Why does the differential wave equation of EM in dielectrics use the permittivity and permeability of vacuum?

Question

I have been reading some basic book to understand how to derive the wave equation of light from the Maxwell equations, but those equations use the permittivity and permeability of vacuum.

The books usually tell you that those constants should be replaced from the specific one of the medium. But when I start reading how to derive the behavior of a EM wave in a dielectric (isotropic) they start with the differential wave equation deduced from Maxwell, but with the vacuum constants.

$$\nabla^2\mathbf{E}-\epsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t^2}=\mu_0\frac{\partial^2\mathbf{P}}{\partial t^2}$$

Is there a reason for that? At the end, by doing some steps with the previous equations in order to get the index of refraction of the dielectric, you get as one of it in terms the ε of the vacuum (is that correct?).

$$\tilde{n}=\sqrt{1+\frac{Ne^2}{m_e\epsilon_0(w_0^2-w^2+i\gamma w)}}$$

And also the wave number (k) will be in terms of the speed of light in vacuum (c) as a k=nw/c, but if from the first moment (in the differential wave equation) we use the permittivity and permeability of the material, we will end up with k=nw/v, where v is the velocity of the wave in the material (isn't that correct?)

I know that I'm not taking into account something (in terms of theory), but I can't figure out what it is.

Answer 1

In principle you can always use the vacuum (or microscopic) Maxwell-equations. There is no theoretical necessity for the macroscopic Maxwell equations (those with the material dielectric constants). The consequence is however that you need to explicitly include all the charges within your system in your calculation, e.g. if you want to describe the propagation of light through glass you need to include all the positive and negative charges (nuclei and electrons) that make up the glass.

This is pretty cumbersome and ineffective and therefore the macroscopic Maxwell equations were introduced. The macroscopic Maxwell equations are based on the Polarization $\mathbf{P}$ and Magnetization $\mathbf{M}$, which desrcibe the effective behaviour of a in total neutral collection of charges. Those charges can (and must be) be discarded from the explicit calculation and only enter by their Polarization and Magnetization. The equation

$$\nabla^2\mathbf{E}-\epsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t^2}=\mu_0\frac{\partial^2\mathbf{P}}{\partial t^2}$$

you stated is therefore already a macroscopic equation and implicitly includes the relative permittivity and permeability. The permittivity explicitly enters if we use the typical assumption that the Polarization of the material depends linearly on the external electric field. Then $$ \mathbf{P} = \chi_e \varepsilon_0 \mathbf{E} $$

$$\nabla^2\mathbf{E}-\epsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t^2}=\mu_0\varepsilon_0 \chi_e\frac{\partial^2\mathbf{E}}{\partial t^2}$$ $$\nabla^2\mathbf{E}-\epsilon_0\mu_0\underbrace{(1-\chi_e)}_{\varepsilon_r} \frac{\partial\mathbf{E}}{\partial t^2}=0$$

To wrap this up: The macroscopic Maxwell equation with the relative permittivity and permeability allow to exclude part of the system from the explicit calculation by encapsulating their effective behaviour within the polarization and magnetization. In the example with light within glass this means that we can use the Maxwell equations as if there were no charges present, i.e. set $\rho=0,\;\mathbf{j}=0$. Only the permittivity and permeability are changed.