From 6934423fd5b1428d7f340ba7cb3952caa23cf783 Mon Sep 17 00:00:00 2001 From: Juergen Gilg Date: Tue, 26 Jun 2012 22:39:44 +0200 Subject: [PATCH] =?utf8?q?...=20potentiel=20coulombien=20--=20l'exp=C3=A9r?= =?utf8?q?iment=20de=20RUTHERFORD=20...?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- gravitation/LISTE.txt | 2 + .../potentiel_coulombien_distiller.pdf | Bin 0 -> 79922 bytes .../potentiel_coulombien_distiller.tex | 144 ++++++++++++++++++ 3 files changed, 146 insertions(+) create mode 100644 gravitation/potentiel_coulombien_distiller.pdf create mode 100644 gravitation/potentiel_coulombien_distiller.tex diff --git a/gravitation/LISTE.txt b/gravitation/LISTE.txt index 5753434..4f8f8e3 100644 --- a/gravitation/LISTE.txt +++ b/gravitation/LISTE.txt @@ -6,6 +6,8 @@ gravitation_02.pdf == Exemples avec Adobe Distiller gravitation_01_distiller.tex gravitation_01_distiller.pdf +potentiel_coulombien_distiller.tex 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+\def\datRoot{C:/Users/J\"{u}rgen/Desktop/gravitation/} + +\begin{document} + +\section{La d\'{e}couverte de la diffusion des particules $\alpha$ par le noyau} +\subsection{L'exp\'{e}riment d'Ernest Rutherford en 1909} + +Rutherford bombarde des particules $\alpha$ (noyau d'h\'{e}lium : 2 neutrons et 2 protons) au travers d'une feuille d'or. +\begin{center} +\begin{pspicture}(-6,-4)(6,5) +\psset{lightsrc=10 -20 50,viewpoint=10 -10 10,Decran=20} +\psSolid[object=cube,a=1,fillcolor=orange,incolor=red!20](0,7.5,0) +\psLineIIID[linecolor=red](0,7,0)(0,0,0) +\psLineIIID[linecolor=red](0,0,0)(45 sin 4 mul,45 cos 4 mul,0) +\psPolygonIIID[fillcolor=yellow,fillstyle=solid,opacity=0.5](-1.2,0,-1.2)(1.2,0,-1.2)(1.2,0,1.2)(-1.2,0,1.2) +\psLineIIID[linecolor=red](0,0,0)(140 sin 4 mul,140 cos 4 mul,0) +\psLineIIID[linecolor=red](0,0,0)(160 sin 4 mul,160 cos 4 mul,0) +\defFunction{G}(t) + {t sin 4 mul} + {t cos 4 mul} + {-0.5} +\psSolid[object=cylindre, + range=10 350, + h=1, + function=G, + axe=0 0 1, %% valeur par d\'{e}faut + ngrid=4 108, + incolor=green!20, + fillcolor=green!20, + linecolor=gray!50, + opacity=0.7] +\psPoint(0,7.5,1){emit} +\uput[u](emit){Emitteur des particules $\alpha$} +\psPoint(45 sin 4.3 mul,45 cos 4.3 mul,0){fluor} +\uput[r](fluor){\'{E}cran fluorescent} +\psSolid[object=plan,definition=equation,args={[0 -1 0 0]},name=pro,action=none] +\psProjection[object=texte,text=Feuille d'or,plan=pro](0,0.7) +\end{pspicture} +\end{center} + +\def\eqRuth{% +y[2]|y[3]|COU*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|COU*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% + +\section{Le potentiel coulombien -- donnant des trajectoires