maxwell.tex

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\title{electromagnetism \& wave optics}
\author{baptiste}
\date{\the\year, T2}


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\begin{document}



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\begin{frame}{Inconsistency for time-varying currents}
\protect\hypertarget{inconsistency-for-time-varying-currents}{}
By time of James Clerk Maxwell (mid-19th century):

\[\begin{aligned}
\Div\vecE& =\rho / \varepsilon_0\qquad\text{(Gau\ss)}\\
\Div\vecB&=0\qquad\text{(no magnetic monopoles)}\\
\Curl\vecE&=-\partial_t \vecB\qquad\text{(Faraday)}\\
\Curl\vecB&=\mu_0 \vecJ\qquad\text{(Ampère)}
\end{aligned}
\]

\hrule

BUT\dots~an inconsistency arises, as \(\Div(\Curl\vecV)\equiv 0\) for
any vector \(\vecV\): \[
\Div(\Curl\vecE)=-\partial_t (\Div\vecB)=0 \qquad(\text{all good})
\] Yet, \[
\Div(\Curl\vecB)=\mu_0 \Div\vecJ \neq 0 \qquad(\text{problem!})
\]

\(\Div\vecJ = -\partial_t \rho\), so \(\Div\vecJ = 0\) only for steady
currents (magnetostatics).
\end{frame}

\begin{frame}{Maxwell's contribution}
\protect\hypertarget{maxwells-contribution}{}
Maxwell realised that the last equation needed an additional term.
Applying continuity of current/charge: \[
\begin{aligned}
\Div\vecJ &=-\frac{\partial \rho}{\partial t} \\
&=-\frac{\partial}{\partial t}\left(\varepsilon_0\Div\vecE\right)=-\Div\left(\varepsilon_0 \frac{\partial \vecE}{\partial t}\right)
\end{aligned}
\] If we add \(\mu_0 \varepsilon_0 \frac{\partial \vecE}{\partial t}\)
to the current \(\vecJ\), the inconsistency disappears and the last curl
equation becomes: \[
\Curl\vecB=\mu_0 \vecJ+\mu_0 \varepsilon_0 \frac{\partial \vecE}{\partial t} \quad \text { (Ampère's law with Maxwell's correction) }
\] \(\varepsilon_0 \frac{\partial \vecE}{\partial t}\) is called the
``\emph{displacement current density}''.
\end{frame}

\begin{frame}{Electrodynamics}
\protect\hypertarget{electrodynamics}{}
Our starting point: the electromagnetic field obeys Maxwell's equations:
\[
\begin{aligned}
\Div\vecD &= \rho_f\\
\Div\vecB &= 0\\
\Curl\vecE &= -\frac{\partial\vecB}{\partial t}\\
\Curl\vecH &= \frac{\partial\vecD}{\partial t} + \vecJ_f
\end{aligned}
\] Lorentz force: \[
\vecF = q\left(\vecE + \vecv \times \vecB\right)
\] + \emph{constitutive relations} (\(\sigma, \varepsilon, \mu,\dots\))
and \emph{boundary conditions}
\end{frame}

\begin{frame}{Remark on constitutive relations}
\protect\hypertarget{remark-on-constitutive-relations}{}
Material responses are very diverse (anisotropic, gyrotropic, nonlinear,
nonlocal, hysteresis, \dots). We'll only discuss the most common
approximations,

\begin{itemize}
\tightlist
\item
  \emph{Linear approximation}: \[
  \begin{array}{l}
  \mathbf{P}(\mathbf{r}, t)=\varepsilon_{0}\left(\overline{\bar{\chi}}_{e}^{(1)} \mathbf{E}(\mathbf{r}, t)+\overline{\bar{\chi}}_{e}^{(2)} \mathbf{E}^{2}(\mathbf{r}, t)+\overline{\bar{\chi}}_{e}^{(3)} \mathbf{E}^{3}(\mathbf{r}, t)+\ldots\right) \\
  \mathbf{P}(\mathbf{r}, t)=\varepsilon_{0} \overline{\bar{\chi}}_{e} \mathbf{E}(\mathbf{r}, t)
  \end{array}
  \]
\item
  \emph{Isotropic medium}: \(\chi_e\) is a scalar (\(\mathbf{P}\) and
  \(\mathbf{E}\) have the same orientation)
\item
  \emph{Non-locality in space}: \(\vecP\) depends on the electric field
  in the vicinity of a point
  \(\mathbf{P}(\mathbf{r}, t)=\varepsilon_{0} \int d \mathbf{r}^{\prime} \chi\left(\mathbf{r}^{\prime}-\mathbf{r}, t\right) \mathbf{E}\left(\mathbf{r}^{\prime}, t\right)\)
  If we use spatial Fourier Transform, this convolution product simply
  becomes:
  \(\underline{\mathbf{P}}(\mathbf{k}, t)=\varepsilon_{0} \underline{\chi_{e}}(\mathbf{k}, t) \underline{\mathbf{E}}(\mathbf{k}, t)\)
  \(\to \varepsilon(\veck,\omega)\) (cf Kittel, etc.)
\end{itemize}
\end{frame}

\end{document}