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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 0 "" }{TEXT 258 35 "Magnetic \+
Field of a Hollow Cylinder" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 0 "" }{TEXT 259 26 "\251 1998 Waterloo Maple Inc." }{TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 31 "NOTE: \+
This worksheet shows how " }{TEXT -1 86 "Maple V can be used to determ
ine the magnetic screening provided by a hollow cylinder." }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 260 12 "Introduction" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 329 "A \+
ferromagnetic tube can be used to shield its interior from static magn
etic fields.  To calculate the screening constant, the tube is presume
d to be set up so that its axis is  perpendicular to the direction of \+
an outer homogeneous magnetic field H0 [A/m]. The outer radius of the \+
tube is Ro [m], and the inner radius is Ri [m]." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 8 "        " }{OLE 1 7204 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 6 
"Theory" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 318 "Under the influence of
 the outside magnetic field, the ferromagnetic material is magnetized.
 The resulting magnetic field can be calculated as a superposition of \+
the outside magnetic field and the fields of magnetic line dipoles. Th
e free constants in the equations are calculated with the help of boun
dary conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "The magnetic scalar potential V [A] is defined by " }
{TEXT 265 1 "H" }{TEXT -1 7 "=-grad(" }{TEXT 266 1 "V" }{TEXT -1 2 ").
" }}{PARA 0 "" 0 "" {TEXT -1 91 "The scalar magnetic potential of a ho
mogeneous magnetic field in polar coordinates (r [m], " }{XPPEDIT 18 
0 "phi" "6#%$phiG" }{TEXT -1 19 " [rad]) is given by" }}{PARA 0 "" 0 "
" {TEXT -1 6 "      " }{XPPEDIT 18 0 "VH = A-B*r*cos(phi);" "6#/%#VHG,
&%\"AG\"\"\"*(%\"BGF'%\"rGF'-%$cosG6#%$phiGF'!\"\"" }{TEXT -1 44 "  (c
haracteristic constants A [A], B [A/m])." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The magnetic field of a line dipol
e is given by" }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "
VD = C*cos(phi)/r;" "6#/%#VDG*(%\"CG\"\"\"-%$cosG6#%$phiGF'%\"rG!\"\"
" }{TEXT -1 41 "       (characteristic constant C [A*m])." }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 262 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 227 "There are three equations that represent the superpositi
on of a homogeneous magnetic scalar potential and the scalar potential
 of a line dipole: one for the outside, a second for the wall and a th
ird for the inside of the tube." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
45 "The permeabitiy of the material is defined as" }}}{EXCHG {PARA 0 "
> " 0 "" {XPPEDIT 19 1 "mu:=mur*mu0:" "6#>%#muG*&%$murG\"\"\"%$mu0GF'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "mu0;" "6#%$
mu0G" }{TEXT -1 64 " : permeability of vacuum [H/m], mur: relative per
meability [-])" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 8 "Outside:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
V1:=A1-B1*r*cos(phi)+C1/r*cos(phi):" "6#>%#V1G,(%#A1G\"\"\"*(%#B1GF'%
\"rGF'-%$cosG6#%$phiGF'!\"\"*(%#C1GF'F*F/-F,6#F.F'F'" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 5 "Wall:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "V2:=A2-B2*r*cos(phi)+C2/r*cos(phi):" "6#>%#V2G,(%#A2G\"\"\"*(%#B2GF
'%\"rGF'-%$cosG6#%$phiGF'!\"\"*(%#C2GF'F*F/-F,6#F.F'F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 7 "Inside:" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "V3:=A3-B3*r*cos(phi)+C3/r*cos(phi):" "6#>%#V3G,(%#A3G\"
\"\"*(%#B3GF'%\"rGF'-%$cosG6#%$phiGF'!\"\"*(%#C3GF'F*F/-F,6#F.F'F'" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "On the inside, the potential must
 be finite so that C3 = 0. On the axis where r = 0 and where " }
{XPPEDIT 18 0 "phi = 1/2*Pi;" "6#/%$phiG*(\"\"\"\"\"\"\"\"#!\"\"%#PiGF
'" }{TEXT -1 64 " , the potential has the value V3 = V2 = V1 = V0 = A1
 = A2 = A3." }}{PARA 0 "" 0 "" {TEXT -1 118 "Choose V0 = 0. The result
 on the inside is a homogeneous magnetic field with a field intensity \+