hyperboles} + +Prenons un rep\`{e}re cart\'{e}sien. La masse $m_0$ d'un particule $\alpha$ a les coordonn\'{e}s $(x/y)$ et la vitesse $v=\sqrt{\dot{x}^2+\dot{y}^2}$. Le particule d'or est mis dans l'origine $O(0/0)$. Soit $r=\sqrt{x^2+y^2}$ la distance entre l'origine et la masse $m_0$. + +Pour l'en\'{e}rgie cin\'{e}tique on re\c{c}oit : +\[ +T(\dot{x},\dot{y})=\frac{1}{2}m_0v^2=\frac{1}{2}m_0(\dot{x}^2+\dot{y}^2) +\] +Le potentiel coulombien, ce qui est conservatif : +\[ +U(x,y)=\frac{Z_1Z_2e_0^2}{4\pi\epsilon_0 \sqrt{x^2+y^2}}=\frac{k}{\sqrt{x^2+y^2}}\quad \text{dont} \quad k=\frac{Z_1Z_2e_0^2}{4\pi\epsilon_0} +\] +Le Lagrangien est : +\[ +L(x,y,\dot{x},\dot{y})=T-U=\frac{1}{2}m_0(\dot{x}^2+\dot{y}^2)-\frac{k}{\sqrt{x^2+y^2}} +\] +Pour $x,\,\dot{x}$ : +\begin{align*} +\frac{\partial L}{\partial x}&=k\frac{x}{\left(\sqrt{x^2+y^2}\right)^3}\\ +\frac{\partial L}{\partial \dot{x}}&=m_0\dot{x}\\ +\frac{\text{d}}{\text{d} t}\frac{\partial L}{\partial \dot{x}}&=m_0\ddot{x} +\end{align*} +Alors +\[ +m_0\ddot{x}-k\frac{x}{\left(\sqrt{x^2+y^2}\right)^3}=0 +\] +Division par $m_0$ +\[ +\psframebox[linestyle=none,fillstyle=solid,fillcolor=cyan!20]{% +\ddot{x}=\dfrac{k}{m_0}\dfrac{x}{\left(\sqrt{x^2+y^2}\right)^3} +} +\] +Le Lagrangien est sym\'{e}trique en $x$ et $y$, alors : +\[ +\psframebox[linestyle=none,fillstyle=solid,fillcolor=cyan!20]{% +\ddot{y}=\dfrac{k}{m_0}\dfrac{y}{\left(\sqrt{x^2+y^2}\right)^3} +} +\] + +\newpage +\section{Les trajectoires des particules $\alpha$} + +L'exp\'{e}riment original donne les param\`{e}tres suivants : +\begin{align*} +m_0&=6,64\cdot 10^{-27}\,\text{kg}\\ +e_0&=1,6\cdot10^{-19}\,\text{C}\\ +\varepsilon_0&=8,85\cdot10^{-12}\,\frac{\text{A\,s}}{\text{kg\,m}^3}\\ +Z_1&=2\\ +Z_2&=79\\ +v_{0x}&=2,1\cdot 10^7\,\text{m\,s}^{-1} +\end{align*} + +\begin{center} +\begin{pspicture*}(-6,-9)(6,9) +%\psset{unit=2}% +\psgrid[subgriddiv=2,gridcolor=lightgray,subgridcolor=lightgray,gridlabels=8pt](-6,-9)(6,9) +\pscircle*[linecolor=gray](0,0){0.3} +\psaxes[ticks=none,labels=none]{->}(0,0)(-5,-8)(5,8) +\uput[0](5,0){$x$} +\uput[90](0,8){$y$} +\multido{\rA=-5+0.25}{41}{% +\pstVerb{% + /Pi 3.1415 def + /m0 6.64e-27 def + /Z1 2 def + /Z2 79 def + /e0 1.6e-19 def + /epsil 8.85e-12 def + /COU1 Z1 Z2 mul e0 mul e0 mul 4 div Pi div epsil div m0 div def + /COU 7.5 def + /x0 6 neg def + /y0 \rA\space def +% /v0x 2.1e7 def + /v0x 2 def + /v0y 0 def + /r1 COU 2 mul v0x 2 exp div def +}% +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi +% ,saveData,filename=XiYi.dat +]{0}{43}{x0 y0 v0x v0y}{\eqRuth}% +\listplot[linecolor=red]{XiYi aload pop} +} + +\end{pspicture*} +\end{center} + + +\end{document} -- 2.20.1