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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 259 12 "Ampere's law" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 375 "Gauss' law rel
ates the integral over a closed surface of the electric field vector t
o the charge contained in the volume bounded by the surface. Ampere's \+
law represents the equivalent for magnetic fields generated by electri
c currents. It is based on the Lorentz force, which forms the basis of
 understanding the force between two parallel wires that carry electri
c currents." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 55 "The law states that the integral of the magnetic field " }
{TEXT 258 1 "B" }{TEXT -1 128 " over a closed contour that encloses a \+
current is proportional to the current. The proportionality constant i
s the permeability " }{XPPEDIT 18 0 "mu[0]" "6#&%#muG6#\"\"!" }{TEXT 
-1 38 " which is related to the permittivity " }{XPPEDIT 18 0 "epsilon
[0]" "6#&%(epsilonG6#\"\"!" }{TEXT -1 15 " (the product  " }{XPPEDIT 
18 0 "epsilon[0]*mu[0]=1/c^2" "6#/*&&%(epsilonG6#\"\"!\"\"\"&%#muG6#F(
F)*&F)F)*$%\"cG\"\"#!\"\"" }{TEXT -1 18 " ). In MKSA units " }
{XPPEDIT 18 0 "mu[0]" "6#&%#muG6#\"\"!" }{TEXT -1 8 " equals " }
{XPPEDIT 18 0 "4Pi/10^7 " "6#*(\"\"%\"\"\"%#PiGF%*$\"#5\"\"(!\"\"" }
{TEXT -1 7 " Tm/A ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 102 "The right-hand rule that underlies the Lorentz force d
etermines also the orientation of the generated " }{TEXT 257 1 "B" }
{TEXT -1 35 " field with respect to the current." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 352 "As with Gauss' law, the \+
usefulness of Ampere's law is restricted by the fact that a high degre
e of symmetry must be present, so that simple contours can be found on
 which the magnetic field is constant. If this is the case, the field \+
can be brought outside the line integral and a simple product of its s
trength times the length of the contour results." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "For a straight wire it i
s apparent that the right contour is given by concentric circles in th
e plane perpendicular to the wire. Thus" }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Ampere:=B*2*Pi*r=mu[0
]*I[0];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Outside of the conduct
or we have for the field:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "Brout:=solve(Ampere,B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "T
he strength of the magnetic field decreases with the distance from the
 wire. As before with Gauss' law we can also use Ampere's law inside t
he current distribution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 49 "Suppose the cylindrically shaped wire has radius \+
" }{TEXT 260 1 "R" }{TEXT -1 97 " and the current density inside the w
ire is constant. What is the magnetic field inside the wire?" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We use a circle
 with radius " }{TEXT 262 1 "r" }{TEXT -1 2 " <" }{TEXT 261 2 " R" }
{TEXT -1 141 ", which encloses a fraction of the current. The constant
 current density is given as total current divided by the total cross-
sectional area:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "j:=I[0]/
(Pi*R^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The amount of curren
t enclosed by a cross-section of radius " }{TEXT 263 2 "r " }{TEXT -1 
10 "is called " }{TEXT 19 2 "Ip" }{TEXT -1 19 " (partial current):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Ip:=j*Pi*r^2;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 160 "Ampere's law now provides a statement ab
out the constant magnetic field strength on the circle surrounding the
 enclosed area, which is permeated by the current " }{TEXT 19 2 "Ip" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Ampere:=B*
2*Pi*r=mu[0]*Ip;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "We observe t
hat the magnetic field strength inside the conductor grows with the ra
dius (of the circle on which it is constant):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "Brin:=solve(Ampere,B);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "P1:=plot(subs(mu[0]=1,I[0]=1,R=1,Brin),r=0..1):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P2:=plot(subs(mu[0]=1,I
[0]=1,R=1,Brout),r=1..5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display
(\{P1,P2\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 328 "Experimentally n
o sources or sinks of magnetism, i.e., no magnetic monopoles are known
 to exist. Magnetic field lines are always closed. One can formulate G
auss' law for magnetism, but given that there is no equivalent of char
ges (permanent magnets are always dipoles), the magnetic flux through \+
closed surfaces always vanishes." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 24 "Gauss' law f
or magnetism" }}{PARA 0 "" 0 "" {TEXT -1 345 "We can use Gauss' law fo
r magnetic fields to calculate the magnetic flux through a square wire
 loop that is placed between the poles of a permanent magnet (or an el
ectromagnet). The magnet produces a constant field, e.g., of  B = 0.01
 T (Tesla). The square wire loop with sides l = 0.01 m is tilted with \+
an angle of 20 degrees w.r. to the field." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "We need to find the oriented surf
ace element and calculate its scalar product with the field. The vecto
rial surface element d" }{TEXT 256 1 "A" }{TEXT -1 115 " is perpendicu
lar to the loop, and thus stands at an angle of theta=90-20 degrees w.