of Hi, so that B3 = Hi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "
A3:=0: A2:=0: A1:=0: C3:=0: B3:=Hi:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 121 "On the outside, the magnetic field in very big distances is un
disturbed by the magnetic field caused by the tube, so that" }}{PARA 
0 "" 0 "" {TEXT -1 6 "      " }{TEXT 256 2 "V1" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "V1 = B1*r*cos(phi)+C1*cos(phi)/r;" "6#/%#V1G,&*(%#B1G\"
\"\"%\"rGF(-%$cosG6#%$phiGF(F(*(%#C1GF(-F+6#F-F(F)!\"\"F(" }{TEXT -1 
3 " ~ " }{XPPEDIT 18 0 "-B1*r*cos(phi);" "6#,$*(%#B1G\"\"\"%\"rGF&-%$c
osG6#%$phiGF&!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 24 "Th
is results in B1 = H0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B1
:=H0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "The conditions at the i
nterfaces are that the tangential components of the vector H and the n
ormal components of the vector B (B = " }{XPPEDIT 18 0 "mu;" "6#%#muG
" }{TEXT -1 19 "*H) are continuous." }}{PARA 0 "" 0 "" {TEXT -1 5 "Fro
m " }{XPPEDIT 18 0 "H = -grad(V);" "6#/%\"HG,$-%%gradG6#%\"VG!\"\"" }
{TEXT -1 62 " in cylinder coordinates, one obtains the following equat
ions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "v := [r, phi, z]:" "6#>%\"vG7%
%\"rG%$phiG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "h1:=gra
d(V1, v, coords=cylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1
G-%'vectorG6#7%,&*&%#H0G\"\"\"-%$cosG6#%$phiGF,!\"\"*&*&%#C1GF,F-\"\"
\"F5*$)%\"rG\"\"#F5!\"\"F1*&,&*(F+F5F8F,-%$sinGF/F,F,*&*&F4F5F>F5F5F8F
:F1F5F8F:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "h2:=grad(
V2, v, coords=cylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h2G-
%'vectorG6#7%,&*&%#B2G\"\"\"-%$cosG6#%$phiGF,!\"\"*&*&%#C2GF,F-\"\"\"F
5*$)%\"rG\"\"#F5!\"\"F1*&,&*(F+F5F8F,-%$sinGF/F,F,*&*&F4F5F>F5F5F8F:F1
F5F8F:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "h3:=grad(V3,
v,coords=cylindrical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h3G-%'vec
torG6#7%,$*&%#HiG\"\"\"-%$cosG6#%$phiGF,!\"\"*&F+\"\"\"-%$sinGF/F,\"\"
!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "At r = Ri, the tangential co
mponents of H must be continuous (component of h1, h2 and h3 with 'd/d
(" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 4 ")')." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "eqHi:=simplify(evalf(subs(r=Ri,h2[2
]))=evalf(subs(r=Ri,h3[2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq
HiG/*&*&-%$sinG6#%$phiG\"\"\",&*&%#B2GF,)%#RiG\"\"#\"\"\"F,%#C2G$!\"\"
\"\"!F,F3*$)F1\"\"#F3!\"\"*&%#HiGF,F(F3" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 61 "At r = Ro, the tangential components of H must be continu
ous." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "eqHa:=simplify(eval
f(subs(r=Ro,h1[2]))=evalf(subs(r=Ro,h2[2])));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqHaG/*&*&-%$sinG6#%$phiG\"\"\",&*&%#H0GF,)%#RoG\"\"
#\"\"\"F,%#C1G$!\"\"\"\"!F,F3*$)F1\"\"#F3!\"\"*&*&F(F3,&*&%#B2GF,F0F3F
,%#C2GF5F,F3*$)F1\"\"#F3F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "At
 r = Ri, the normal components of B must be continuous (component of h
1, h2 and h3 with 'd/d(r)')." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 71 "eqBi:=simplify(evalf(subs(r=Ri,mu*h2[1]))=evalf(subs(r=Ri,mu0*h3
[1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqBiG/,$*&**%$murG\"\"\"
%$mu0GF*-%$cosG6#%$phiGF*,&*&%#B2GF*)%#RiG\"\"#\"\"\"F*%#C2GF*F*F6*$)F
4\"\"#F6!\"\"$!\"\"\"\"!,$*(F+F6%#HiGF*F,F6F<" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 57 "At r = Ro, the normal components of B must be continuo
us." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "eqBa:=simplify(evalf
(subs(r=Ro,mu0*h1[1]))=evalf(subs(r=Ro,mu*h2[1])));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%eqBaG/,$*&*(%$mu0G\"\"\"-%$cosG6#%$phiGF*,&*&%#H0G
F*)%#RoG\"\"#\"\"\"F*%#C1GF*F*F5*$)F3\"\"#F5!\"\"$!\"\"\"\"!,$*&**%$mu
rGF*F)F5F+F5,&*&%#B2GF*F2F5F*%#C2GF*F*F5*$)F3\"\"#F5F:F;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "Solving these equations for C1, C2, B2 an
d Hi gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol:=simplify
