\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{\RR}{\mathbb{R}} \newcommand{\MM}{\mathbb{M}} \newcommand{\NN}{\mathbb{N}} \newcommand{\J}{\mathrm{J}} \newcommand{\K}{\mathcal{K}} \newcommand{\QQ}{\mathcal{Q}} \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{\tr}{\mathrm{tr}} \newcommand{\dx}{\dd x} \newcommand{\s}{\mathrm{s}} \newcommand{\MeV}{\mathrm{MeV}} \newcommand{\dr}{\dd r} \newcommand{\dtr}{\dd^3 r} \newcommand{\dt}{\dd t} \newcommand{\cte}{\mathrm{cte}} \newcommand{\ds}{\dd s} % \newcommand{\parcial}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\deriv}[2][]{\frac{\dd #1}{\dd #2}} \newcommand{\E}[1]{\times 10^{#1}} \newcommand{\AAA}{\ensuremath{\,\mathrm{\AA}}} \newcommand{\cm}{\mathrm{cm}} % \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}} \newcommand{\vL}{\vec{L}} \newcommand{\vr}{\vec{r}} \newcommand{\vp}{\vec{p}} \newcommand{\vsigma}{\vec{\sigma}} \newcommand{\vR}{\vec{R}} \newcommand{\vA}{\vec{A}} \newcommand{\vepsilon}{\vec{\varepsilon}} \newcommand{\vJ}{\vec{J}} \newcommand{\vk}{\vec{k}} \newcommand{\vv}{\vec{v}} \newcommand{\vX}{\vec{X}} \newcommand{\vY}{\vec{Y}} \newcommand{\ve}{\vec{e}} \newcommand{\vx}{\vec{x}} \newcommand{\vu}{\vec{u}} \newcommand{\vy}{\vec{y}} \newcommand{\vxi}{\vec{\xi}} \newcommand{\vbeta}{\vec{\beta}} \newcommand{\vn}{\vec{n}} \newcommand{\vomega}{\vec{\omega}}\newcommand{\vK}{\vec{K}} \newcommand{\vF}{\vec{F}} \newcommand{\vz}{\vec{z}} \newcommand{\vm}{\vec{m} } \newcommand{\vq}{\vec{q}} \newcommand{\parcte}[3]{ \left( \parcial{#1}{#2} \right)_{\!#3} } \newcommand{\sep}{\vspace{1mm} \hrule \vspace{1mm} \noindent} \newcommand{\inv}[1]{{#1}^{-1}} % \newcommand{\la}{\left<} \newcommand{\ra}{\right>} \newcommand{\bok}[3]{ \la #1 \right| #2 \left| #3 \ra } \newcommand{\ketbra}[2]{ \left| #1 \ra\la #2 \right| } \newcommand{\commut}[2]{\left[ #1 , #2 \right]} \author{pod} \title{Electrodinàmica clàssica} \date{Otoño, 2001} \begin{document} \maketitle \noindent $ \vec{r'}_\bot =\vec{r}_\bot \: , \qquad r'_\| =\gamma ( r_\| - v t ) \: , \qquad t' =\gamma(t-\frac{1}{c}\vr\vbeta)$\\ $ x' = x \cosh\xi - c t \sinh \xi \: , \qquad c t' = ct \cosh\xi - x \sinh\xi $ \\ $ \vec{r'} = \vec{r} + \left( \frac{\gamma-1}{v^2} \vec{r}\cdot\vec{v} - \gamma t \right) \vec{v} \: , \qquad t' = \gamma \left( t - \frac{\vec{v}\cdot\vec{r}}{c^2} \right) $ \\ $\begin{array}{rll} i,j \leq 3 & L^{i'}_{\,j} = & \delta^{i'}_{\,j} + \frac{\gamma-1}{\beta^2} \beta^{i'} \beta_{\,j} \\ i \leq 3 & L^{i'}_{\,4} = & L^{4'}_{\,i} = - \gamma \beta^i \\ & L^{4'}_{\,4} = & \gamma \end{array} $ \sep Contracció longituds: $ \Delta\vec{x'}_\bot = \Delta\vec{x}_\bot \; , \qquad \Delta x'_\| = \gamma \Delta x_\| $ \\ $ l = l' \sqrt{1-\left(\frac{v}{c}\right)^2 \cos^2\alpha'} = \frac{l'}{\sqrt{\gamma^2 \cos^2\alpha+\sin^2\alpha}} \ ,$ $ \tg\alpha' = \frac{1}{\gamma} \tg\alpha $ \\ Temps: $ \Delta t = \gamma \Delta t' = \gamma \tau $ \sep Velocitats: $ \vec{u'}_\bot = \frac{\vec{u}_\bot}{\gamma\left(1-\frac{\vec{v}\cdot\vec{u}}{c^2}\right)} \: , \quad u'_\| = \frac{u_\| - v}{1-\frac{\vec{u}\cdot\vec{v}}{c^2}} \ , \quad \phi' = \phi - \xi $ \\ $ \vec{u'} = \frac{\vec{u}+\frac{\gamma-1}{v^2}\vec{u}\cdot\vec{v}-\gamma\vec{v}}{\gamma\left(1-\frac{\vec{v}\cdot\vec{u}} {c^2}\right)} \ , \quad \tg\theta' = \frac{u\sin\theta}{\gamma \big( u \cos\theta - v \big)} \ , \ v = c\, \tgh \xi$ \\ $ u' = \frac{\sqrt{u^2+v^2-2uv\cos\theta - \left(\frac{vu}{c}\sin\theta\right)^2}} {1-\frac{\vec{v}\cdot\vec{u}}{c^2}} \ , \quad \gamma'_u = \gamma_u \gamma_v \left( 1 - \frac{uv}{c^2} \right) $ \\ \sep Ones planes: $\vn = \vk / 2\pi = \hatn / \lambda$ \\ $ \vec{n'} = \vec{n} + \frac{\gamma-1}{c^2}(\vec{v}\cdot\vec{n})\vec{v} - \frac{\gamma\nu}{c^2}\vec{v} \ , \quad \nu' = \gamma \big( \nu - \vec{v}\cdot\vec{n} \big) $ \\ Doppler: $ \nu_R = {\nu_E}\Big/\, {\gamma\left(1-\frac{\vec{v}\cdot\hat{n}_R}{c_{1,R}}\right)} $ \\ D. rad (lluny): $ \nu_R = \nu_E \sqrt{\frac{1-v/c}{1+v/c}} $ , tg: $\nu_R = {\nu_E} / {\gamma}$\\ Aberració: $\sin\alpha' = \frac{\sin\alpha}{\gamma ( 1 + \beta \cos\alpha)} \ , \qquad$ $\cos\alpha' = \frac{\cos\alpha + \beta}{1+\beta \cos\alpha}$ \\ Fresnel: $c'_1 =c_1 \frac{1-\frac{v}{c_1}\cos\alpha}{\sqrt{1-2\cos\alpha \frac{v c_1}{c^2}- \left(\frac{v}{c}\sin\alpha\right)^2}} \approx c_1 - v \left(1-\frac{1}{r_1^2}\right)$ \sep Quadrivectors: $ v^{\mu'} = \Lambda^{\mu'}_{\,\nu} v^\nu \ , \ \Lambda = L\cdot R $ \\ t. propi $\Delta\tau = \gamma^{-1}\Delta t = \frac{1}{c} \int^\lambda_{\lambda_0} \dd\lambda \sqrt{-\dot{X}^\mu \dot{X}_\mu}$ \\ 4-vel: $u^\mu = \frac{\dd X^\mu}{\dd\tau}=\gamma(\vec{\omega},c) \ , \ (u)^2 = -c^2 \ , \ \vec{\beta} = \frac{\vec{\omega}}{c} = \frac{\vec{u}}{u^4}$ \\ 4-acc: $b^\mu = \frac{\dd u^\mu}{\dd\tau} \ , \ b^4 = \gamma^4 \frac{\vec{\omega}\cdot\vec{a}}{c} = \frac{\vec{b}\cdot \vec{\omega}}{c} = \frac{\vec{u}\cdot \vec{b}}{u^4} \ , \ b^\mu u_\mu = 0$ \\ $\vec{b} = \gamma^2\vec{a} + \gamma^4\left(\frac{\vec{\omega}\cdot\vec{a}}{c^2}\right)\vec{\omega} = \gamma^4\big(\vec{a} + \frac{1}{c^2} \vec{\omega}\times(\vec{a}\times\vec{\omega})\big)$ \\ $\frac{\dd\gamma}{\dd t} = \frac{1}{c^2}\vec{\omega}\cdot\vec{a}\gamma^3 \ , \quad b^\mu b_\mu = \gamma^4 \left( \vec{a}^2 + \gamma^4 \frac{\vec{\omega}\cdot\vec{a}}{c^2}\right) $ \\ \sep 4-p: $ p^\mu = m u^\mu = m \gamma ( \vomega , c ) \ , \ p^\mu p_\mu = - m^2 c^2 \ , \ E = c p^4$ \\ $ E^2 = (c\vp)^2+(mc^2)^2 \ ,\ \vbeta = \vp / p^4 \ , \ c |\vp| = \sqrt{T(T+2mc^2)} $ \\ $\vp_{cm} = 0 \ , \ \vv_{cm} = c \vp / p^4 \ , \qquad \vK = h\nu/c (\hatn , 1)$ \sep Dinàmica: $f^\mu = \deriv{\tau} p^\mu = \gamma ( \vF , \vv \vF / c ) $ \\ $\vec{F}= m\gamma\vec{a} + m\gamma^3\frac{\vec{v}\cdot\vec{a}}{c^2}\vec{v} \longrightarrow_{(\vv\bot\va)} m \gamma^3 \va $ \sep $ L = -mc^2\sqrt{1-\frac{v^2}{c^2}} \: , \ H = c\sqrt{m^2c^2 + \vec{p}^2}$ \\ Acció $S = \int L\dd t = -mc^2\int\dd\tau$ \sep\sep $\vnabla\cdot\vB = 0 \ ,\quad \vnabla\times\vE+\partial_t\vB = 0 \ ,\quad \vD =\epsilon_0\vE+\vec{P} = \epsilon \vE $ \\ $\vnabla\cdot\vD =\rho\ ,\quad \vnabla\times\vH-\partial_t\vD = \vec{\jmath} \ ,\quad \vH =\frac{\vB}{\mu_0}-\vec{M} = \mu \vB $ \\ 4-corrent: $ j^\mu = (\vj , c\rho) \ ,\quad \textrm{Continuïtat:} \parcial{x^\mu} j^\mu = 0 $ \\ Força de Lorentz $ f^\mu = q F^\mu_{\ \nu} u^\nu$ \\ Tensor electromagnètic $F_{i4} = - F_{4i} = E_i/c $ \\ $ F_{ij} = \epsilon_{ijk} B^k \ ;\quad E_i = c F_{i4} = - c F_{4i} \ , \quad B^k = \frac{1}{2}\epsilon^{ijk} F_{ij}$ \\ Camp E: $ E'_\| = E_\| \ , \ \vE'_\bot = \gamma \left( \vE_\bot + \vv \times \vB \right) $ \\ Camp B: $ B'_\| = B_\| \ , \ \vB'_\bot = \gamma \left( \vB_\bot - \frac{1}{c^2}\vv \times \vE \right)$ \\ $\to \vE' = \vE + \gamma(\vv \times \vB) + \frac{1-\gamma}{v^2} \vv \times (\vv \times \vE) $ \\ $\to \vB' = \vB - \frac{\gamma}{c^2}(\vv \times \vE) + \frac{1-\gamma}{v^2} \vv \times (\vv \times \vB) $ \\ Invariants $ \vE\cdot\vB \qquad \ , \quad \vB^2 - \frac{1}{c^2}\vE^2 $ \sep Maxwell $ \partial_\rho F^{\nu\rho} = \mu_0 j^\nu \ , \quad \partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta} = 0 $ \\ 4-potecial $A^\nu = (\vec{A},\phi/c) \ ,\quad F_{\mu\nu} =\partial_\mu A_\nu - \partial_\nu A_\mu $ \\ $ \to $ Maxwell $\partial^\nu\partial_\rho A^\rho - \partial_\rho\partial^\rho A^\nu = \mu_0 j^\nu$ \\ Galga Coulomb $\vnabla\cdot\vA = 0$ \\ Lorentz $\partial_\mu A^\mu = 0$ \sep Energía $\vnabla\cdot\vS + \parcial[U]{t} = - \vj \cdot \vE \ ,\quad \vS = \vE\times\vH$ \\ Energia-impuls $\theta^{\mu\alpha} = \epsilon_0 c^2 \left( F^{\nu\mu} F_\nu^\alpha - \frac{\eta}{4} F^{\sigma\rho} F_{\sigma\rho} \right)$ \\ $\theta^{44} = U \: , \qquad \theta^{i4} = \theta^{4i} = S_i/c $ \\ $\qquad \theta^{ij} = -\epsilon_0 \left[ E^i E^j + c^2 B^i B^j - \frac{1}{2}\delta^{ij}(\vE^2+c^2\vB^2)\right] =-T^{ij}$\\ Conservació $\partial_\mu \theta^{\mu\alpha} = j_\nu F^{\nu\alpha}$ \\ Moment $ \deriv{t}\int_v (\rho E_i+(\vj \times \vB)_i)\dd v = -\frac{1}{c^2} \deriv{t}\int_v S_i \dd v + \int_A T^{ij} \hat{n}_i \dd^2 A $ \sep $ L(\vx,t) = - m c^2 \sqrt{1-\frac{v^2}{c^2}} + q (\vv \times \vA) - q \phi $ \\ $ P_i = \parcial[L]{v^i} = m\gamma v_i + qA_i$ \\ $ H (P,\vx,t) = \sqrt{m^2c^4 + c^2(\vP-q\vA)^2} + q\phi $ \sep\sep Telegrafia $\nabla^2 \vE - \mu\epsilon \parcial[^2]{t^2}\vE - \sigma \parcial{t} \vE = 0 $ \\ si $\vE = \vE(\vr) \expon{-i\omega t} \to \nabla^2 \vE + \left(1+\frac{\sigma}{\mu\epsilon}\right) \mu\epsilon\vE = 0$\\ Transversalitat $ c_1 \vB = \hatn \times \vE $ \\ Temps retardat $ \tau_r = t- \frac{1}{c} |\vx-\vy| \ ,\ c^2 (t-t_r)^2 -(\vx-\vz_r)^2=0$ \sep Radiació multipolar \\ $ A^\mu(\vx, t) = \frac{\mu_0}{4\pi} \int_{R^3} \dd^3\vy \frac{j^\mu\left(\vy, t \mp \frac{|\vy - \vx|}{c}\right)}{|\vx - \vy|} \theta\left(\pm t - \frac{|\vx - \vy|}{c} \right) \pm $\\ $ \pm \frac{1}{4\pi}\int_{R^3} \frac{\dd^3\vy}{|\vx - \vy|} \left[\frac{1}{c} \partial_t A^\mu(\vy,0) \delta( c t \mp |\vx - \vy| ) +\right.$\\ $\left. + A^\mu(\vy,0) \delta'(c t \mp |\vx - \vy|) \right] $ \\ Part. lliures a $t\to - \infty$ \\ $ \to A^\mu = \frac{\mu_0}{4\pi} \int_{R^3} \dd^3\vy \frac{j^\mu \left(\vy, t \mp \frac{|\vy - \vx|}{c}\right)}{|\vx - \vy|} $ \sep Font localitzada. Camps pròxims \\ $ \vE_I = \frac{1}{4\pi\epsilon_0} \int_{\RR^3} \dd^3\vy \frac{\rho(\vy)}{|x-y|^2} \expon{-i k|\vx-\vy|} $ \\ $ \vB_I =-\frac{\mu_0}{4\pi} \int_{\RR^3} \dd^3\vy \frac{\hatn\times\vj(\vy)}{|\vx-\vy|^2}\expon{-i k|\vx-\vy|} $ \\ Camps de radiació. Camps de radiació \\ $ \vE_{II} = \frac{i