Générateur Van de Graaf et effets spéciaux

!"#$%&#'()##'*+','*+(# # # # # # ###########!-./012.34#56#7891.01#
EXAMEN&PARTIEL&D’&ELECTROMAGNETISME&&
:4-41;3091#<;-#=0#:1;;>#03#0>>032#2?4@.;9A##
Durée&:&1h30.&Documents&non&autorisés,&calculatrice&autorisée.&Les&durées&de&
chaque&partie&sont&données&à&titre&indicatif.&
#
B;# C;@D.-0# =0# <;-# =0# :1;;># 023# 9-# E4-41;3091#
4F0@31823;3.G90# ?01C033;-3# =0# ?18=9.10# =02# ?830-3.0F2#
4F0@31823;3.G902# 31H2# 4F0/426# B0# C8=0# =0# >[email protected]#
023# 2@D4C;3.24# @.,@8-310#I# J# FK;.=0# =K9-0# D;930# 30-2.8-# 89#
?;1# >18330C0-3L# 9-0# @89118.0# 023# @D;1E40# [email protected]#
?;1# 9-# ?0.E-0# @D;1E4# MN#@D;1E0=# @8CO#PQ6# "-# >;.2;-3#
3891-01# F;# @89118.0# MN#@8-/0R81# O0F3#PQL# F02# @D;1E02# 28-3#
31;-2?813402#S92G9KJ# 9-# ?0.E-0# @8FF0@3091# MN#@[email protected]#
@8CO#PQ# G9.# 10=.231.O90# F02# @D;1E02# =;-2# 9-0# 2?DH10#
C43;FF.G906#
#
T-# 20# ?18?820# .@.# =K439=.01#=02#0>>032#4F0@31823;3.G902#
2?0@3;@9F;.102#F.42#J#@02#D;9302#30-2.8-26#
#
$;13.0#+#I#@D;C?#03#?830-3.0F#4F0@31823;3.G902#@1442#?;1#F0#E4-41;3091#I#M∼ 45&minutes)
$891#2.C?F.>.01#F0#?18OFHC0#8-#-4EF.E0#F;#?;13.0#@8C?10-;-3#F;#@89118.0#I#8-#@8-2.=H10#F0#
E4-41;3091#@8CC0#9-0#2?DH10#@8-=9@31.@0#=0#1;R8-#UL#J#FK4G9.F.O10#4F0@31823;3.G906#
+6+Q V8CC0-3#28-3#14?;13.02#F02#@D;1E02#=;-2#9-#@8-=9@3091#J#FK4G9.F.O10#W#X90#?093,
8-#0-#=4=9.10#?891#F0#@D;C?#4F0@31823;3.G90#J#FK.-341.091#=9#@8-=9@3091#W#"3#?891#
F0#?830-3.0F#W#
+6'Q Y#FK;.=0#=9#3D481HC0#=0#:;922#F8@;F#03#=0#F;#G9023.8-#?14@4=0-30L#C8-310Z#G90#F;#
=0-2.34#/8F9C.G90#=0#@D;1E02#[#J#FK.-341.091#=0#F;#2?DH10#023#-9FF06##
+6(Q T-#29??820#G90#F;#=0-2.34#291>;@.G90#=0#@D;1E02#023#9-.>81C0#I#\]\*6#Y#FK;.=0#=9#
3D481HC0#=0#29?01?82.3.8-L#C8-310Z#G90#F0#?830-3.0F#4F0@31823;3.G90#;9#@0-310#=0#
F;#2?DH10#[email protected]#
€
V0=
σ
0R
ε
0
6##
+6^Q _4=9.20Z,0-# F;# @;?;@.34# =0# F;# 2?DH10# @D;1E40# ;.-2.# G90# FK4-01E.0# 4F0@31823;3.G90#
0CC;E;2.-406#
+6)Q "39=.0Z#F02#2RC431.02#=0#F;#=.231.O93.8-#=0#@D;1E02#03#=4=9.20Z,0-#F;#[email protected]#=9#
@D;C?# Y# BK"`a"Ub"!U#=0#F;#2?DH10#@D;1E40L#;.-2.#G90#F02#/;1.;OF02#=8-3#2;#
-81C0#=4?0-=6#X90FF0#291>;@0#=0#:;922#201;.3#;=;?340#J#@0#?18OFHC0#W#
+6cQ V;F@9F0Z# ;F812# F0# @D;C?# 4F0@31823;3.G90# J# FK0A341.091#=0#F;#2?DH10#@D;1E40L#03#
C8-310Z# G9K.F# 023# 4G9./;F0-3# ;9# @D;C?# @144# ?;1# 9-0# @D;1E0# ?8-@390FF0# 0>>0@3./0#
X0>>#?F;@40# ;9# @0-310# =0# F;# 2?DH10L# G90# FK8-# 0A?1.C01;# 0-# >[email protected]# =0# \*# 03# U6#
V8C?;10Z#/8310#1429F3;3#?891#X0>>#;/0@#F02#1429F3;32#=02#G9023.8-#+6(#03#+6^6#

+6dQ U;??0F01#M2;-2#F0#=4C8-3101Q#F0#3D481HC0#=0#V89F8CO6#"23,.F#/41.>.4#;9#/8.2.-;E0#
=0#F;#291>;@0#=0#F;#2?DH10#W#
+6eQ _K;?1H2#F;#G9023.8-#+6cL#@;F@9F0Z#F0#?830-3.0F#4F0@31823;3.G90#<M1Q#J#FK0A341.091#=0#
F;# 2?DH106# "A?1.C0Z# F0# 0-# >[email protected]# =0# <*L# U# 03# 16#On& donne& en& coordonnées&
sphériques&
€
dl =drur+rd θ u
θ
+rsin θ d ϕ u
ϕ
.#
+6fQ Y??F.@;3.8-2#-9C41.G902#I##
;Q# V;F@9F0Z# F;# =0-2.34# 291>;@.G90# =0# @D;1E02# J# E4-4101# 2.# FK8-# /093# C03310# F0#
E4-41;3091#J#9-#?830-3.0F#=0#+*)<6#B0#1;R8-#=0#F;#2?DH10#023#U]'*@C6#T-#=8--0#
g*]eLe)6+*,+'#hb6#
OQ#"-#=4=9.10#F;#@D;1E0#383;F0#?81340#?;1#F;#2?DH10#=;-2#@02#@8-=.3.8-26#
@Q#!-0#O;3301.0#=0#34F4?D8-0#?813;OF0#383;F0C0-3#@D;1E40#@8-3.0-3#9-0#@D;1E0#=0##
+'**#CY6D6#_0#G90F#?891@0-3;E0#@0330#@D;1E0#O;.2201;.3,0FF0#2.#F;#O;3301.0#201/;.3#
J#@D;1E01#F0#E4-41;3091#W#[email protected]#W#
#
$;13.0#'#I#0A?41.0-@02#2?0@3;@9F;.102#I (∼ 15&minutes)#
'6+Q "-#/892#O;2;-3#291#F02#1429F3;32#=0#
F;# ?;13.0# ?14@4=0-30L# 0A?F.G90Z#
G9;F.3;3./0C0-3# @0# G9.# ;11./0# J# F;#
@D0/0F910# =0# F;# S09-0# >.FF0# =0# F;#
?D838# @.,@8-310#M8-#@8-2.=H10#F;#
S09-0# >.FF0# @8CC0# 9-# @8-=9@3091#
?;1>;.3# 0-# O8--0# ;??18A.C;3.8-Q6#
5923.>.01# FK81.0-3;3.8-# =0# 202#
@D0/09A# 03# 31;@01#
2@D4C;3.G90C0-3# FK;FF910# =02#
4G9.?830-3.0FF02# 03# =02# F.E-02# =0#
@D;C?#?1H2#=0#F;#2?DH10#03#?1H2#=0#
F;#3i30#=0#F;#S09-0#>.FF06##
#
'6'Q T-# ;??18@D0# 9-# 39O0# >F98102@0-3# M;O92./0C0-3# ;??0F4# N#-48-#PQ# =9# E4-41;3091#
?8134#J#9-#?830-3.0F#=0#<*](6+*^<6#B0#@0-310#TK#=9#39O0#023#81.0-34#1;=.;F0C0-3L#
=0#C;-.H10#J#@0#G90#F;#=18.30#TTK#10F.;-3#F0#@0-310#=0#F;#2?DH10#;9#@0-310#=9#39O0#
28.3#D81.Z8-3;F06#B0#39O0#F9.,CiC0#>;.3#9-#;-EF0#j#;/0@#FKD81.Z8-3;F0L#03#202#=09A#
0A314C.342#Y#03#kL#02?;@402#=0#'FL#28-3#2.39402#J#9-0#=.23;-@0#1Y#03#1k#=0#F;#2?DH106#
#
##
;Q# h.# F;# =.>>410-@0# =0# ?830-3.0F# 0-310# F02# 4F0@318=02# <Y,<k#023#29?41.0910#J#9-0#
30-2.8-#209.F#M3R?.G90C0-3#+***#<QL#F0#39O0#@8CC0-@0#J#4C03310#=0#F;#F9C.H106#h.#
8-#;??18@D0#F0#39O0#81.0-34#1;=.;F0C0-3#Mj]*QL#@;F@9F0Z#F;#=.23;-@0#C;A.C;F0#1C;A#
0-310#F;#2?DH10#03#F0#39O0#0-#=022892#=0#F;G90FF0#F0#39O0#2K;FF9C0#M@8CC0#.FF92314#
O
A
B
α
rO'
rB
rA
(σ)V
A
VB
V0
2l

?;1# F;# ?D838# @.,=02292Q6# B0# 39O0# 023# 30F# G90# 'F]+CL# 03# F0# 1;R8-# =0# F;# 2?DH10# 023#
U]*L'#C6#
OQ _.2@930Z#F0#@;2#?;[email protected]#j]lm'6#B0#39O0#?8911;,3,.F#2K;FF9C01#W#
&
$;13.0#(#I#?8F;1.2;O.F.34#=02#;38C02#I#M∼ 30&minutes)#
CETTE&PARTIE&PEUT&ETRE&TRAITEE&INDEPENDAMMENT#
!-# 39O0# >F98102@0-3# M@8CC9-4C0-3# ;??0F4# N#-48-#PQ# 2K;FF9C0# F812G90# F0# @D;C?#
[email protected]#;??F.G94#0-310#202#=09A#4F0@318=02#023#29>>.2;CC0-3#4F0/4#?891#.8-.201#F0#E;Z#
G9K.F# @8-3.0-36# $891# =02# @D;C?2# [email protected]# ?F92# >;.OF02L# FK0>>03# 201;# =0# =4?F;@01# F02#
@D;1E02#?;1#1;??813#J#F091#?82.3.8-#=K4G9.F.O10#M8-#29??8201;#F02#-8R;9A#.CC8O.F02Q6#B0#
@D;C?#/;#=8-@#.-=9.10#F;#>81C;3.8-#=0#=.?nF02#4F0@31823;3.G902#;9#20.-#=0#@D;G90#;38C06##
o892# C8=4F.28-2# .@.# 9-# ;38C0#=KDR=18EH-0#@8CC0# 9-# ?1838-# ?8-@390F# =0# @D;1E0# p0#
0-38914# =K9-# -9;E0# [email protected]# 2?D41.G90# =0# 1;R8-# ;# 03# =0# @D;1E0# 383;F0# q0# 14?;13.0#
9-.>81C4C0-3# MF;# =0-2.34# /8F9C.G90# =0# @D;1E02# @81102?8-=;-30# [#023#@8-23;-30Q6# !-#
@D;C?# 0A341.091#
€
E0
#@8-23;-3#023#;??F.G94#291#FK;38C06# B0# -9;E0# [email protected]# /;# ;F812#
;330.-=10#9-0#-89/0FF0#?82.3.8-#=K4G9.F.O10L#@0-3140#J#9-0#=.23;-@0#14G#=9#?1838-6#
#
(6+Q _4301C.-01# F;# =0-2.34# /8F9C.G90# =0# @D;1E02#
[email protected]#[6#
(6'Q X90FF0# 023# F;# ?82.3.8-# =K4G9.F.O10# =9# -9;E0#
[email protected]# 03# =9# ?1838-# 0-# FK;O20-@0# =0#
@D;C?#0A341.091#W#
(6(Q T-# @8-2.=H10# =;-2# 9-# ?10C.01# 30C?2# F0# -9;E0#
4F0@318-.G90# 209F# 03# 2;-2# @D;C?# 0A341.0916# Y#
FK;.=0#=9#3D481HC0# =0# :;922L#@;F@9F0Z#F0#@D;C?#
€
Ee−(r)
#@144#?;1#@0# -9;E0# [email protected]# J# 9-0#
=.23;-@0# 1# =0# 28-# @0-310#TL# 03# C8-310Z# G90
€
Ee−(r)=er
4 πε
0a3ur
6##
(6^Q V;F@9F0Z# 4E;F0C0-3# F0# ?830-3.0FL# 0-# >.A;-3#
<M1]*Q]*6#
(6)Q BK4G9.F.O10# =0# FK;38C0# 023# ;330.-3# F812G90# F;#
>81@0# 4F0@31823;3.G90# 383;F0# 2K0A01r;-3# 291# F0# -9;E0# [email protected]# # 2K;--9F06#
_4=9.20Z,0-#G90#F;#=.23;-@0#=K4G9.F.O10#0-310#F0#?1838-#03#F0#-9;E0#[email protected]#
/;93#
€
réq =E04
πε
0a3
e
6#
(6cQ "A?1.C0Z#;F812#F0#C8C0-3#=.?8F;.10#.-=9.3#
€
p
#0-#>8-@3.8-#=0#0#03#14GL#03#C8-310Z#
G90#
€
p= αε
0E0
L# 8s# j# 023# ;??0F4# ?8F;1.2;O.F.34# =0# FK;38C0# 03# 2K0A?1.C0# 0-#
>[email protected]#=0#;6##
(6dQ _;-2# @0# C8=HF0L# FK;38C0# 023# .8-.24# 2.# F0# ?1838-# N#2813#P#=9#-9;E0#4F0@318-.G906#
X90FF0# 023# F;# =.23;-@0# =K4G9.F.O10# C;A.C;F0#=9# ?1838-#W# V;F@9F0Z# ;F812# FK4-01E.0#
?830-3.0FF0# 4F0@31823;3.G90# =0# FK;38C06# 7;.302# FK;??F.@;3.8-# -9C41.G90# ?891#
FK;38C0# =KDR=18EH-0# M?891# F0G90F# ;]*L)'f6+*,+*# CQL# 03# @8C?;10Z# J# FK4-01E.0#
=K.8-.2;3.8-#=0#@03#;38C0#".]+(Lc#0<6#
O+e
-e
ρ=cte
a
réq
E0