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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 115 "2000 Oregon State Univ
ersity              Capstones in Physics: Electromagnetism            \+
      Philip J. Siemens" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "
" 0 "" {TEXT -1 10 "MULTIPOLES" }}{PARA 0 "" 0 "" {TEXT -1 86 "This Ma
plesheet is about the electric field of a localized static charge dist
ribution." }}{PARA 0 "" 0 "" {TEXT -1 76 "We are mainly looking at the
 pattern of the field far away from the sources." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: with (lina
lg): with (plots):" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 26 "ELECTRIC \+
FIELD OF A DIPOLE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 81 "The simplest electric dipole consists of two charges, sep
arated by some distance " }{TEXT 263 1 "D" }{TEXT -1 174 ".\n\nFrom th
e principle of superposition we know that the electric field for the d
ipole is just the vector sum of the electric fields from each of the i
ndividual point charges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 13 "For a charge " }{TEXT 264 1 "Q" }{TEXT -1 56 ", th
e constant of proportionality for the electric field" }{TEXT 260 1 " \+
" }{TEXT -1 2 "is" }{TEXT 261 2 " k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 
"Q/(4*pi*epsilon[0])" "*&%\"QG\"\"\"*(\"\"%F$%#piGF$&%(epsilonG6#\"\"!
F$!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 50 "We will choos
e to consider charges of a magnitude " }{TEXT 262 1 "k" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "d^2" "*$%\"dG\"\"#" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 23 " is a unit of distance." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "\nIn cartesian coordinates, the e
lectric field at the poin" }{TEXT 265 2 "t " }{TEXT -1 1 "(" }{TEXT 
267 5 "x,y,z" }{TEXT -1 26 ") due to a point charge at" }{TEXT 269 1 "
 " }{TEXT -1 1 "(" }{TEXT 270 13 "x=a, y=b, z=c" }{TEXT -1 13 ") is gi
ven by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rabc:=sqrt((x-a)^2+(y-b)^
2+(z-c)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Exyz:=[k*(x-
a)/(rabc^3),k*(y-b)/(rabc^3),k*(z-c)/(rabc^3)];" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Let's look at the fi
eld of a point charge at the location (" }{TEXT 268 1 "x" }{TEXT -1 4 
"=0, " }{TEXT 271 1 "y" }{TEXT -1 4 "=0, " }{TEXT 272 1 "z" }{TEXT -1 
4 "=0.9" }{TEXT 258 1 "d" }{TEXT -1 35 ").  We can factor out the unit
s of " }{TEXT 259 1 "d" }{TEXT -1 24 " to cancel the factor of" }
{TEXT 273 2 " k" }{TEXT -1 29 ".  Then the electric field is" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r1:=sqrt(
(x-0)^2+(y-0)^2+(z-0.9)^2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "E1:=[(x-0)/(r1^3),(y-0)/(r1^3),(z-0.9)/(r
1^3)];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "We can view this with the fieldplot3d command" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "fieldplot3d
(E1,x=-10..9,y=-10..9,z=-10..10,grid=[10,10,10],axes=boxed,shading=zhu
e);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "Close to the edge of the box, \+
the arrows get so small you can't see them.  Use your menu items to lo
ok at the picture with a larger scale. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 88 "To see more clearly what happens far
 from the sources, we can use the asymptotic theory." }}{PARA 0 "" 0 "
" {TEXT -1 47 "It says that the potential is proportional to  " }
{XPPEDIT 18 0 "1/r" "*&\"\"\"F#%\"rG!\"\"" }{TEXT -1 26 " , so the fie
ld goes like " }{XPPEDIT 18 0 "1/r^2" "*&\"\"\"F#*$%\"rG\"\"#!\"\"" }
{TEXT -1 9 " , where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 1 " " }{MPLTEXT 1 0 21 "r:=sqrt(x^2+y^2+z^2);" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Let's take out thi
s factor " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "E1r2:=[r^2*(x-0)/(r1^3),r^2*(y-0)/(r1^3),r^2*(z-0.9)/
(r1^3)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "fieldplot3d(E1r2,x=-10.
.9,y=-10..9,z=-10..10,grid=[10,10,10],axes=boxed,shading=zhue);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "N
ow you can see that, far from the sources, the arrows all have the sam
e lengths.  " }}{PARA 0 "" 0 "" {TEXT -1 69 "As you go away from the s
ource, they neither get shorter nor longer. " }}{PARA 0 "" 0 "" {TEXT 
-1 68 "Their directions change, though: they always point radially out
ward." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "
This is the pattern of a MONOPOLE FIELD. " }}{PARA 0 "" 0 "" {TEXT -1 
72 "It is the dominant pattern of ANY distribution with a total net ch
arge, " }}{PARA 0 "" 0 "" {TEXT -1 56 "   as long as you get far enoug
h away from the sources.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "\nN
ow change the scale to look at a closer view." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "fieldplot3d(E1r2,x=-1.5..1.5,y=-1.5..1.5,z=-1.5..1.5,
grid=[10,10,10],axes=boxed,shading=zhue);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "You can see that, closer \+
to the sources, the field pattern differs from the monopole." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now let
's look at a distribution with no net charge." }}{PARA 0 "" 0 "" 
{TEXT -1 35 "We can place a negative charge at (" }{TEXT 276 1 "x" }
{TEXT -1 4 "=0, " }{TEXT 277 1 "y" }{TEXT -1 4 "=0, " }{TEXT 278 1 "z
" }{TEXT -1 5 "=-0.9" }{TEXT 279 1 "d" }{TEXT -1 1 ")" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r2:=sqrt((x-0)^2
+(y-0)^2+(z+0.9)^2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "E2:=[-(x-0)/(r2^3),-(y-0)/(r2^3),-(z+0.9)/(r2^3)
];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 66 "Then we combine the fields of the two charges, and plot the res
ult" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "E12:=E1+E2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "fieldplot3d(E12,
x=-1.5..1.5,y=-1.5..1.5,z=-1.5..1.5,grid=[10,10,10],axes=boxed,shading
=zhue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "This falls off even faster at large distances.  The facto
r " }{XPPEDIT 18 0 "r^2" "*$%\"rG\"\"#" }{TEXT -1 41 " isn't enough to
 keep it from falling off" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "E2r2:=
[-r^2*(x-0)/(r2^3),-r^2*(y-0)/(r2^3),-r^2*(z+0.9)/(r2^3)];" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E12r2:=E1r2
+E2r2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "fieldplot3d(E12r2,x=-10..
9,y=-10..9,z=-10..10,grid=[10,10,10],axes=boxed,shading=zhue);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Instead
, we need an extra power of " }{TEXT 280 1 "r" }{TEXT -1 1 ":" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "E1r
3:=[r^3*(x-0)/(r1^3),r^3*(y-0)/(r1^3),r^3*(z-0.9)/(r1^3)];" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 64 "E2r3:=[-r^3*(x-0)/(r2^3),-r^3*(y-0)/(r2^3),
-r^3*(z+0.9)/(r2^3)];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "E12r3:=E1r3+E2r3;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 87 "fieldplot3d(E12r3,x=-10..9,y=-10..9,z=-10..10,grid=[10,10,10],
axes=boxed,shading=zhue);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Now, if you look at an enlarged p
icture, you can see that the length and direction of the arrows doesn'
t change as you go farther from the center.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "However, the lengths and
 directions of the arrows do depend on the direction from the field po
int to the charge distribution.  The pattern is called a DIPOLE FIELD.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The d
ipole field we have here is oriented along the " }{TEXT 275 1 "z" }
{TEXT -1 13 "-direction.  " }}{PARA 0 "" 0 "" {TEXT -1 79 "Far away fr
om the source, any net-neutral charge distribution looks like this, " 
}}{PARA 0 "" 0 "" {TEXT -1 65 "except that the pattern may be oriented
 along a different axis.  " }}{PARA 0 "" 0 "" {TEXT -1 112 "The direct
ion of the axis of the field pattern is the direction of the dipole mo
ment of the charge distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Y
our homework exercise E1 is a continuation of this theme." }}{PARA 0 "
" 0 "" {TEXT -1 76 "You can adapt the commands in this worksheet to so
lve the homework exercise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 261 "" 0 "" {TEXT -1 22 "version 6 January 2000" }}}}{MARK "0 6 \+
0" 37 }{VIEWOPTS 1 1 0 1 1 1803 }