k}{4\pi\epsilon_0} \int_{\RR^3} \dd^3\vy \frac{\expon{-i k|\vx-\vy|}}{|\vx-\vy|} ( \rho \hatn - \frac{1}{c} \vj(\vy)) = i \omega \hat{r} \times (\hat{r}\times\vA $) \\ $ \vB_{II} =-\frac{i k \mu_0}{4\pi} \int_{\RR^3} \dd^3\vy \frac{\expon{-i k|\vx-\vy|}}{|\vx-\vy|}\hatn\times\vj(\vy)) = - \frac{i \omega}{c} \hat{r} \times \vA $ \\ $ \vA = \frac{\mu_0}{4\pi} \frac{\expon{-i k r}}{r} \int_V \dd^3\vy \expon{i k \hat{r}\vy} \vj(\vy) $\\ \sep Radiació dipolar magnètica ($\expon{i k \hat{r} \vy} \approx 1$)\\ $ \vA = i \frac{\mu_0 \omega}{4\pi} \frac{\expon{-i k r}}{r} \vp \ ,\quad \vp = \int_v \dd^3\vr \rho(\vy) \vy $ \\ Moment dipolar magnètic (part antisimétrica) \\ $ \vA = -\frac{ik\mu_0}{4\pi} \frac{\expon{-i k r}}{r} \hat{r} \times \vm \ ,\ \vm = \int_v \dd^3\vy\ \vM = \int_v \dd^3\vy\ \frac{1}{2} \vy \times \vj(\vy) $ \\ Moment quadrupolar elèctric (part simètrica) \\ $ \vB = -i\frac{c k^2}{24\pi}\mu_0 \frac{\expon{-ikr}}{r} \hat{r}\times\vq \ ,\ \vE = -i\frac{c^2k^3}{24\pi} \mu_0 \frac{\expon{-ikr}}{r} (\hat{r}\times\vq)\times\hat{r} $\\ $ a^{ij} = \int_v \dd^3 \vy ( 3 y^i y^j - \vy^2 \delta^{ij} ) \rho(\vy) \ ,\ q^j = \hat{r}_i a^{ij} $ \\ $\hat{r} \times \vq = 3 \hat{r} \times \int_v \dd^3 \vy (\vy \hat{r}) \vy \rho(\vy)$ \sep Partícules en moviment $j^\mu = q \delta(\vy-\vz(t)) (\vv(t),c) \ ,\ R^\rho = x^\rho - y^\rho \ ,\ \rho_r = |\vx-\vz|$ \\ $ A^\nu(\vx,t_r) = -\frac{\mu_0c}{4\pi} q \frac{\dot{z}^nu}{(\vx-\vz)^\rho \dot{z}_\rho} $\\ $ F^{\mu\nu}_I =-\frac{q}{4\pi\epsilon_0 c} \frac{\dot{z}^\mu r^\nu - \dot{z}^\nu r^\mu}{\rho_r^2} $ \\ $ F^{\mu\nu}_{II}=-\frac{q}{4\pi\epsilon_0 c} \left\{ \frac{\ddot{z}^\mu r^\nu - \ddot{z}^\nu r^\mu}{c \rho_r} + \frac{\dot{z}^\mu r^\nu - \dot{z}^\nu r^\mu}{c \rho_r} (r \ddot{z}) \right\}$ \\ a $t, \tau_r \to -\infty \ ,\ \ddot{z} = 0 \to F_{II} = 0 $ \\ $$ \vE_{II} = \frac{q}{4\pi\epsilon_0 c |\vx-\vz| (1-\hatn\vbeta)^3} \hatn \times (\hatn-\vbeta)\times\dot{\vbeta} $$ $$ \vB_{II} = \frac{q}{4\pi\epsilon_0 c^3 |\vx-\vz| (1-\hatn\vbeta)} \left( (1-\vbeta\hatn)(\va\times\hatn) \right. $$ $$+ \left. (\va\hatn) (\vbeta\times\hatn) \right) = \frac{1}{c} \hatn \times \vE_{II} $$ 4-moment radiat $ \deriv{\tau} p^\mu = \frac{q^2}{6\pi\epsilon_0 c^5} (\ddot{z}^\nu\ddot{z}_\nu) \dot{z}^\mu $ \\ Energía: $ \deriv{t} \varepsilon = \frac{q \gamma^6}{6\pi\epsilon c^3} ( \va^2 - (\vbeta\times\va)^2 ) $ \\ formula de Larmor $ \deriv{t}\varepsilon = \frac{q^2}{6\pi\epsilon_0 c^3} \va^2 $ \sep Distribució angular $ \deriv{t} \varepsilon = \vS \dd^2 \vA $ \\ observador $\deriv[^2 \epsilon]{t \dd^2\Omega} = \frac{q^2}{16\pi^2 \epsilon_0 c} \frac{1}{(1-\hatn\vbeta)^6} \left( \hatn \times \left( (\hatn-\vbeta) \times \dot{\vbeta} \right) \right)^2$ càrrega (t. retardat) $$\deriv[^2 \epsilon]{t \dd^2\Omega} = \frac{q^2}{16\pi^2 \epsilon_0 c} \frac{1}{(1-\hatn\vbeta)^5} \left( \hatn \times \left( (\hatn-\vbeta) \times \dot{\vbeta} \right) \right)^2$$ \sep Accelerador lineal $\deriv{t} \varepsilon = \frac{q^2 \gamma^6}{6\pi\epsilon_0 c^3} \va^2 \ ,\quad \deriv{t}\varepsilon = m \gamma^3 \vv \va$ \\ $ \deriv[\varepsilon_{rad}]{\varepsilon}=\frac{q^2}{6\pi\epsilon_0 c^3 m^2 v} \deriv[\varepsilon]{x} \ , \quad \dd\varepsilon = m\gamma^3 v a \dt = m \gamma^3 a \dx$ \\ Distribució angular $ \deriv[^2W]{\Omega^2} = \frac{q^2 \dot{v}^2}{16\pi^2 \epsilon_0 c^3} \frac{\sin^2\theta}{(1-\beta\cos\theta)^5} $ \\ $ f(\theta) = \frac{3(1-\beta^2)}{8\pi} \frac{\sin^2\theta}{(1-\beta \cos\theta)^5} \ ,\quad \cos\theta_\textrm{max} = \frac{-1+\sqrt{1+15\beta^2}}{3\beta}$ \\ resultats $ \deriv[^2W]{\Omega^2} \propto \gamma^8 \ ,\quad \Delta\theta \propto \gamma^{-1} $ \\ triem $\vbeta = \beta \hat{k} \ , \quad \hatn = ( \sin\theta \cos\varphi , \sin\theta \cos\varphi , \sin\varphi)$ \sep Accelerador circular $\deriv{t}\omega = \frac{q^2 \gamma^4}{6\pi\epsilon_0 c^3} a^2$ \\ $ \frac{\Delta \varepsilon_{rad,1rev}}{\varepsilon} = \frac{\Delta \varepsilon}{m\gamma c^2} = \frac{q^2 \epsilon^2 \beta^3}{3 \epsilon_0 R m^4 c^8} $ \\ $ \deriv[^2W]{\Omega^2} = \frac{q^2 \dot{\beta}^2}{16\pi^2 \epsilon_0 c (1-\beta\cos\theta)^5} \left[ (1-\beta\cos\theta)^2 - (1-\beta^2) \sin^\theta \cos^\varphi \right] $ \\ $ \rho = \frac{3}{8\pi\gamma^4(1-\beta\cos\theta)^3} \left[ 1 - \frac{\sin^2\theta \cos^2\varphi}{\gamma^2 (1-\beta \cos\theta)^2} \right] $ \\ $\textrm{si} {\gamma\uparrow} \to \rho \approx \frac{1}{(1+\gamma^2\theta^2)} \left( 1-\gamma^2 \frac{\theta^2 \cos^2\varphi}{(1+\gamma^2\theta^2)^2} \right)$ \\ triem $\vbeta = \beta \hat{k} \ ,\quad \dot{\vbeta} = \dot{\beta} \hat{\imath} $ \sep Sincrotó $\delta t = 2 \delta \theta R (1/\beta-1= \approx \gamma^{-3} / \omega_0 $ \\ Radi de radiació $ \delta \omega \approx \omega_0 \gamma^3 $ \end{document}