r. to the field. Thus, we have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "theta:=(90-20)*Pi/180;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "l:=0.01;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Phi:=B*c
os(theta)*l^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "B:=0.01;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(Phi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 87 "The magnetic flux is calculated in Tesla*
m^2, a unit which carries the name Weber (Wb)." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 28 "Magne
tic field of a solenoid" }}{PARA 0 "" 0 "" {TEXT -1 294 "Ampere's law \+
permits to calculate approximately the magnetic field strength inside \+
a solenoid. Many turns are contributing their identical magnetic field
s to add up (superposition principle for magnetic fields). We consider
 an ideal solenoid, i.e., a long coil for which we ignore edge effects
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 437 "Con
sider a cross section through the length of the solenoid and observe h
ow the top and bottom parts of each turn contribute with their circula
r B fields to the overall magnetic field inside and outside of the coi
l. The net result is a very strong nearly homogeneous field inside the
 solenoid, with diverging field lines at the two ends to close in larg
e loops around the length of the solenoid, i.e., a very weak field out
side the coil." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 257 "To calculate the magnetic field inside the coil we imagi
ne a rectangular loop with one long side inside the solenoid, the othe
r outside with the two short legs making the connection. The solenoid \+
consists of a single layer and the rectangular loop encloses " }{TEXT 
265 1 "N" }{TEXT -1 34 " turns of wire carrying a current " }{TEXT 
266 1 "I" }{TEXT -1 306 ". The line integral along the rectangular loo
p has to be carried out in a specific direction consistent with the ri
ght-hand rule. Only the long leg inside the solenoid contributes to th
e line integral over the magnetic field: outside the solenoid the fiel
d is weak. The short legs are perpendicular to the " }{TEXT 267 1 "B" 
}{TEXT -1 47 " field, and therefore do not contribute at all." }}
{PARA 0 "" 0 "" {TEXT -1 28 "The enclosed current equals " }{TEXT 269 
1 "N" }{TEXT -1 1 " " }{TEXT 268 1 "I" }{TEXT -1 16 ", and therefore:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "B:='B': l:='l':" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Ampere:=B*l=mu[0]*N*I[0];" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "B:=solve(Ampere,B);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This is the magnetic field strengt
h inside the coil." }}{PARA 0 "" 0 "" {TEXT -1 57 "One can also introd
uce a number of turns per unit length " }{TEXT 270 1 "n" }{TEXT -1 16 
", and eliminate " }{TEXT 271 1 "l" }{TEXT -1 19 " upon substitution \+
" }{TEXT 272 2 "N " }{TEXT -1 2 "= " }{TEXT 273 1 "n" }{TEXT -1 1 " " 
}{TEXT 274 1 "l" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 267 "Note \+
that the field is homogeneous inside the coil. This result follows fro
m the fact that the short legs do not contribute at all to the line in
tegral, i.e., the same result is obtained independent of the location \+
of the contributing leg of length l inside the coil." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "We can use the above f
inding together with some basics about conductivity to understand the \+
design of electromagnets. We consider a single-layer coil with wire of
 thickness (diameter) " }{TEXT 275 1 "d" }{TEXT -1 13 ". Obviously, " 
}{TEXT 276 4 "N d " }{TEXT -1 2 "= " }{TEXT 277 2 "l." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "l:=N*d;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 2 "B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Now suppo
se a wire with diameter " }{TEXT 278 1 "d" }{TEXT -1 124 "=0.6 mm can \+
carry at most a current of 30 mA. We can calculate the maximum possibl
e field strength inside the electromagnet:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 53 "evalf(subs(I[0]=0.03,d=0.0006,mu[0]=4*Pi*10^(-7),B)
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "The field of 0.06 mT is ra
ther weak. How can we increase it? The amount of current that one can \+
pass through the wire depends on its cross sectional area, i.e., the s
quare of the diameter. On the other hand the number of turns for a giv
en length " }{TEXT 279 1 "l" }{TEXT -1 109 " decreases with the diamet
er. We find that the maximum value of B is proportional to the diamete
r as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(I[0]=
c1*d^2,B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Exercise: an inter
esting variation of the above is the toroidal coil: a torus defined by
 the big radius " }{TEXT 280 1 "R" }{TEXT -1 29 " (diameter) and small
 radius " }{TEXT 281 1 "r" }{TEXT -1 115 " (cross section) has no edge
 effects due to the circular hsape. However, the field is not constant
 inside the coil." }}{PARA 0 "" 0 "" {TEXT -1 45 "Calculate the variat
ion in B with the radius " }{TEXT 282 2 "R'" }{TEXT -1 6 " for  " }
{TEXT 283 1 "R" }{TEXT -1 1 "-" }{TEXT 284 1 "r" }{TEXT -1 5 "  <  " }
{TEXT 285 2 "R'" }{TEXT -1 5 "  <  " }{TEXT 286 1 "R" }{TEXT -1 1 "+" 
}{TEXT 287 1 "r" }{TEXT -1 63 "  using Ampere's law. Again, ignore the
 field outside the coil." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 288 26 "A semi-realistic solenoid:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 527 "We can use Maple'
