} \times \mu_o \overrightarrow{M}) \label{5} \]

This generalization of Faraday's law is the postulated equation, Equation 2.3.25b.