\documentclass[11pt,a4paper,twocolumn,catalan]{article} \usepackage{babel} \usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage[dvips]{epsfig} \usepackage{indentfirst} \usepackage[a4paper,right=2cm,top=2.5cm]{geometry} \setlength\arraycolsep{2pt} \newcommand{\exptem}{\expon{-j(k_g z - \omega t)}} \newcommand{\expcav}{\expon{j\omega t}} \newcommand{\J}{\mathcal{J}} \newcommand{\K}{\mathcal{K}} \newcommand{\dd}{\mathrm{d}} \newcommand{\tg}{\mathrm{tg}} \newcommand{\tgh}{\mathrm{tgh}} \newcommand{\cotg}{\mathrm{cotg}} \newcommand{\dB}{\mathrm{dB}} \newcommand{\Np}{\mathrm{Np}} \newcommand{\eV}{\mathrm{eV}} \newcommand{\hatn}{\hat{n}} \newcommand{\expon}[1]{\mathrm{e}^{#1}} \newcommand{\parcial}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\deriv}[2][]{\frac{\dd #1}{\dd #2}} \newcommand{\E}[1]{\times 10^{#1}} \newcommand{\vD}{\vec{D}} \newcommand{\vH}{\vec{H}} \newcommand{\vj}{\vec{\jmath}} \newcommand{\vE}{\vec{E}} \newcommand{\vB}{\vec{B}} \newcommand{\vP}{\vec{P}} \newcommand{\vM}{\vec{M}} \newcommand{\vN}{\vec{N}} \newcommand{\vS}{\vec{S}} \newcommand{\va}{\vec{a}} \newcommand{\vnabla}{\vec{\nabla}} \author{pod} \title{\bf Ampliación de electromagnetismo} \date{Otoño, 2001} \begin{document} \maketitle \section{Propagación del campo} \noindent Maxwell $$ \vnabla \cdot \vD = \rho \ , \ \vnabla \times \vH = \vj + \epsilon_0 + \parcial[\vD]{t}$$ $$\vnabla \cdot \vB = 0 \ , \ \vnabla \times \vE = - \parcial[\vB]{t} $$ $$ \hatn \cdot ( \vD_+ - \vD_-) = \sigma \ , \ \hatn\cdot( \vB_+ - \vB_-) = 0 $$ $$ \hatn\times(\vE_+ - \vE_-)=0 \ , \ \hatn\times (\vH_+ - \vH_-) = \vec{\kappa} $$ $$\vD = \epsilon_0 \vE + \vP = \epsilon \ , \ \vB = \mu_0 (\vH + \vM) = \mu \vH $$ \noindent Poyting $\vN = \vE \times \vH$, conservación \begin{align*} \int_v \vE'\cdot\vj \dd v & = \int_v \frac{j^2}{\gamma}\dd v \\&+ \parcial{t} \int_v \frac{1}{2} (\vE\vD + \vB\vH)\dd v + \oint_S \vN \dd\vS \end{align*} \noindent Propagación $$ \Delta\vE - \mu\gamma \parcial[\vE]{t} - \mu\epsilon \parcial[^2\vE]{t^2} = 0 \ , $$ $$ \Delta\vH - \mu\gamma\parcial[\vH]{t} - \mu\epsilon \parcial[^2\vH]{t^2} = 0 $$ $$ \Delta \vE + \omega^2 \epsilon \mu \left( 1 - j \frac{\gamma}{\omega\epsilon} \right) \vE = 0 $$ \noindent Onda progresiva $$ E_i = E_o \expon{j(\omega t - k x)} \ , \ \vB = \frac{k}{\omega} \hatn \times \vE \ , \ k = \omega \sqrt{\epsilon\mu} $$ \noindent Conductores $ K^2 = {\epsilon\mu\omega^2-j \omega\mu\gamma} = \tilde{\epsilon}\mu\omega^2$ $$ \beta = k \sqrt{\frac{1}{2} \left[ \sqrt{1+\left(\frac{\gamma}{ \epsilon\mu}\right)^2}+1\right]} \ ,$$ $$ K = \beta - j \alpha \ ,\ \tg\delta = \frac{\gamma}{\epsilon\mu} $$ $$ \alpha = k \sqrt{\frac{1}{2} \left[ \sqrt{1+\left(\frac{\gamma}{\epsilon\mu}\right)^2}-1\right]} \ , $$ $$\lambda = \frac{2\pi}{\beta} \ , \ K = M \expon{j\phi} $$ $$ K =_{(\gamma\gg)} \frac{1}{\sqrt{2}} (1-j) \sqrt{\omega \mu\gamma} $$ \noindent Impedancia intrínseca $$ \eta = \frac{E_x}{H_y} = \frac{\omega\mu}{K} =_{(\gamma=0)}\sqrt{\frac{\mu}{ \epsilon}} = \sqrt{\frac{\omega\mu}{2\gamma}}(1+j)$$ \section{Líneas de transmisión} \noindent Ecuaciones generales $$\parcial[V]{x} = R i + L \parcial[i]{t} \ , \ \parcial[i]{x} = G V + C \parcial[V]{t} $$ Sin pérdidas ($ R = G = 0 $) $$ \parcial[^2V]{x^2} = L C \parcial[^2V]{t^2} \ , \ \parcial[^2i]{x^2} = L C \parcial[^2i]{t^2} \ , \ v = \frac{1}{\sqrt{L C}} = \frac{1}{\sqrt{\epsilon\mu}} $$ \noindent Sinusoidal $$ \deriv[V]{x} = ( R + j\omega L) I = z I \ , \ \deriv[I]{x} = (G+j\omega C) V = y V$$ Cte. propagación y impedancia característica $$ \gamma^2 = y z\ ,\ \gamma = \alpha + j\beta\ , \ \lambda = \frac{2\pi}{\beta} \ , \quad \frac{1}{Z_0} = \frac{\gamma}{z} = \sqrt{\frac{y}{z}} $$ $$ V(x) = V_R \left\{ \cosh \gamma x + \frac{Z_0}{Z_r} \sinh\gamma x \right\}$$ $$ I(x) = I_r \left\{ \cosh \gamma x + \frac{Z_0}{Z_r} \sinh\gamma x \right\} $$ Impedancia de línea $$ Z(x) = \frac{V(x)}{I(x)} = Z_0 \frac{Z_R + Z_0 \tgh \gamma x}{Z_0+Z_R \tgh \gamma x} $$ \noindent Pérdidas (1r orden) $$ \alpha = \frac{R}{2Z_0} + \frac{1}{2} G Z_0 \ , \ \beta = \omega \sqrt{LC} \ , Z_0 = \sqrt{L/C}$$ Vientres i nodos de tensión (intensidad, intercambiados) $$ \phi_0 - \beta x_M = 2 n \pi \ , \phi_0 - 2 \beta x_m = (2n+1) \pi \ , $$ $$ |x_M - x_m| = \lambda/4$$ \noindent Coef. reflexión $$ r_R = \frac{Z_R - Z_0}{Z_R + Z_0} = | r_R | \expon{i\phi_0} \ , $$ $$ Z(x) = Z_0 \frac{1+r_R \expon{-2\gamma x}}{1-r_R \expon{-2\gamma x}} $$ $$V(x) = V'_R \expon{\gamma x}\left\{ 1 + |r_R| \expon{j\phi_0} \expon{-2\gamma x}\right\}$$ $$I(x) = \frac{V'_r}{Z_0}\expon{\gamma x}\left\{ 1 - |r_R| \expon{j\phi_0} \expon{-2\gamma x} \right\}$$ Relación ondas estacionarias $$ \rho = S = \frac{V_M}{V_m} = \frac{1+|r_R|}{1-|r_R|} \ , \ \bar{z}_M = \rho = \frac{1}{\bar{z}_m}$$ \noindent Adaptación impedancias $ Z(l) = Z^*_g $ \\ Cuarto de onda $$ Z(\lambda/4) = Z_0^2/Z_R = Z_g^* \ , \ \bar{z}_\textrm{ent} = \frac{1}{\bar{z}_R}$$ Media onda $$ Z(\lambda / 2) = Z_R $$ Stub $\bar{Y}(l_1) = 1 + \sigma_l$ $$ \bar{y} = 1/\bar{z} = \rho + j \sigma \ , \ Y = Y_\textrm{línea}(l_1) + j(\sigma_\textrm{línea} + \sigma_\textrm{stub}) $$ $$ l_1 =\frac{\lambda}{ 2\pi} \left\{ \phi_0 - \arccos(-|r_R|) \right\} $$ $$ \textrm{c.c.}\ \cotg \frac{2\pi}{\lambda} l_2 = \frac{-2|r_R|}{\sqrt{1-|r_R|^2}} \ , $$ $$ \textrm{c.o.}\ \tg \frac{2\pi}{\lambda} l_2 = \frac{2|r_R|}{\sqrt{1-|r_R|^2}} $$ \noindent Smith $$ \psi = \phi_0 - 2\beta x \ , \ M \expon{j\psi} = p+j q \ , \ \bar{z} = \frac{1+p+jq}{1-(p+jq)} = r + j x $$ $$ r = \frac{(1+p)(1-p) - q^2}{(1-p)^2 + q^2} \ , \ x = \frac{2q}{(1-p)^2 + q^2} $$ \noindent Atenuación $ 1\dB = 1\Np / 8,686$ $$ \alpha \Delta z = 10 \log \frac{P_0}{P} = 20\log\frac{E_0}{E} \ , $$ $$ E = E_0 \expon{-\alpha(z-z_0)}\ , \ P = P_0 \expon{-2\alpha(z-z_0)} $$ \section{Guias conductores} \noindent Condición de contorno $ \parcial[H_\tau]{n} = 0 $ \\ Impedancia $$ Z(z) = Z_{g1} \frac{Z_{g2} - Z_{g1} \tgh (j k_g z)}{ Z_{g1} - Z_{g2} \tgh(j k_g z)} = Z_{g1} \frac{1 + \Gamma \expon{2j k_{g1} z}}{1- \Gamma \expon{j2 k_{g1} z}}$$ Coef. reflexión $$ \Gamma = \frac{E_r}{E_i} = \frac{Z_{g2} - Z_{g1}}{Z_{g2} + Z_{g1}} $$ \noindent Modos $TE$ $$ E_z = 0 \ , \ E_s = -j \frac{\omega\mu}{k_s^2} \vnabla_s \times \vH_z = - \frac{\omega\mu}{k_s^2 } \vH_s \times \va_z \, $$ $$ \vH_s = - j\frac{k_g}{k_s} \vnabla_s H_z $$ \noindent Modos $TM$ $$ H_z = 0 \ , \\vE_s = -j \frac{k_g}{k_s} \vnabla_s E_z \ , \ \vH_s = \frac{\omega\epsilon}{k_s} \vE_s \times \va_z $$ \noindent {\bf Guia rectangular:} $$ k_s^2 = \pi^2 \left[ \left( \frac{m}{a} \right)^2 + \left( \frac{n}{b} \right)^2 \right] \ , \ \lambda_c = \frac{2\pi}{k_s} = \frac{2}{\sqrt{\left( \frac{m}{a} \right)^2 + \left( \frac{n}{b} \right)^2}} $$ $$ k_x = m\pi/a \ , \ k_y = m\pi/b \ , \ \lambda_{\max} = 2\max(a,b) $$ \noindent G. rectangular, modos $TE_{mn}$ \begin{eqnarray*} H_x & = & j \left(\frac{\lambda_c}{2\pi}\right)^2 k_x H_0 \sin k_x x \cos k_y y\ \expon{-j(k_g z - \omega t)}\\ H_y & = & -j \left(\frac{\lambda_c}{2\pi}\right)^2 k_y H_0 \cos k_x x \sin k_y y\ \expon{-j(k_g z - \omega t)}\\ H_z & = & H_0 \cos k_x x \cos k_y y \expon{-j(k_g z - \omega t)} \\ E_x & = & j\omega\mu H_0 \left(\frac{\lambda_c}{2\pi}\right)^2 k_y \cos k_x x \cos k_y y\ \expon{-j(k_g z - \omega t)}\\ E_y & = & -j\omega\mu H_0 \left(\frac{\lambda_c}{2\pi}\right)^2 k_x \sin k_x x \cos k_y y\ \expon{-j(k_g z - \omega t)}\\ E_z & = & 0 \end{eqnarray*} \noindent G. rectangular, modos $TM_{mn}$ ($\not\exists\ TM_{m0}, TM_{0n}$) \begin{eqnarray*} E_x & = & -j k_g k_x \left(\frac{\lambda_c}{2\pi}\right)^2 E_0 \cos k_x x \sin k_y x\ \exptem\\ E_y & = & -j k_g k_y \left(\frac{\lambda_c}{2\pi}\right)^2 E_0 \sin k_x x \cos k_y y\ \exptem\\ E_z & = & E_0 \sin k_x \sin k_y y\ \expon{-j(k_g z - \omega t)} \\ H_x & = & j \omega\epsilon k_y\left(\frac{\lambda_c}{2\pi}\right)^2E_0\sin k_x x\cos k_y y\ \exptem\\ H_y & = & -j\omega\epsilon k_x\left(\frac{\lambda_c}{2\pi}\right)^2E_0\cos k_x x\sin k_y y\ \exptem\\ H_z & = & 0 \\ \end{eqnarray*} \noindent {\bf Guias planes:} $TEM = TM_0\ (f_c = 0)$ $$ k_y = \frac{n \pi}{b} \ ,\ k_g = \sqrt{\omega^2\epsilon\mu - \left(\frac{n\pi}{b}\right)^2} \ ,\ \lambda_c = \frac{2 b}{n} $$ \noindent G. plana, modos $TM_n$ \begin{eqnarray*} E_x & = & 0 \\ E_y & = & -j k_g \frac{\lambda_c}{2\pi} E_0 \cos k_y y\ \exptem \\ E_z & = & E_0 \sin k_y y\ \exptem \\ H_x & = & -j k_g \frac{\lambda_c}{2\pi} E_0 \cos k_y y\ \exptem \\ H_y & = & 0 \\ H_z & = & 0 \end{eqnarray*} \noindent G. plana, modos $TE_n$ ($\not\exists TE_0$) \begin{eqnarray*} E_x & = & j \omega \mu \frac{\lambda_c}{2\pi} H_0 \sin k_y y\ \exptem \\ E_y & = & 0 \\ E_z & = & 0 \\ H_x & = & 0 \\ H_y & = & -j k_g \frac{\lambda_c}{2\pi} H_0 \sin k_y y\ \exptem \\ H_z & = & H_0 \cos k_y y \exptem \end{eqnarray*} \noindent {\bf Guia cilíndrica circular:} Modos $TE_{ln}$ $$ k_s \Big|_{TE_{ln}} = \frac{Z^l_\textrm{ero} \J'_n}{r_0} \ , \ Z^1_\textrm{ero} J'_1 = 1,84 \ ,\ \lambda_{\min} = 3,41 r_0 $$ \begin{eqnarray*} E_r & = & j \frac{\omega\mu}{r k_s^2}n H_0 \J_n (k_s r) \sin n\varphi\ \exptem \\ E_\varphi & = & j \frac{\omega\mu}{k_s} H_0 \J'_n (k_s r) \cos n\varphi\ \exptem \\ E_z & = & 0 \\ H_r & = & -j \frac{k_g}{k_s} H_0 \J'_n (k_s r) \cos n\varphi\ \exptem\\ H_\varphi & = & j \frac{k_g}{r k_s^2} n H_0 \J_n(k_s r) \sin n\varphi\ \exptem\\ H_z & = & H_0 \J_n ( k_s r ) \cos n\varphi\ \exptem \end{eqnarray*} \noindent Guia cilíndrica circular. Modos $TM_{ln}$ $$ k_s \Big|_{TM_{ln}} = \frac{Z^l_\textrm{ero} \J_n}{r_0} \ , \ \lambda_c = \frac{2\pi}{k_s} $$ \begin{eqnarray*} E_r & = & -j \frac{k_g}{k_s} E_0 \J'_n (k_s r) \cos n\varphi\ \exptem \\ E_\varphi & = & j \frac{k_g}{r k_s^2} n E_0 \J_n (k_s r) \sin n\varphi\ \exptem \\ E_z & = & E_0 \J_n(k_s r) \cos n\varphi \exptem \\ H_r & = & -j \frac{\omega\epsilon}{r k_s^2} n E_0 \J_n (k_s r) \sin n\varphi\ \exptem\\ H_\varphi & = & j \frac{\omega\epsilon}{k_s} E_0 \J'_n(k_s r) \cos n\varphi\ \exptem\\ H_z & = & 0 \end{eqnarray*} \noindent {\bf Guia coaxial:} ($a < b$, $ a k_s , b k_s \ll 1$) Modo $TE_l$ i $TM_l$ $$ k_s \sim \frac{l \pi}{b-a} \ , \ \lambda_c = {2}{l} (b-a) $$ Modo TEM ($\lambda_c \to \infty$)$$ k_g = \omega \sqrt{\epsilon\mu} \ , \ v = 1/ \sqrt{\epsilon\mu} \ , \ Z_h = \eta = \sqrt{\mu/\epsilon}$$ \section{Cavidades resonantes} $$ \deriv[H_\tau]{n} = 0 \ ,\ \vE \| S \ ,\ \vH \bot S $$ Factor calidad $$ Q = \omega \frac{W}{P_\textrm{perd}} \ ,\ W = \frac{1}{2} \int_v (\epsilon E^2 + \mu H^2)\dd v $$ \noindent {\bf Cavidad rectangular:} ($\not\exists TE_{000})$ $$ k_x = \frac{l \pi}{a} \ , \ k_y = \frac{m\pi}{b} \ , \ k_z = \frac{n\pi}{c} \ ,$$ $$ \omega_{lmn} = v k_{lmn} = 2\pi f_{lmn} $$ $$ k_{lmn} = \pi \sqrt{ \left(\frac{l}{a}\right)^2 +\left(\frac{m}{b}\right)^2 +\left(\frac{n}{c}\right)^2 } $$ $$ Q = \frac{\omega \mu V_\textrm{olum}}{2 R_s S_\textrm{total}} \ , \ R_s = \frac{1}{\delta\gamma} = \sqrt{\frac{\omega\mu}{2\gamma}} $$ \begin{eqnarray*} E_x & = & E_x \cos k_x x \sin k_y y \sin k_z z \ \expcav \\ E_y & = & E_y \sin k_x x \cos k_y y \sin k_z z \ \expcav \\ E_z & = & E_z \sin k_x x \sin k_y y \cos k_z z \ \expcav \\ H_x & = & \frac{k_3 E_2 - k_2 E_3}{j\omega\mu} \sin k_x \cos k_y y \cos k_z z \ \expcav \\ H_y & = & \frac{k_1 E_3 - k_1 E_3}{j\omega\mu} \cos k_x \sin k_y y \cos k_z z \ \expcav \\ H_z & = & \frac{k_2 E_1 - k_1 E_2}{j\omega\mu} \cos k_x \cos k_y y \sin k_z z \ \expcav \end{eqnarray*} \noindent {\bf Cavidad cilíndrica circular:} \\ Modos $TM_{mln}$ $$ k_s = \frac{Z^l_\textrm{ero} \J_m}{r_0} \ , \ k_z = \frac{n\pi}{c} \ , \ \omega_{mln} = v \sqrt{ k_z^2 + k_s^2 } $$ $$ Q = \frac{\pi\eta r_0 c}{\lambda R_s (c+r_0)} \ , \ R_s = \frac{1}{\delta\gamma} = \frac{\alpha}{\gamma} $$ \begin{eqnarray*} E_r & = & - k_s k_z \J'_m(k_s r) \cos m\varphi \cos k_z z \ \expcav\\ E_\varphi & = & \frac{k_z m}{r} \J_m(k_s r) \sin m\varphi \sin k_z z \ \expcav \\ E_z & = & k_s^2 \J_m (k_s r) \cos m\varphi \cos k_z z \ \expcav \\ H_r & = & -j \frac{\omega\epsilon m}{r} \J_m(k_s r) \sin m\varphi \cos k_z z\ \expcav \\ H_\varphi & = & -j\omega\epsilon k_s \J'_m(k_s r) \cos m\varphi \cos k_z z \ \expcav \\ H_z & = & 0 \end{eqnarray*} \noindent Modos $TE_{mln}$ ($\not\exists TM_{0l0}$) $$ k_s = \frac{Z^l_\textrm{ero} \J'_m}{r_0} \ , \ k_z = \frac{n\pi}{c} \ , \ \omega_{mln} = v \sqrt{ k_z^2 + k_s^2 } $$ \begin{eqnarray*} E_r & = & j \frac{\omega\mu m}{r} \J_m(k_s r) \sin m\varphi \sin k_z z \ \expcav\\ E_\varphi & = & j \omega \mu k_s \J'_m(k_s r) \cos m\varphi \sin k_z z \ \expcav \\ E_z & = & 0 \\ H_r & = & k_z k_s \J'_m(k_s r) \cos m\varphi \cos k_z z \ \expcav \\ H_\varphi & = & - \frac{k_z m}{r} \J_m(k_s r) \sin m\varphi \cos k_z z \ \expcav \\ H_z & = & k^2_s \J_m(k_s r) \cos m\varphi \sin k_z z \ \expcav \end{eqnarray*} \section{Guias dieléctricas} \noindent {\bf Guia plana:} Modos $TE$ $$ \alpha_c = \omega^2 (\mu_1 \epsilon_1 - \mu_2 \epsilon_2) \ , \ \alpha_c < \frac{\pi}{a} $$ \noindent Modos $TE$, par $$ E_y\Big|_1 = E_1 \cos \alpha x \expon{j(\omega t - k_z z)} \ ,$$ $$E_y\Big|_2 = E_2 \expon{-\beta x} \expon{j(\omega t - k_z z)}$$ $$ H_x\Big|_1 = - \frac{k_z}{\omega\mu} E_y \ ,$$ $$H_z\Big|_1 = -j\frac{\alpha}{\omega\mu} E_1 \sin\alpha x \ \expon{j(\omega t - k_z z)} $$ $$ H_x\Big|_1 = - \frac{k_z}{\omega\mu} E_y \ , $$ $$H_z\Big|_2 = - j \frac{\beta}{\mu \omega} E_y $$ condiciones de propagación $$ \epsilon_1 > \epsilon_2 \ , \tg \alpha \frac{a}{2} = \frac{\beta}{\alpha} = \sqrt{\frac{\omega^(\mu_1 \epsilon_1 - \mu_2 \epsilon_2)}{\alpha^2}-1} $$ \noindent Modos $TE$, impar $$ E_y\Big|_1 = E_1 \sin \alpha x \expon{j(\omega t - k_z z)} \ , $$ $$ H_z\Big|_1 -j\frac{\alpha}{\omega\mu} E_1 \sin\alpha x \ \expon{j(\omega t - k_z z)} $$ condiciones de propagación $$ \epsilon_1 > \epsilon_2 \ , -\cotg \alpha \frac{a}{2} = \sqrt{\frac{\omega^2(\mu_1 \epsilon_1 - \mu_2 \epsilon_2)}{\alpha^2}-1} $$ \noindent Modos $TM$, par $$ H_y\Big|_1 = H_1 \cos \alpha x \expon{j(\omega t - k_z z)} \ ,$$ $$ H_y\Big|_2 = H_2 \expon{-\beta x} \expon{j(\omega t - k_z z)} $$ $$ \vnabla \times \vH = \parcial[\vE]{t}\ , \ \tg\alpha \frac{a}{2} = \frac{\epsilon_1}{\epsilon_2} \frac{\beta}{\alpha} $$ Impar $$ H_y\Big|_1 = H_1 \sin \alpha x \expon{j(\omega t - k_z z)} \ , \ -\cotg\alpha\frac{a}{2} =\frac{\epsilon_1}{\epsilon_2} \frac{\beta}{\alpha} $$ \noindent {\bf Fibras ópticas:} Modo $TM$ $$ E_z\Big|_1 = E_1 \J_n(\alpha r) \cos n\varphi\ \expon{j(\omega t - k_z z)}\ ,$$ $$ \vE_s = -j \frac{k_g}{k_s^2} \vnabla_s \vE_z $$ $$ E_z\Big|_3 = E_2 K_n (\beta r) \cos n\varphi E_\varphi\Big|_1 \ ,$$ $$ \vH = \frac{\omega\epsilon}{ k_g } \va_z \times \vE_s $$ \begin{align*} \frac{\epsilon_1}{\alpha} \frac{\J_{n-1}(\alpha a)-\J_{n+1}(\alpha a)}{\J_n(\alpha a)} = \\ \quad - \frac{\epsilon_2}{\beta} \frac{\K_{n-1}(\beta a) + \K_{n+1}(\beta a)}{\K_n(\beta a)} \end{align*} $$ \alpha^2 + \beta^2 = \omega^2 (\epsilon_1 \mu_1 - \epsilon_2 \mu_2) $$ Corte $\lambda_c = 2\pi / \alpha_c$, un modo $\alpha_c > \pi / a$ \\ Multimodo ($a \gg \lambda$, $\epsilon_1 \gg \epsilon_2$) $$ \frac{\epsilon_1}{\alpha} \tg\left(\alpha a - \frac{n\pi}{2 } - \frac{\pi}{2}\right) = - \frac{\epsilon_2}{\beta} $$ \noindent {\bf Óptica integrada:} $$ v_{ph} = \frac{\omega}{k} = \frac{1}{\sqrt{\epsilon_1 \mu_1}} \left[1- \frac{\alpha^2}{\omega^2\mu_1\epsilon_1}\right]^{-1/2} \qquad\qquad $$ $$ \qquad\qquad \approx v_1 \left[ 1 + \frac{{k'}^2}{8\pi^2}\frac{\lambda^2}{a^2} \right] $$ $$ n_{ef} = \frac{c}{v_1}\frac{v_1}{v_{ph}} \approx \frac{n_1}{1+\frac{\alpha^2}{\omega^2\mu_1\epsilon_1}}$$ \section{Movimiento de cargas} \noindent {\bf Óptica electrostática:} $$ \sqrt{V_1} \sin \alpha_1 = \sqrt{V_2} \sin \alpha_2 \ ,\ \Delta V = 0 \ ,\ E_r = \frac{r}{2} V''$$ Ec. básica $$ \deriv[^2 r]{z^2} + \frac{V'}{2V} \deriv[r]{z} + \frac{r}{4} \frac{V''}{V} = 0 \ ,$$ $$ r(z) = C_1 r_1(z) + C_2 r_2(r) $$ Ec. Helmholtz-Lagrange $$h_a \gamma_a \sqrt{V(a)}= h_b \gamma_b \sqrt{V(b)}\ ,$$ $$ r'_1(b) = \Gamma' \ ,\ r_2(b) = \beta' $$ Lente delgada: $$ \frac{1}{a} + \frac{1}{b} = \frac{1}{f} = \frac{1}{4\sqrt{V_0}} \int^{b'}_{a'} \frac{V''}{V} \dd z \ , $$ $$ \frac{1}{f_i} = \frac{1}{4\sqrt{V_i}} \int^{b'}_{a'} \frac{V''}{V} \dd z $$ $$ \frac{f_a}{a}+\frac{f_b}{b}=1 \ ,\ \frac{f_a}{f_b} = \sqrt{\frac{V_a}{V_b}} $$ \noindent {\bf Campos magnéticos} $$ \rho = \frac{m v_\bot}{q B} \ ,\ T = \frac{2\pi m}{q B} \ ,\ l = \frac{2\pi m}{q B} v_0 \cos\alpha $$ $$ \Delta = BC = 4 \rho \sin^2\frac{\psi}{2} \approx \rho \psi^2 $$ \noindent {\bf Campo axial} $$ B_r = -\frac{r}{2} \parcial[B_z]{z} \ ,$$ $$\Delta\varphi = \frac{e}{2mv} \int_a^b B\dd z \ ,\ \deriv[^2 r]{z^2} + \frac{e B^2}{m V} r = 0$$ Lente delgada $$ \frac{1}{a} + \frac{1}{b} = \frac{e}{8mV} \int_a^b B^2 \dd z = \frac{1}{f} > 0 $$ Lentamente variables (invar. adiabático) $$ \rho \frac{|\nabla\vB|}{B} \ll 1 \ ,\ m_b = \frac{W_\bot}{B} = \frac{1}{2}m v_\bot^2 = cte \ ,\ \frac{\sin^2\alpha}{B} = cte $$ Botella magnética $$ \sqrt{\frac{B_0}{B_m}} < \sin \alpha_0 $$ \noindent {\bf Campo transversal:} Condensador plano\\ ($\vE = -E \va_y$) $$ y = x \tg\beta - \frac{q E}{2 m} \frac{x^2}{(v_0\cos\beta)^2} \ ,$$ $$ (x,y)_\textrm{foc} = \frac{v_0^2}{a} \left( \cotg\beta , \ 1-\frac{1}{\cos^2\beta}\right) $$ Condensador cilíndrico ($r = r_0+u$): $$ E=\alpha/r = \frac{\Delta V}{r \ln (r_2/r_1)} \ ,\ \ddot{u} + \frac{2q\alpha}{ m r_0^2} u =0 \ ,$$ $$ u = \frac{\epsilon r_0}{\sqrt{2}} \sin \sqrt{2}\varphi $$ $$ \varphi_\textrm{foc} = \frac{\pi}{\sqrt{2}} \ , \ \varphi_\textrm{par} = \frac{\pi}{2\sqrt{2}} $$ \noindent {\bf lente cuadrupolar eléctrica:} $$ V = \frac{a}{2} (x^2-y^2) \ ,\ a = \frac{V_k}{h^2} \ ,\ p = \sqrt{\frac{a}{2V_0}} $$ $$ f_x = \frac{1}{p \sin p l} \ ,\ f_y = \frac{-1}{p \sinh p l} $$ Lente delgada (cruzadas) $$ f = f_x \approx -f_y \approx = \frac{1}{p^2 l} \ ,\ F = F_x = F_y = \frac{f^1}{d}$$ \section{Física de Plasmas} \noindent Temperatura $$ T(\eV) = K_B T(K) \ ,\ K_B = 1,38\E{-23} J/K $$ Grado ionización $\chi = \frac{n_e}{n_e+n_g}$ \\ Est. Maxwell-Boltzmann $$ n_e(x) = n_0 \exp\left[ \frac{e V(x)}{k_B T_e} \right] \ ,\ E_c = \frac{1}{2} k T = \frac{1}{2} m v^2 $$ Longitud de Debye $$ \lambda_\textrm{DE} = \sqrt{\frac{\epsilon_0 k T_e}{n_0 e^2}} \ ,\ V = V_0 \exp\left[-\frac{|x|}{ \lambda_\textrm{DE}}\right] $$ Frecuencia de plasma $$ \omega_p^2 = \frac{e^2 n_0}{\epsilon_0 m} \ ,\ \nu > \frac{\omega_p}{2\pi} $$ Sonda de Langmuir plana ($A \gg l_\textrm{vaina}^2$) $$ V_B \ll 0 \ ,\ I = -I_i = -e n_s v_B A\ ,$$ $$ v_B \approx \sqrt{\frac{k T_e}{M}} \ ,\ n_0 \approx \frac{n_S}{0,61} $$ $$ V_B < V_p \ ,\ \qquad\qquad\qquad$$ $$I + I_i = I_e = \frac{1}{4} e n_S \bar{v_e} A \exp \left[ \frac{e (V_B - V_p)}{k T_e}\right] $$ $$ I_e = I_{e,sat} \exp \left[ \frac{e (V_B - V_p)}{k T_e}\right] \ ,\ \bar{v}_e = \sqrt{\frac{8k T-e}{m_e \pi}}$$ \end{document}