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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 6 "FLUX  " }}
{PARA 0 "" 0 "" {TEXT 257 34 "by Shannon Mayer & Corinne Manogue" }}
{PARA 0 "" 0 "" {TEXT -1 30 "Copyright 1997 Corinne Manogue" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "In this worksh
eet, you will examine the flux of the electric field due to a point ch
arge through a cube, with sides of length 2L, centered at the origin. \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "restart:with(linalg):with(plots):\nsetoptions3d(scali
ng=constrained,axes=boxed):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "SE
T-UP THE PROBLEM\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Define the \+
dimensions of the square box to be L=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 5 "L:=1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 51 "Define the unit vectors in rectangular coordinates:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ihat:=[1,0,0];jhat:=[0,1,0]
;khat:=[0,0,1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 106 "Initially, define the location of the point charge \+
to be at the origin of a rectangular coordinate system:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "Point:=[0,0,0];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "a:=Point[1];b:=Point[2];c:=Point[3];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 44 "Find the electric field of the point char
ge:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r:=sqrt((x-a)^2+(y-b)^2+(z-c
)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Efield:=(q/(4*Pi*e
psilon))*grad(-1/r,[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "P
lot the three dimensional vector field.  For plotting purposes, choose
 the charge so that " }{XPPEDIT 18 0 "q/(4*Pi*epsilon)=1" "6#/*&%\"qG
\"\"\"*(\"\"%F&%#PiGF&%(epsilonGF&!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "Esubs:=subs(q=4*Pi*epsilon,Efield);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "fieldplot3d(Esubs,x=-L..L,y=-L..L,z
=-L..L,\ngrid=[6,6,6],shading=none,arrows=SLIM,thickness=2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "CALCULATE THE FLUX THROUGH ONE SUR
FACE (" }{XPPEDIT 18 0 "(z=L)" "6#/%\"zG%\"LG" }{TEXT -1 3 ").\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "E1:=subs(z=L,[Esubs[1],Esubs[2],Esu
bs[3]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Integrand1:=dot
prod(E1,khat);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Examine the int
egrand.  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot3d(Integrand1,x=-L
..L,y=-L..L,axes=framed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Perf
orm the integration.  Put back in the factor of " }{XPPEDIT 18 0 "q/(4
*Pi*epsilon" "6#*&%\"qG\"\"\"*(\"\"%F%%#PiGF%%(epsilonGF%!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "q/(4*Pi*epsilon)*
Int(Int(Integrand1,x=-L..L),y=-L..L);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "int(int(q/(4*Pi*epsilon)*Integrand1,x=-L..L),\ny=-L..
L);" }}{PARA 0 "" 0 "" {TEXT -1 98 "The Maple command \"Int leaves the
 integral in unevaluated form, while\"int\" evaluates the integral." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "
CALCULATE THE TOTAL FLUX (summed over all six sides of the cube).\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "Total:=simplify(q/(4*Pi*epsilon)
\n*(int(int(dotprod(subs(x=L,\n[Esubs[1],Esubs[2],Esubs[3]]),ihat),\ny
=-L..L),z=-L..L)\n+int(int(dotprod(subs(x=-L,\n[Esubs[1],Esubs[2],Esub
s[3]]),-ihat),\ny=-L..L),z=-L..L)\n+int(int(dotprod(subs(y=L,\n[Esubs[
1],Esubs[2],Esubs[3]]),jhat),\nx=-L..L),z=-L..L)\n+int(int(dotprod(sub
s(y=-L,\n[Esubs[1],Esubs[2],Esubs[3]]),-jhat),\nx=-L..L),z=-L..L)\n+in
t(int(dotprod(subs(z=L,\n[Esubs[1],Esubs[2],Esubs[3]]),khat),\nx=-L..L
),y=-L..L)\n+int(int(dotprod(subs(z=-L,\n[Esubs[1],Esubs[2],Esubs[3]])
,-khat),\nx=-L..L),y=-L..L)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "0 0 1" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
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