\documentclass[11pt,a4paper,twocolumn,catalan]{article}
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\newcommand{\expcav}{\expon{j\omega t}}
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\newcommand{\cotg}{\mathrm{cotg}}	\newcommand{\dB}{\mathrm{dB}}		\newcommand{\Np}{\mathrm{Np}}
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\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}