(solve(\{eqHi,eqHa,eqBi,eqBa\},\{C1,C2,B2,Hi\}));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$solG<&/%#C2G,$*&**%#H0G\"\"\")%#RoG\"\"#\"\"\")%#RiG
F/F0,&%$murGF,$!\"\"\"\"!F,F,F0,.*&F-F0)F4F/F0F,*&F4F,F-F0$F/F7*&F:F0F
1F0F5*&F4F0F1F0F<*$F-F0F,*$F1F0F5!\"\"$!\"#F7/%#C1G*&*(F+F0F-F0,*F9F,F
=F5F?F5F@F,F,F0F8FA/%#HiG,$*&*(F+F0F-F0F4F0F0F8FA$\"\"%F7/%#B2G,$*&*(F
+F0F-F0,&F4F,$F,F7F,F,F0F8FAF<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The resul
t for the screening factor 'Screen' (= Hi/H0) is then given by" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Screen:=simplify(Hi/H0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ScreenG,$*&*&)%#RoG\"\"#\"\"\"%$mur
GF+F+,.*&F(F+)F,F*F+\"\"\"*&F,F0F(F+$F*\"\"!*&F/F+)%#RiGF*F+$!\"\"F3*&
F,F+F5F+F2*$F(F+F0*$F5F+F7!\"\"$\"\"%F3" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 263 33 "Graphical Display of the Solution" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 141 "For the plots, values are assigned to the parameter
s. The relative permeability for ferromagnetic materials is a very sma
ll value of mur = 6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Ri:
=1: Ro:=2: H0:=10: mur:=6:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Fo
r the contour plot, the plotted function must be written in cartesian \+
coordinates so the radius r and the angle phi are substituted by x and
 y." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Vc1:=simplify(subs(phi
=arctan(y/x),subs(r=sqrt(x^2+y^2),V1))):" "6#>%$Vc1G-%)simplifyG6#-%%s
ubsG6$/%$phiG-%'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"-F)6$/%\"rG-%%sqrtG6#,
&*$F3\"\"#F2*$F1\"\"#F2%#V1G" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "Vc2:=simplify(subs(phi=arctan(y/x),subs(r=sqrt(x^2+y^2),V2))):" "6#
>%$Vc2G-%)simplifyG6#-%%subsG6$/%$phiG-%'arctanG6#*&%\"yG\"\"\"%\"xG!
\"\"-F)6$/%\"rG-%%sqrtG6#,&*$F3\"\"#F2*$F1\"\"#F2%#V2G" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "Vc3:=simplify(subs(phi=arctan(y/x),su
bs(r=sqrt(x^2+y^2),V3))):" "6#>%$Vc3G-%)simplifyG6#-%%subsG6$/%$phiG-%
'arctanG6#*&%\"yG\"\"\"%\"xG!\"\"-F)6$/%\"rG-%%sqrtG6#,&*$F3\"\"#F2*$F
1\"\"#F2%#V3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The plot a piece
wise function is defined." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
Pot:=piecewise(x^2+y^2<Ri^2,Vc3,                                      \+
                     x^2+y^2>Ri^2 and x^2+y^2<Ro^2,Vc2,               \+
                     x^2+y^2>Ro^2,Vc1):" "6#>%$PotG-%*piecewiseG6(2,&*
$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F-*$%#RiG\"\"#%$Vc3G32*$F2\"\"#,&*$F+\"\"
#F-*$F/\"\"#F-2,&*$F+\"\"#F-*$F/\"\"#F-*$%#RoG\"\"#%$Vc2G2*$FE\"\"#,&*
$F+\"\"#F-*$F/\"\"#F-%$Vc1G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "N
ow the equipotential lines can be plotted. In addition, two circles ar
e included so that the location of the tube is indicated." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "with(plottools):\nc1 := circle([0,
0], Ri, color=red):                                     c2 := circle([
0,0], Ro, color=red): p:=plots[contourplot](Pot,x=-3..3,y=-3..3,scalin
g=CONSTRAINED,        color=blue,numpoints=4000):\nplots[display](c1,c
2,p);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6--%'CURV
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{PARA 0 "" 0 "" {TEXT -1 145 "In this case, the screening factor is ab
out 0.56, so that the magnetic field inside the tube is still more tha
n half of the value on the outside." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "Screen;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)3NSh&!
#5" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 7 "Summary" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 186 "For some applications, magnetic fields need to
 be shielded. In the preceeding calculation, an illustrative example i
s given on how this can be accomplished. A ferromagnetic tube (with a \+
" }{TEXT 257 4 "very" }{TEXT -1 104 " low relative permeability) is pl
aced in a static magnetic field and the screening factor is calculated
." }}}}}{MARK "1 1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }