Pre-Quantum Electrodynamics

Propagation in Linear Media emdm.emwm.plm

  • Gr 9.3.1

In matter regions without free charge and free current: Maxwell's equations are

\begin{align*} (i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = 0, \nonumber \\ (ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\ (iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\ (iv)~~ &{\boldsymbol \nabla} \times {\bf H} = \frac{\partial {\bf D}}{\partial t}. \end{align*}

For linear medium: \[ {\boldsymbol D} = \varepsilon {\boldsymbol E}, \hspace{1cm} {\boldsymbol H} = \frac{1}{\mu} {\boldsymbol B}. \label{Gr(9.66)} \] If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),

  • Gr (9.65)
\begin{align} (i)~~ &{\boldsymbol \nabla} \cdot {\bf E} = 0, \nonumber \\ (ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\ (iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\ (iv)~~ &{\boldsymbol \nabla} \times {\bf B} = \mu \varepsilon \frac{\partial {\bf E}}{\partial t}. \tag{Max_lh}\label{Max_lh} \end{align}

These are the same equations as in vacuum, except for the substitution of \(\mu_0 \varepsilon_0\) by \(\mu \varepsilon\).

Speed of propagation: \[ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n} \label{Gr(9.68)} \] where the index of refraction of the material is defined as

Index of refraction

  • Gr (9.69)

\[ n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}} \tag{n}\label{n} \]

Fact: for most materials, \(\mu\) is very close to \(\mu_0\), so \[ n \simeq \sqrt{\varepsilon_r} \label{Gr(9.70)} \] with \(\varepsilon_r\) being the dielectric constant epsr.

Energy density: \[ u = \frac{1}{2} \left( \varepsilon E^2 + \frac{1}{\mu} B^2 \right) \label{Gr(9.71)} \] Poynting vector: \[ {\boldsymbol S} = \frac{1}{\mu} {\boldsymbol E} \times {\boldsymbol B} \label{Gr(9.72)} \] Wave intensity \[ I = \frac{1}{2} \varepsilon v E_0^2 \label{Gr(9.73)} \]



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Author: Jean-Sébastien Caux

Created: 2024-02-27 Tue 10:31