From d0a3ee143a0ea12f6163d92b03eb9e80a1387244 Mon Sep 17 00:00:00 2001 From: Luca Zoppetti Date: Tue, 30 Apr 2024 12:18:50 +0200 Subject: [PATCH] lec18 first hour --- .../screenshots/2024-04-30-11-26-26.png | Bin 0 -> 13562 bytes .../screenshots/2024-04-30-11-27-49.png | Bin 0 -> 3082 bytes .../screenshots/2024-04-30-11-33-08.png | Bin 0 -> 3817 bytes .../screenshots/2024-04-30-11-45-24.png | Bin 0 -> 7839 bytes .../screenshots/2024-04-30-11-49-36.png | Bin 0 -> 8797 bytes .../screenshots/2024-04-30-11-50-50.png | Bin 0 -> 10406 bytes .../screenshots/2024-04-30-11-53-50.png | Bin 0 -> 5256 bytes .../screenshots/2024-04-30-12-00-57.png | Bin 0 -> 28803 bytes .../screenshots/2024-04-30-12-05-26.png | Bin 0 -> 47425 bytes .../screenshots/2024-04-30-12-08-10.png | Bin 0 -> 29972 bytes .../screenshots/2024-04-30-12-12-53.png | Bin 0 -> 10426 bytes .../screenshots/2024-04-30-12-16-51.png | Bin 0 -> 7225 bytes tex/waves/lectures/lec_13.tex | 5 +- tex/waves/lectures/lec_18.tex | 113 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In futuro rappresenteranno gli orbitali). Queste funzioni soddisfano equazioni del tipo: @@ -35,6 +35,9 @@ \subsection{Soluzione delle onde sferiche} \centering \includegraphics[width=0.7\textwidth]{screenshots/2024-04-09-11-41-43.png} \end{figure} +\begin{note} + È molto importante notare che le soluzioni nell'equazione \eqref{eq:sol_wave_sphere} sono esatte solo nel caso di \(l=0\), mentre negli altri casi valgono nel limite \(r \to +\infty \). Si vede inserendole nell'equazione di D'Alembert. +\end{note} Noi generalmente guarderemo le armoniche sferiche con \(l=0,1\). Le funzioni sono ortonormali sulla superficie della sfera (nello spazio delle funzioni definite su \((\theta ,\varphi )\)). Guardiamo ora i casi che si possono presentare per il termine dipendente da \(r\mp vt\). \begin{itemize} diff --git a/tex/waves/lectures/lec_18.tex b/tex/waves/lectures/lec_18.tex new file mode 100644 index 0000000..cebabe4 --- /dev/null +++ b/tex/waves/lectures/lec_18.tex @@ -0,0 +1,113 @@ +\lecture{18}{30 aprile 2024} +\section{Propagazione nei materiali} +Per i materiali la velocità è \(\quotient{1}{\sqrt{\mu \varepsilon } } \), che nella maggior parte dei casi si riduce a \(\quotient{c}{\sqrt{\varepsilon _r} }=\quotient{c}{n} \) perché \(\mu _r \thickapprox 1\) (i mezzi così sono detti mezzi ottici). +\begin{definition} + [Indice di rifrazione] + \begin{equation} + n = \frac{c}{v} \thickapprox \sqrt{\varepsilon _r} \geq 1 + \end{equation} +\end{definition} +Considerando la luce che viaggia in acqua (\(\varepsilon _r \thickapprox 80\)), trovo che la velocità della luce dovrebbe essere \(v\thickapprox \quotient{c}{9} \), mentre se faccio una misura trovo \(v \thickapprox \quotient{3}{4} c\). La soluzione potrebbe essere che la costante dielettrica relativa varia al passaggio di un'onda elettromagnetica! Nei mezzi ottici, senza cariche e senza correnti, \(\vert \vec{E} \vert = v \vert \vec{B} \vert = v \mu \vert \vec{H} \vert \). Inoltre +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-11-26-26.png} +\end{figure} + +\subsection{Cosa succede nei dielettrici?