Electromagnetic field in a general medium

In a material medium whose constitutive relations are as yet unspecified, we can always set, without loss of generality:

$$\underline{\mathcal{D}}=\varepsilon_0 \underline{\mathcal{E}} + \underline{\mathcal{P}} \, \underline{\mathcal{B}}=\mu_0 \underline{\mathcal{H}} + \underline{\mathcal{M}} \space ,$$ (1.123)

where the polarization vector $\underline{\mathcal{P}}$ and the magnetization vector $\underline{\mathcal{P}}$ are unspecified functions of the field vectors:

$$\underline{\mathcal{P}}=\underline{\mathcal{P}}(\underline{\mathc... ...erline{\mathcal{D}}, \underline{\mathcal{H}}, \underline{\mathcal{B}}) \space , \underline{\mathcal{M}}=\underline{\mathcal{M}} ( \underline{\mat... ...erline{\mathcal{D}}, \underline{\mathcal{H}}, \underline{\mathcal{B}}) \space ;$$ (1.124)

in free space, $\underline{\mathcal{P}}$ and $\underline{\mathcal{M}}$ are both zero. It should be noted that many authors use $\mu_0 \underline{\mathcal{M}}$ instead of $\underline{\mathcal{M}}$ in the second of (1.125), thereby destroying the symmetry between electric and magnetic quantities.

Equations (1.125 allow us to rewrite Maxwell's equations (1.1-1.4) in terms of equivalent sources (indicated by [ ]) in free space, as follows:

$$\nabla \times \underline{\mathcal{H}} =\left[ \underline{\mathcal{J}}_e\right] +\varepsilon_0 \frac{\partial \underline{\mathcal{E}}}{\partial t} \space ,$$ (1.125)

$$\nabla \times \underline{\mathcal{E}} =-\left[ \underline{\mathcal{J}}_m\right] -\mu_0 \frac{\partial \underline{\mathcal{H}}}{\partial t} \space ,$$ (1.126)

$$\nabla \cdot \underline{\mathcal{E}} = \frac{1}{\varepsilon_0} \left[ \rho_e \right] \space ,$$ (1.127)

$$\nabla \cdot \underline{\mathcal{H}} = \frac{1}{\mu_0} \left[ \rho_m \right] \space ,$$ (1.128)

where

$$\left[ \underline{\mathcal{J}}_e \right] = \underline{\mathcal{J}}_e + \frac{\partial \underline{\mathcal{P}}}{\partial t} \space ,$$ (1.129)

$$\left[ \underline{\mathcal{J}}_m \right] = \underline{\mathcal{J}}_m + \frac{\partial \underline{\mathcal{M}}}{\partial t} \space ,$$ (1.130)

$$\left[ \rho_e \right] = \rho_e - \nabla \cdot \underline{\mathcal{P}} \space ,$$ (1.131)

$$\left[ \rho_m \right] = \rho_m - \nabla \cdot \underline{\mathcal{M}} \space ,$$ (1.132)

Equations (1.127-1.130) are obviously equivalent to (1.1-1.4), but represent a different viewpoint. In (1.1-1.4), a macroscopic approach is taken in which the bulk properties of the medium are characterized by relations among the field quantities (constitutive relations). Alternatively, in (1.127-1.130), the material is seen as a collection of electric and magnetic charge and current densities (1.131-1.134) in free space; from this microscopic viewpoint, the equivalent sources (1.131-1.134) include the ``true'' sources which appear in (1.1-1.4) as well as fictitious sources which allow us to replace the material medium with free space. In other words, any material medium may be seen as electromagnetically equivalent to free space in which the following sources exist: an electric charge density $-\nabla \cdot \underline{\mathcal{P}}$, a magnetic charge density $-\nabla \cdot \underline{\mathcal{P}}$, an electric current density $\partial \underline{\mathcal{P}} / \partial t$, and a magnetic current density $\partial \underline{\mathcal{M}} / \partial t$.

The formulation (1.127-1.130) allows us to apply the formalism developed in the previous two sections for a field in free space. Thus, the electromagnetic potentials in the time domain are, according to (1.103-1.106):

$$\Phi_e(\underline{r},t) =\frac{1}{4\pi \varepsilon_0} \int_{v_s} \frac{\rho_e ( \underlin... ...dot \underline{\mathcal{P}}( \underline{r}',t')}{R} \, d\underline{r}' \space ,$$ (1.133)

$$\Phi_m(\underline{r},t) =\frac{1}{4\pi \mu_0} \int_{v_s} \frac{\rho_m ( \underline{r}', t... ...dot \underline{\mathcal{M}}( \underline{r}',t')}{R} \, d\underline{r}' \space ,$$ (1.134)

$$\underline{\mathcal{A}}_e(\underline{r},t) = \frac{\mu_0}{4 \pi} \int_{v_s} \frac{\underline{\mathcal{J}}_e(... ...\partial t'} \underline{\mathcal{P}}(ul{r}',t')}{R} \, d\underline{r}' \space ,$$ (1.135)

$$\underline{\mathcal{A}}_m(\underline{r},t) = \frac{\varepsilon_0}{4 \pi} \int_{v_s} \frac{\underline{\mathca... ...{\partial t'} \underline{\mathcal{M}}(ul{r}',t')}{R} \, d\underline{r}'\space ;$$ (1.136)

the corresponding potentials in the frequency domain are, according to (1.111-1.114):

$$\Phi_e(\underline{r}) =\frac{1}{\varepsilon_0} \int_{v_s} \left[ \rho_e(\underline{r}')... ...a' \cdot \underline{\mathcal{P}}(\underline{r}') \right] G \, d\underline{r}' ,$$ (1.137)

$$\Phi_m(\underline{r}) =\frac{1}{\mu_0} \int_{v_s} \left[ \rho_m(\underline{r}') - \nabla' \cdot \underline{\mathcal{M}}(\underline{r}') \right] G \, d\underline{r}' ,$$ (1.138)

$$\underline{A}_e(\underline{r}) =\mu_0 \int_{v_s} \left[ \underline{\mathcal{J}}_e(\underline{r}'... ...mega \underline{\mathcal{P}}(\underline{r}') \right] G d\underline{r}' \space ,$$ (1.139)

$$\underline{A}_m(\underline{r}) =\mu_0 \int_{v_s} \left[ \underline{\mathcal{J}}_m(\underline{r}'... ...omega \underline{\mathcal{M}}(\underline{r}') \right] G d\underline{r}'\space .$$ (1.140)

Substitution of (1.135-1.138) into (1.96) in the time-domain case, or of (1.139-1.142) into (1.109) in the frequency-domain case, accompanied by a specified dependence of polarization and magnetization upon the electric and magnetic fields, yields two coupled integro-differential equations for the electric and magnetic field vectors. From these two equations, all integral equations commonly used in electromagnetism can be derived, as we shall see in the following chapters.