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\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}