} +Considero un mezzo dielettrico fra le lastre di un condensatore: +\begin{figure}[H] + \centering + \includegraphics[width=0.3\textwidth]{screenshots/2024-04-30-11-33-08.png} +\end{figure} +Detta \(V_0\) la differenza di potenziale applicata alle lastre del condensatore, il campo elettrico senza dielettrico sarà \(\vert \vec{E} \vert = \quotient{V_0}{d} \) e in presenza di dielettrico sarà \(\vert \vec{E} \vert = \quotient{\vert \vec{E}_0 \vert }{\varepsilon _r}\). Il campo elettrico deforma le molecole e nasce un momento di dipolo indotto: \(\vec{p}= \alpha \vec{E}\) dove \(\alpha \) è la polarizzabilità atomica/molecolare. È come se ogni atomo sentisse una forza elettrica approssimabile \(\vec{f}= q \vec{E}\), in modo che le forze interne rispondano con una forza elastica \(f_e = -kx\). + +Consideriamo ora cosa succede quando il potenziale è variabile: \(\Delta V (t) = V_0 \cos (\omega t)\). L'asse x è diretto come il campo elettrico e ha origine sull'atomo. L'elettrone si comporta come un oscillatore armonico forzato. Sia \(f=-kx\) la forza di richiamo elastica, \(f_v = -\beta \dot{x}\) la forza viscosa e \( =E_0 e^{i \omega t}\) l'andamento del campo elettrico. +\begin{equation} + m \ddot{x} + \beta \dot{x} + kx = - e E_0 e^{i \omega t} +\end{equation} +Poniamo \(\gamma = \quotient{\beta }{m}, \omega _0 ^{2} = \quotient{k}{m} \). Siamo bravissimi e sappiamo già risolvere questo problema! (Guardiamo solo la soluzione stazionaria quando il transiente non ha più effetto). +\begin{equation} + x(t)= A e^{i \omega t} \text{ con } A= \frac{-e E_0}{m[(- \omega ^{2} + \omega ^{2} _0) + i \gamma \omega ]} +\end{equation} +Ma a livello macroscopico cosa succede? Ogni atomo ha un dipolo atomico \(\vec{p}=-e x \vec{\hat{i}}\). La polarizzazione del dielettrico è \(\vec{P}= n_a \vec{p}\) dove \(n_a\) è la densità atomica. Da elettromagnetismo sappiamo che c'è una relazione fra \(\vec{P}\) ed \(\vec{E}\): \(\vec{P}= \varepsilon _0 \chi _e \vec{E} = \varepsilon _0 \chi _e E_0 e^{i \omega t} \vec{\hat{i}}\). Dalla soluzione precedente ho anche: +\begin{align} + \vec{P}&= \varepsilon _0 \chi _e \vec{E} = n_a \vec{p} = -e n_a x(t) \vec{i}=\\ + &= - e n_a \frac{- e E_0}{m[(- \omega ^{2} + \omega ^{2} _0) + i \gamma \omega ]}e^{i \omega t} \vec{\hat{i}} +\end{align} +Da cui posso risolvere per \(\chi _e\) (la suscettività elettrica del materiale): +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-11-45-24.png} +\end{figure} +Da cui si ottiene che \(\varepsilon (\omega )= \varepsilon _0 + \varepsilon _0 \chi _e(\omega )\) e l'indice di rifrazione che sarà: +\begin{align} + n (\omega ) &= \sqrt{\varepsilon _r (\omega )} = \sqrt{1 + \chi _e (\omega )} \thickapprox 1 + \frac{1}{2}\chi _e(\omega )\\ + &= 1 + \frac{n_a e^{2} }{2 m \varepsilon _0} \frac{1}{(-\omega ^{2} + \omega ^{2} _0)+ i \gamma \omega } +\end{align} +L'indice di rifrazione ha una componente immaginaria, possiamo scriverlo come \(n(\omega ) = n_r - i n_I\) con +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-11-49-36.png} +\end{figure} +La parte reale dell'indice di rifrazione dipende dalla pulsazione, quindi la velocità di fase dipende dalla pulsazione dell'onda e il mezzo è dispersivo! Anche \(k\) dipende dall'indice di rifrazione: +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{screenshots/2024-04-30-11-50-50.png} +\end{figure} +\paragraph{Legge di Lambert} +Che significato ha il numero d'onda complesso \(k= k_r - i k_I\)? Considero un'onda elettrica che si propaga nella direzione z e che oscilla come il campo elettrico di prima: \(E(z,t) = E_0 e^{i( \omega t - kz)}\). +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-11-53-50.png} +\end{figure} +L'onda appare muoversi con una velocità \(v= \quotient{\omega }{k_r} = \quotient{c}{n_r} \), ma la sua ampiezza si riduce man mano che avanza nel mezzo dielettrico. Pongo \(\beta = k_I\), l'ampiezza cala come \(E_0(z)=E_0 e^{-\beta z}\). L'intensità dell'onda cambia con il progredire dell'onda nel mezzo (\(I \propto E_0 ^{2} \)): +\begin{formula} + [Legge di Lambert] + \begin{equation} + I(z) = I_0 e^{-2 \beta z} = I_0 e^{- \mu z} + \end{equation} + dove si è posto \(\mu = 2 \beta \) che è detto \emph{coefficiente di assorbimento}. +\end{formula} +La dissipazione di energia è dovuta al termine viscoso dell'oscillatore forzato usato nel modellino. + +\paragraph{Velocità di gruppo} +Essendo in un mezzo dispersivo, la velocità di fase e la velocità di gruppo saranno diverse. La velocità di gruppo si trova come segue: +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-12-00-57.png} +\end{figure} +\begin{remark} + Si possono avere due casi di dispersione: la dispersione normale e la dispersione anomala. Nel primo caso, \(\frac{\mathrm{d}n_r}{\mathrm{d}\omega } \geq 0 \implies v_g \leq v_f\), mentre nel secondo \(\frac{\mathrm{d}n_r}{\mathrm{d}\omega } \implies v_g \geq v_f \). +\end{remark} +Si possono studiare gli andamenti della parte reale e immaginaria dell'indice di rifrazione di un singolo elettrone: +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-12-05-26.png} +\end{figure} +% TODO mettere reference per ampiezze +Ricordano molto l'ampiezza elastica e l'ampiezza assorbitiva! La regione del picco nella parte immaginaria dell'indice di rifrazione indica la regione in cui l'energia viene maggiormente assorbita dal materiale e quindi in cui l'onda fa più fatica ad attraversarlo. Notiamo che questi andamenti descrivono un atomo di idrogeno con un singolo elettrone. Nel caso generico in cui un atomo può avere più elettroni con diverse pulsazioni proprie si generalizza come segue: +\begin{figure}[H] + \centering + \includegraphics[width=0.45\textwidth]{screenshots/2024-04-30-12-08-10.png} +\end{figure} +Si nota che ad alte pulsazioni l'indice di rifrazione si avvicina a 1. + +\paragraph{Limite delle alte frequenze} +Consideriamo il caso \(\omega \gg \omega _{0j} \forall j\), cioè il limite di raggi X energetici o raggi gamma: +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-12-12-53.png} +\end{figure} +Quindi si trova che la velocità di fase è maggiore di \(c\)! +\begin{equation} + v_f = \frac{c}{n_r} = \frac{c}{1- \frac{e^{2} }{2m \varepsilon _0 \omega ^{2} }\sum_{j}n_{aj} } > c +\end{equation} +Ma ci salviamo in corner: l'importante è che la velocità a cui viaggia l'informazione sia minore di \(c\). Per fortuna è questo il caso: +\begin{equation} + n_r(\omega )=1- \frac{e^{2} }{2 m \varepsilon _0 \omega ^{2} }\sum_{j} n_{aj} \rightsquigarrow \frac{\mathrm{d}n_r}{\mathrm{d}\omega } = \frac{e ^{2} }{m \varepsilon _0 \omega ^{3}} \sum_{j}n_{aj} +\end{equation} +da cui +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{screenshots/2024-04-30-12-16-51.png} +\end{figure} \ No newline at end of file diff --git a/tex/waves/waves.tex b/tex/waves/waves.tex index a5d879f..e27d420 100644 --- a/tex/waves/waves.tex +++ b/tex/waves/waves.tex @@ -29,6 +29,6 @@ \newpage -\lec{3}{17} +\lec{3}{18} \end{document} \ No newline at end of file