s graphing capabilities to draw the magnetic field vectors for a solen
oid with a finite number of turns in a cross section through the cente
r of the coil. We make use of the result of the dependence of the magn
etic field strength on the radial distance from the wire obtained abov
e. We use the fact that the sign of the field lines around the cross s
ections of the wires in the top half is opposite to the bottom half, s
ince the current enters the plane in one half and exits from the plane
 for the other half." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 23 "We use arbitrary units." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B:='B':" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Ampere:=B*2*Pi*r=mu[0]*I[0];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Brout:=solve(subs(I[0]=1,mu[0]=1,Am
pere),B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The following proced
ure takes a point in the plane with coordinates (" }{TEXT 297 1 "x" }
{TEXT -1 2 ", " }{TEXT 298 1 "y" }{TEXT -1 220 ") and performs a sum o
ver the contributions to the magnetic field at that point that origina
te from 15 turns of the coil. Note that the expression derived from Am
pere's law provides the field strength at radial distance " }{TEXT 
289 1 "r" }{TEXT -1 81 " from the wire. The direction of the field is \+
tangential to the circle of radius " }{TEXT 290 1 "r" }{TEXT -1 164 " \+
at that point. The direction of the field is taken into account by eit
her adding positive or negative 90 degrees (Pi/2) to the angle of the \+
line connecting point (" }{TEXT 292 1 "x" }{TEXT -1 2 ", " }{TEXT 291 
1 "y" }{TEXT -1 180 ") with the point where the wire intersects the pl
ane. Maple's arctangent function with two arguments covers a range of \+
2 Pi! The procedure returns a 2dimensional 'vector' with the " }{TEXT 
293 1 "x" }{TEXT -1 5 " and " }{TEXT 294 1 "y" }{TEXT -1 53 " componen
ts of the total magnetic field at location (" }{TEXT 296 1 "x" }{TEXT 
-1 2 ", " }{TEXT 295 1 "y" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 448 "Imagine that the coil is place
d horizontally: in a cross-sectional diagram the top row of current cr
ossing through the plane corresponds to current going 'into', and the \+
bottom row to current coming 'out' of the plane. Note that the sign of
 the current has been picked together with the orientation of the wind
ing. To switch either of the two one would switch the convention of ad
ding/subtracting for the top and bottom rows of intersection points." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 "The re
sulting field vector consist of a significant number of 'cancellations
': imagine a location directly between two intersections of the wire i
n the top (or bottom) half and what the effect of circular field lines
 associated with the two wires is at that point." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Field:=proc
(x,y) local i,B,xi,yi,phi,Bbotx,Bboty,Btopx,Btopy; global Brout;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Btopx:=0: Btopy:=0: #top wires firs
t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1 to 15 do:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xi:=i*0.2: yi:=0.5:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "B:=evalf(subs(r=sqrt((x-xi)^2+(y-yi)^2),Brout)
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "phi:=arctan(yi-y,xi-x);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Btopx:=evalf(Btopx+B*cos(phi+Pi/2))
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Btopy:=evalf(Btopy+B*sin(phi+P
i/2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "Bbotx:=0: Bboty:=0: #bottom wires now:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "for i from 1 to 15 do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "xi:=i*0.2: yi:=-0.5:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "B:=evalf(subs(r=sqrt((x-xi)^2+(y-yi)^2),Brout));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "phi:=arctan(yi-y,xi-x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "Bboty:=evalf(Bboty+B*sin(phi-Pi/2));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "Bbotx:=evalf(Bbotx+B*cos(phi-Pi/2));" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 35 "od: [Btopx+Bbotx,Btopy+Bboty]; end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Field(1,0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 73 "fieldplot(Field(x,y),x=-1..4,y=-2..2,color=blu
e,arrows=thick,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "The homogeneity of the ma
gnetic field inside the coil is apparent. The fieldplot shows also ver
y clearly how the field weakens outside the solenoid." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 300 10 "Exercises:" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 83 "1) Vary the dimensions of the coil, i.e., number, \+
diameter and spacing of windings." }}{PARA 0 "" 0 "" {TEXT -1 140 "2) \+
Do the results change if the top and bottom rows of intersections are \+
shifted by one half of the spacing (as is the case in a real coil)?" }
}{PARA 0 "" 0 "" {TEXT -1 43 "3) Draw a fieldplot for a single-turn co
il." }}{PARA 0 "" 0 "" {TEXT -1 123 "4) Write a procedure that allows \+
to vary these parameters through the argument list or by modifiaction \+
of global variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 217 "In another worksheet a more accurate calculation of th
e magnetic field is performed by summing the exact magnetic field comp
onents for current loops (Currentloop.mws), and constructing magnetic \+
field lines from these." }}}}{MARK "0 0 0" 12 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }