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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "PHYSICS RESOURCE PACKETS P
ROJECT                               File: HOLTPLOT.MWS" }}{PARA 0 "" 
0 "" {TEXT -1 110 "Rose-Hulman Institute of Technology                \+
                       Authors: Perry Peters & Greg Williby" }}{PARA 
0 "" 0 "" {TEXT -1 97 "http://www.rose-hulman.edu/~moloney            \+
                       Software: Maple V Release 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 249 "Make a 3D plot of the ma
gnitude of the magnetic field between a pair of helmholtz coils. (See \+
problem 'loopaxis' for the setup of helmholtz coils.)  [In parts a) an
d b) below, try to use the contours of the 3D plot to help you answer \+
the questions.]" }}{PARA 0 "" 0 "" {TEXT -1 204 "a)  How far from the \+
center of the coils do we have to go laterally before the field magnit
ude changes by 10%.  Express your answer in terms of the coil radius (
 0.25 radii, 0.32 coil radii, or whatever)." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "b) How far from the center do \+
we have to go along the axis before the magnitude of the field changes
 by 10%?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
8 "Solution" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "
Vector to point in 3d space." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P:=
vector([x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 29 "Vector to point on wire loop." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "s:=vector([R*cos(theta),R*sin(theta),0]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Direction around wire loop." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ds:=map(diff,s,theta);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Vector from loop to 3d point." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "r:=evalm(P-s);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "Biot-Savart law" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dB:
=mu[0]*i*crossprod(ds,r)/(4*Pi*sqrt(dotprod(r,r))^3);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "Distribute constant over the three compon
ents of the vector." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dBvect:=eval
m(dB);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Substitute in the value
s of the known constants." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dBvect
sub:=subs(\{R=1,mu[0]=4*Pi*10^(-7),i=1\},op(dBvect));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "Evaluate the integrals using Simpson's me
thod.  \":\" supresses the output." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "sol:=map(simpson,dBvectsub,theta=0..2*Pi,20):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Evaluate the Sigma
 notation in sol." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Bxyz:=map(valu
e,sol):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Position a second coil
 1m from the first coil." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Bxyz2:=
subs(z=z-1,op(Bxyz)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Add the \+
fields of the two coils together." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "Bxyztot:=evalm(Bxyz+Bxyz2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
41 "Find the magnitude of the combined field." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "Bmag:=(Bxyztot[1]^2+Bxyztot[2]^2+Bxyztot[3]^2)^(1/2):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Take a cross section of the B
 field in the xz plane." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bmagxz:=
subs(y=0,Bmag):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Plot the B fie
ld in the xz plane." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot3d(Bmag
xz,x=-1.5..1.5,z=-0.5..1.5,view=0..2*10^(-6),title=`|B| from helmholtz
 coils`,grid=[40,40]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Animate the same function as i
n the 3d plot to \"walk\" through the shape above.  Notice how little \+
the field in the middle changes." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 
"animate(Bmagxz,x=-1.5..1.5,z=0..1,title=`|B| from helmholtz coils`,co
lor=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Find the field at th
e center of the two coils." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "field
_at_center:=evalf(subs(\{z=0.5,x=0,y=0\},Bmag));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "90% of the field" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "field90:=0.9*field_at_center;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "1
10% of the field" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "field110:=1.1*f
ield_at_center;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Plot the area \+
in the xz plane that has a magnetic field within 10% of the center." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot3d(Bmagxz,x=-1.5..1.5, z=-0.5
..1.5,view=field90..field110,grid=[40,40],title=`B field within 10% of
 that at the center`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Magnitu
de of field along z axis." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Bmagz:
=subs(\{y=0,x=0\},Bmag):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "Magnitude of field between the two coils.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Bmagx:=subs(\{y=0,z=1/2\},Bmag)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot(\{Bmagx,field90,f
ield110\},x=-1.5..1.5,B=0..1.2*10^(-6),title=`|B| along x when z=0.5`)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Part A" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Find lateral distance for
 field to change by 10%" }}{PARA 0 "" 0 "" {TEXT -1 30 "Since R=1, ans
wer is in radii." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fsolve(Bmagx=fi
eld90,x=0.6..0.8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(
\{Bmagz,field90,field110\},z=-.5..1.5,B=0..1.2*10^(-6),title=`|B| alon
g z axis`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Part B" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Find distance alon
g z axis for field to change by 10%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "sol:=fsolve(Bmagz=field90,z=0.9..1.2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 70 "Subtract 1/2 from this answer to measure the distance fro
m the center." }}{PARA 0 "" 0 "" {TEXT -1 29 "Again, the answer is in \+
radii" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dist:=sol-0.5;" }}}}{MARK 
"30 2 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 }