Propagation in a dielectric medium

Consider the propagation of an electromagnetic wave through a uniform dielectric medium of dielectric constant $\epsilon$. According to Eqs. (810) and (812), the dipole moment per unit volume induced in the medium by the wave electric field ${\bf E}$ is

$${\bf P} = \epsilon_0 (\epsilon -1) {\bf E}.$$ (1139)

There are no free charges or free currents in the medium. There is also no bound charge density (since the medium is uniform), and no magnetization current density (since the medium is non-magnetic). However, there is a polarization current due to the time-variation of the induced dipole moment per unit volume. According to Eq. (859), this current is given by

$${\bf j} = \frac{\partial{\bf P}}{\partial t}.$$ (1140)

Hence, Maxwell's equations take the form

$$\nabla\!\cdot\!{\bf E} = 0,$$ (1141)

$$\nabla\!\cdot\!{\bf B} = 0,$$ (1142)

$$\nabla\times{\bf E} = - \frac{\partial {\bf B}}{\partial t} ,$$ (1143)

$$\nabla\times{\bf B} = \mu_0 {\bf j}+ \epsilon_0 \mu_0 \frac{\partial {\bf E}}{\partial t}.$$ (1144)

According to Eqs. (1139) and (1140), the last of the above equations can be rewritten

$$\nabla\times{\bf B} =\epsilon_0 \mu_0 (\epsilon-1) \frac{... ...t}= \frac{\epsilon}{c^2} \frac{\partial {\bf E}}{\partial t},$$ (1145)

since $c= (\epsilon_0 \mu_0)^{-1/2}$. Thus, Maxwell's equations for the propagation of electromagnetic waves through a dielectric medium are the same as Maxwell's equations for the propagation of waves through a vacuum (see Sect. 4.7), except that $c\rightarrow c/n$, where

$$n = \sqrt{\epsilon}$$ (1146)

is called the refractive index of the medium in question. Hence, we conclude that electromagnetic waves propagate through a dielectric medium slower than through a vacuum by a factor $n$ (assuming, of course, that $n>1$). This conclusion (which was reached long before Maxwell's equations were invented) is the basis of all geometric optics involving refraction.