#### Maxwell's Equations in Matter
emdm.Me.Mem

Maxwell's equations in vacuum:

\begin{align*} (i)~~ &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ (ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &\mbox{anonymous} \nonumber \\ (iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\ (iv)~~ &{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &\mbox{Ampère + Maxwell}. \end{align*}

are complete as they stand. In presence of matter: more convenient to write sources in terms of free charges and currents.

From static case: electric polarization \({\bf P}\) produces bound charge density rhob \[ \rho_b = -{\boldsymbol \nabla} \cdot {\bf P} \label{Gr(7.46)} \] and magnetization \({\bf M}\) produces bound current density JbcurlM \[ {\bf J}_b = {\boldsymbol \nabla} \times {\bf M} \label{Gr(7.47)} \] One new feature in nonstatic case: change in electric polarization involves flow of bound charge (call it \({\bf J}_p\)) which must be included in total current. Consider a small chunk of polarized material. Polarization induces charge density \(\sigma_b = P\) at one end and \(-\sigma_b\) at other (refer to sigmab. If \(P\) increases a bit, charge increases, giving net current \[ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t} da_{\perp}. \] We therefore have the

Polarization current density

- Gr (7.48)

\[ {\bf J}_p = \frac{\partial {\bf P}}{\partial t} \tag{Jp}\label{Jp} \]

otherwise simply called the polarization current. This has nothing to do with the bound current \({\bf J}_b\) (the latter is associated to magnetization; the polarization current is the result of linear motion of charge when polarization changes). We can check consistency with the continuity equation associated to the conservation of bound charges:

Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.

\[ {\boldsymbol \nabla} \cdot {\bf J}_p = {\boldsymbol \nabla} \cdot \frac{\partial {\bf P}}{\partial t} = \frac{\partial}{\partial t} ({\boldsymbol \nabla} \cdot {\bf P}) = -\frac{\partial \rho_b}{\partial t} ~~\longrightarrow~~\frac{\partial \rho_b}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J}_p = 0 \] so OK, continuity equation is satisfied. \({\bf J}_p\) is essential to account for conservation of bound charge. Changing magnetization does not lead to analogous accumulation of charge and current.

In view of this: total charge density can be separated into 2 parts, free and bound:

- Gr (7.49)

\[ \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P} \tag{rhofb}\label{rhofb} \]

and current can be separated into three parts, free, bound and polarization:

- Gr (7.50)

\[ {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t}. \tag{Jfbp}\label{Jfbp} \]

Gauss's law: can be rewritten \[ {\boldsymbol \nabla} \cdot {\bf E} = \frac{1}{\varepsilon_0} (\rho_f - {\boldsymbol \nabla} \cdot {\bf P}) \hspace{5mm}\longrightarrow \hspace{5mm} {\boldsymbol \nabla} \cdot {\bf D} = \rho_f \label{Gr(7.51)} \] where (as in static case)

\[ {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P} \label{Gr(7.52)} \]

Ampère's law including Maxwell's term: \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 \left( {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \] or \[ {\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t} \label{Gr(7.53)} \] where as before

\[ {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M} \label{Gr(7.54)} \]

Faraday's law and \({\boldsymbol \nabla} \cdot {\bf B} = 0\) remain unaffected by our separation into free and bound parts, since they don't involve \(\rho\) or \({\bf J}\).

In terms of free charges and currents, we thus get

Maxwell's equations (in matter)

- Gr (7.55)

\begin{align} (i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \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} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}. \tag{Max_mat}\label{Max_mat} \end{align}

Last term: displacement current, \[ {\bf J}_d = \frac{\partial {\bf D}}{\partial t} \label{Gr(7.58)} \]

This must all be complemented by the constitutive relations giving \({\bf D}\) and \({\bf H}\) in terms of \({\bf E}\) and \({\bf B}\). For the restricted case of linear media:

- Gr (7.56,7.57)

\[ {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm} {\bf M} = \chi_m {\bf H} \] so \[ {\bf D} = \varepsilon {\bf E}, \hspace{1cm} {\bf H} = \frac{1}{\mu} {\bf B}, \tag{consrel}\label{consrel} \] where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\).

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42