{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Times" 1 12 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "MYTEXT" -1 256 "Times" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "question" -1 257 "Comic Sans MS" 1 12 255 0 255 1 1 2 2 0 0 2 0 0 0 1 }{CSTYLE "reponse" -1 258 "Comic Sans MS" 1 12 128 0 128 1 0 2 2 0 0 2 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 128 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 128 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 128 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "T imes" 1 18 255 0 255 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0 " -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 " Times" 1 12 255 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 255 0 255 1 2 1 1 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT -1 10 "G PHILIPPE" }}{PARA 18 " " 0 "" {TEXT -1 41 "Sph\350re conductrice dans un champ uniforme" }} {PARA 19 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "Th\350me: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 17 "On utilise Maple " }}{PARA 0 "" 0 "" {TEXT -1 52 " pour repr\351senter graphiquement des \351quipotentielles " }}{PARA 0 "" 0 "" {TEXT -1 29 "et les champs correspondants " }}{PARA 0 "" 0 "" {TEXT -1 45 "dans un probl\350me classique d'\351lectrostatique." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Pour \351 viter l'appel aux instructions utilisant les coordonn\351es sph\351riq ues, " }}{PARA 0 "" 0 "" {TEXT -1 69 "on a fait le choix des coordonn \351es cart\351siennes dans tout le travail." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Enfin, on a fait, bien que le p robl\350me soit plan, " }}{PARA 0 "" 0 "" {TEXT -1 34 "le choix de rep r\351sentations en 3d " }}{PARA 0 "" 0 "" {TEXT -1 48 "plus rapides \+ \340 tracer que les repr\351sentations 2d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Programme" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Champ uniforme" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (plots):" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 "Des constan tes pour la suite" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 29 "C\364t\351 du domaine carr\351 \351tudi\351:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L:=1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Fen\352" }{TEXT 260 0 "" }{TEXT -1 46 "tre de trac\351 dan s le plan xy pour le potentiel" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "FENETRE_POTENTIEL:=(x=-L/2.. L/2,y=-L/2..L/2);\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 0 "" }{TEXT -1 181 "Volume de trac\351 en z \+ quasinul pour le champ. On place cette zone au dessus du trac\351 du \+ potentiel d'o\371 le Vmax correspondant \340 la valeur num\351rique de Vmax sur le trac\351 du potentiel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "FENETRE_CHAMP:=(x=-L/2..L/ 2,y=-L/2..L/2,z=Vmax..Vmax+0.01);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Le maillage du domaine \351tud i\351 dans le plan xy correspond \340 un r\351seau de N*N n\234uds r \351guli\350rement r\351partis." }}{PARA 0 "" 0 "" {TEXT -1 65 "La val eur de N est diff\351rente pour le potentiel et pour le champ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Np:=100; Nc:=20;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 "G" }{TEXT 259 0 "" }{TEXT -1 51 "rille de \+ calcul dans le plan xOy pour le potentiel:" }{MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "GRILLE_POTENTIEL:=(grid=[Np,Np]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "G" }{TEXT 262 0 "" }{TEXT -1 47 "rille de calcul dans le plan z=0 pour le champ:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "GRILLE _CHAMP:=(grid=[Nc,Nc,2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Nombre de lignes \351quipotentielles dess in\351es:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "NBRE_EQUIP:=(contours=11);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 36 "Expressions du champ et du potentiel" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Le champ \+ \351tudi\351 est uniforme selon uz. " }}{PARA 0 "" 0 "" {TEXT -1 32 "O n a vecteur(E)= E0 vecteur(ux)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "On part ici de l'expression du potentiel \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 "pour retrouver le champ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "On introduit le potentiel sous forme d'une fonc tion de x, y. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "V:=(x,y)-> -x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "On obtient le champ " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Ex:=(x,y)->-diff(V(x,y),x);Ex(x,y);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "Ey:=(x,y)->-diff(V(x,y),y);Ey(x,y);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Trac\351 du potentiel" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "On va tracer le po tentiel dans le plan xOy " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "On trace \+ donc ce potentiel en 3D " }}{PARA 0 "" 0 "" {TEXT -1 42 "On repr\351se nte donc V en fonction de x et z" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot3d( V(x,y) ,FENETRE_POTE NTIEL,GRILLE_POTENTIEL,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Remarquer que sur 1 (unit\351 \+ de longueur) selon z, le potentiel d\351croit donc de 1 (unit\351 de p otentiel)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "En cliquant avec la souri s sur le dessin, on peut faire tourner la repr\351sentation." }}{PARA 0 "" 0 "" {TEXT -1 104 "On voit d'ailleurs dans le coin en haut \340 g auche de l'\351cran les valeurs de theta et phi qui se modifient" }} {PARA 0 "" 0 "" {TEXT -1 102 "On d\351termine les angles pour que x s oit horizontal croissant vers la droite, y vertical vers le haut." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Nouveau graphe, en ajoutant une option \+ qui impose l'orientation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3d( V(x,y) ,FENETRE_POTENTIEL,G RILLE_POTENTIEL,axes=boxed,orientation=[-90,0]);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "Avec le menu cont extuel (bouton droit de la souris) , on essaye d'autres styles ou coul eurs pour mieux voir en couleur le potentiel. " }}{PARA 0 "" 0 "" {TEXT -1 81 "Il faudrait que ce soit le potentiel ( en Z ici ) qui soi t la source de couleur. " }}{PARA 0 "" 0 "" {TEXT -1 64 "Il faudrait a ussi des contours au niveau des \351quipotentielles..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "Les potentiels \351lev\351s sont en rouge et les \+ petits en violet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 233 "A:=plot3d(V(x,y),FENETRE_POTENTIEL,GRILLE_P OTENTIEL,axes=BOXED,orientation=[-90,0],style=PATCHCONTOUR,NBRE_EQUIP, shading=ZHUE,scaling=CONSTRAINED):A; \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Le trac\351 est r apide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Trac\351 du champ" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "Vmax:=0.5;B:=fieldplot3d([Ex(x,y),Ey(x,y),0],FENETRE_CHAMP,GRIL LE_CHAMP, arrows=THICK,axes=BOXED,color=BLACK,orientation=[-90,0],scal ing=CONSTRAINED):B;" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 30 "Trac\351 du champ et du potentiel" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([A,B]);" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Champ non \+ uniforme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plo ts):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "On fait le trac\351 du potentiel et du champ " }}{PARA 0 "" 0 " " {TEXT -1 28 "pour un champ non uniforme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "L:=1;FENETRE_POT ENTIEL:=(x=-L/2..L/2,y=-L/2..L/2);FENETRE_CHAMP:=(x=-L/2..L/2,y=-L/2.. L/2,z=Vmax..Vmax+0.01);Np:=100; Nc:=20;GRILLE_POTENTIEL:=(grid=[Np,Np] );GRILLE_CHAMP:=(grid=[Nc,Nc,2]);NBRE_EQUIP:=(contours=11);Vmax:=0.5; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "V:=(x,y)->1/2-4*x^2;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Ex:=(x,y)->-diff(V(x,y),x) ;Ex(x,y);" }{TEXT -1 0 "" }{MPLTEXT 1 0 35 "Ey:=(x,y)->-diff(V(x,y),y) ;Ey(x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "A:=plot3d(V(x ,y),FENETRE_POTENTIEL,GRILLE_POTENTIEL,axes=BOXED,orientation=[-90,0], style=PATCHCONTOUR,NBRE_EQUIP,shading=ZHUE,scaling=CONSTRAINED):A; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "B:=fieldplot3d([Ex(x,y), Ey(x,y),0],FENETRE_CHAMP,GRILLE_CHAMP, arrows=THICK,axes=BOXED,color=B LACK,orientation=[-90,0],scaling=CONSTRAINED):B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([B,A]);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Dip\364le" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "res tart; with(plots):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Les trac\351s son t ici plus complexes.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "L:=1;FENETRE_POTENTIEL:=(x=-L/2..L/2,y=-L/2..L/2);FENETRE_CHAMP:=(x=- L/2..L/2,y=-L/2..L/2,z=Vmax..Vmax+0.01);Np:=100; Nc:=20;GRILLE_POTENTI EL:=(grid=[Np,Np]);GRILLE_CHAMP:=(grid=[Nc,Nc,2]);NBRE_EQUIP:=(contour s=11);Vmax:=0.5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 29 "On s'int\351resse ici au dip\364le." }}{PARA 0 "" 0 "" {TEXT -1 95 "La difficult\351 est dans le fait que le dip\364le e st \351tudi\351 g\351n\351ralement en coordonn\351es sph\351riques." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 167 "Attention l'axe not\351 x ici corres pondra \340 l'axe z_sph\351rique souvent introduit en coordonn\351es s ph\351riques.\non continue \340 travailler dans le plan xy donc z_cart \351sienne =0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "On rappelle l'expressi on du potentiel pour le dip\364le en sph\351riques" }}{PARA 0 "" 0 "" {TEXT -1 37 "Pour l'angle n\351cessaire: \351crire theta" }}{PARA 0 " " 0 "" {TEXT -1 31 "On a aussi besoin de : epsilon0" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "V:= p*cos (theta)/(4*Pi*epsilon0*r^2) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "On d\351signe toutes les constantes p ar beta" }}{PARA 0 "" 0 "" {TEXT -1 61 "R\351\351crire l'expression de V en fonction de beta, cos(theta), r" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V:=beta* cos(theta)/r^ 2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "On \351crit ici la fonction donnant V en cart\351siennes.\nIl \+ faut remplacer cos(theta) et r par du x, du y (z=0)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "V:=(x,y)- >beta* x/ (x^2+y^2)^(3/2) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "On prendra beta=0.008;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "beta:= 0.008;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Ex:=(x,y)->-diff( V(x,y),x);Ex(x,y);Ey:=(x,y)->-diff(V(x,y),y);Ey(x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A:=plot3d(V(x,y),FENETRE_POTENTIEL ,GRILLE_POTENTIEL,axes=BOXED,style=PATCHCONTOUR,NBRE_EQUIP,shading=ZHU E):A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 " Il faut recommencer en ajoutant l'option: view=-0,5..0,5 " }}{PARA 0 "" 0 "" {TEXT -1 90 "qui permet de tronquer le graphe selo n Z (c'est \340 dire V) puisque il y a ici de l'infini." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "A:= plot3d(V(x,y),FENETRE_POTENTIEL,GRILLE_POTENTIEL,axes=BOXED,orientatio n=[-90,0],style=PATCHCONTOUR,NBRE_EQUIP,shading=ZHUE,scaling=CONSTRAIN ED,view=-0.5..0.5):A; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "B:=fieldplot3d([Ex(x,y),Ey(x,y),0],FENETRE_CHAMP,GRILLE_CHAMP, arrows =THICK,axes=BOXED,color=BLACK,orientation=[-90,0],scaling=CONSTRAINED) :B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "On constate que les vecteurs champ deviennent tr\350s vite tr \350s petits de sorte que l'on ne voit plus rien. " }}{PARA 0 "" 0 "" {TEXT -1 62 "On d\351cide alors de renoncer \340 l'information grandeu r du champ " }}{PARA 0 "" 0 "" {TEXT -1 78 "et au lieu de repr\351sent er le champ, on repr\351sente le vecteur unitaire associ\351 " }} {PARA 0 "" 0 "" {TEXT -1 64 "de coordonn\351es u et v qui donnera dire ction et sens uniquement. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "On choisit aussi pour arrows l'option SLIM au lie u de THICK" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "u:=(x,y)->Ex(x,y)/(Ex(x,y)^2+Ey(x,y)^2)^(1/2);v:=(x ,y)->Ey(x,y)/(Ex(x,y)^2+Ey(x,y)^2)^(1/2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 139 "B:=fieldplot3d([u(x,y),v(x,y),0],FENETRE_CHAMP,GRI LLE_CHAMP, arrows=SLIM,axes=BOXED,color=BLACK,orientation=[-90,0],scal ing=CONSTRAINED):B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "disp lay([A,B]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Sph\350re charg\351e en surface par une densit\351 en \+ cos(theta)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(p lots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Prendre Vmax=0.2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "L:=1;FENETRE_POTENTIEL:=(x= -L/2..L/2,y=-L/2..L/2);FENETRE_CHAMP:=(x=-L/2..L/2,y=-L/2..L/2,z=Vmax. .Vmax+0.01);Np:=100; Nc:=20;GRILLE_POTENTIEL:=(grid=[Np,Np]);GRILLE_CH AMP:=(grid=[Nc,Nc,2]);NBRE_EQUIP:=(contours=11);Vmax:=0.2;beta:=0.008; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Dans ce cas, " }}{PARA 0 "" 0 "" {TEXT -1 61 "\340 l'int\351rieur \+ de la sph\350re de rayon R, le champ est uniforme " }}{PARA 0 "" 0 "" {TEXT -1 61 "et \340 l'ext\351rieur de la sph\350re, le champ est celu i d'un dip\364le" }}{PARA 0 "" 0 "" {TEXT -1 40 "Le potentiel est cont inu sur le surface." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Le rayon \340 c onsid\351rer pour la sph\350re vaut 0,2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R:=0.2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Le graphe C (toujours en 3d) repr\351sente la sph\350re de rayon R " }}{PARA 0 "" 0 "" {TEXT -1 54 "(cercle correspondant \340 la projection sur le p lan xy)." }}{PARA 0 "" 0 "" {TEXT -1 89 "Ce dessin est plac\351, comme la repr\351sentation du champ au dessus du graphe 3d du potentiel" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "C:=implicitplot3d(x^2+y^2=R^2,FENETRE_CHAMP,GRILLE_CHAMP,axes=B OXED,color=BLACK,orientation=[-90,0],scaling=CONSTRAINED,thickness=3): C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Le potentiel ( on utilise " }{HYPERLNK 17 "piecewise" 2 "piecew ise" "" }{TEXT -1 57 " ). Ici: si r>R c'est V ext\351rieur sinon c'est V int\351rieur" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "V:=(x,y)->piecewise( (x^2+y^2)>R^2,Vexterieur( x,y),Vinterieur(x,y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 113 "Pour Vinterieur, on prend le sym\351triq ue du champ uniforme \351tudi\351 en premi\350re partie c'est \340 di re x au lieu de -x" }}{PARA 0 "" 0 "" {TEXT -1 62 "Pour Vexterieur, on prend le dip\364le \351tudi\351 en troisi\350me partie" }}{PARA 0 "" 0 "" {TEXT -1 134 "Il faudrait v\351rifier que sur le sph\350re les de ux potentiels donnent la m\352me valeur sur la sph\350re en consid\351 rant les valeurs num\351riques\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Vinterieur:=(x,y)-> x;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Vexterieur:=(x,y)->beta* x/ ( x^2+y^2)^(3/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "A:= plot3d(V(x,y),FENETRE_POTENTIEL,GRILLE_POTENTIEL,axes=BOXED,orientatio n=[-90,0],style=PATCHCONTOUR,NBRE_EQUIP,shading=ZHUE,scaling=CONSTRAIN ED):display([A,C]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "On peut toujours faire tourner la figure \+ pour mieux visualiser le potentiel en 3d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "La repr\351sentation du vecteur unitaire associ\351 au champ:" }}{PARA 0 "" 0 "" {TEXT -1 78 "on rappelle qu'il y a discontinuit\351 \+ de E \340 la travers\351e de la surface charg\351e." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Exint:=( x,y)->-diff(Vinterieur(x,y),x);Eyint:=(x,y)->-diff(Vinterieur(x,y),y); uint:=(x,y)->Exint(x,y)/(Exint(x,y)^2+Eyint(x,y)^2)^(1/2);vint:=(x,y)- >Eyint(x,y)/(Exint(x,y)^2+Eyint(x,y)^2)^(1/2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 194 "Exext:=(x,y)->-diff(Vexterieur(x,y),x);Eyext: =(x,y)->-diff(Vexterieur(x,y),y);uext:=(x,y)->Exext(x,y)/(Exext(x,y)^2 +Eyext(x,y)^2)^(1/2);vext:=(x,y)->Eyext(x,y)/(Exext(x,y)^2+Eyext(x,y)^ 2)^(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "u:=(x,y)->pie cewise( (x^2+y^2)>R^2,uext(x,y),uint(x,y));v:=(x,y)->piecewise( (x^2+y ^2)>R^2,vext(x,y),vint(x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "B:=fieldplot3d([u(x,y),v(x,y),0],FENETRE_CHAMP,GRILLE_CHAMP, ar rows=SLIM,axes=BOXED,color=BLACK,orientation=[-90,0],scaling=CONSTRAIN ED):display([B,C]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "disp lay([A,B,C]);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Sph\350re cond uctrice plac\351e dans un champ uniforme" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "On place une sph\350re ne utre dans un champ uniforme. " }}{PARA 0 "" 0 "" {TEXT -1 42 "Cette sp h\350re va se charger en cos(theta). " }}{PARA 0 "" 0 "" {TEXT -1 75 " A l'\351quilibre \351lectrostatique, le champ est nul \340 l'int\351ri eur de la sph\350re " }}{PARA 0 "" 0 "" {TEXT -1 61 "et pour l'ext\351 rieur cette sph\350re se comporte comme un dip\364le." }}{PARA 0 "" 0 "" {TEXT -1 81 "La solution est obtenue ici en faisant le somme de l'e xemple 1 (champ uniforme ) " }}{PARA 0 "" 0 "" {TEXT -1 56 "et de l'ex emple pr\351c\351dent (sph\350re charg\351e en cos theta )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "res tart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "L:=1; FENETRE_POTENTIEL:=(x=-L/2..L/2,y=-L/2..L/2);FENETRE_CHAMP:=(x=-L/2..L /2,y=-L/2..L/2,z=Vmax..Vmax+0.01);Np:=100; Nc:=20;GRILLE_POTENTIEL:=(g rid=[Np,Np]);GRILLE_CHAMP:=(grid=[Nc,Nc,2]);NBRE_EQUIP:=(contours=11); Vmax:=0.5;beta:=0.008;R:=0.2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "La sph\350re:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "C:=implici tplot3d(x^2+y^2=R^2,FENETRE_CHAMP,GRILLE_CHAMP,axes=BOXED,color=BLACK, orientation=[-90,0],scaling=CONSTRAINED,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Le potent iel:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "V:=(x,y)->piecewise( (x^2+y^2)>R^2,Vexterieur(x,y),Vi nterieur(x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Vinterie ur:=(x,y)-> 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Vexterieur:=(x,y) ->beta* x/ ( x^2+y^2)^(3/2)-x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "A:=plot3d(V(x,y),FENETRE_POTENTIEL,GRILLE_POTENTIEL, axes=BOXED,orientation=[-90,0],style=PATCHCONTOUR,NBRE_EQUIP,shading=Z HUE,scaling=CONSTRAINED):display([A,C]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Le champ:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "uint:=(x ,y)->0;vint:=(x,y)->0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 " Exext:=(x,y)->-diff(Vexterieur(x,y),x);Eyext:=(x,y)->-diff(Vexterieur( x,y),y);uext:=(x,y)->Exext(x,y)/(Exext(x,y)^2+Eyext(x,y)^2)^(1/2);vext :=(x,y)->Eyext(x,y)/(Exext(x,y)^2+Eyext(x,y)^2)^(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "u:=(x,y)->piecewise( (x^2+y^2)>R^2 ,uext(x,y),uint(x,y));v:=(x,y)->piecewise( (x^2+y^2)>R^2,vext(x,y),vin t(x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "B:=fieldplot3d ([u(x,y),v(x,y),0],FENETRE_CHAMP,GRILLE_CHAMP, arrows=SLIM,axes=BOXED, color=BLACK,orientation=[-90,0],scaling=CONSTRAINED):display([B,C]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display([A,B,C]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }