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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Computer Intensive Physic
s" }}{PARA 18 "" 0 "" {TEXT -1 23 "Physics 212E: Fall 1997" }}}{EXCHG 
{PARA 19 "" 0 "" {TEXT -1 16 "Steven R. Dunbar" }}{PARA 19 "" 0 "" 
{TEXT -1 40 "Department of Mathematics and Statistics" }}{PARA 19 "" 
0 "" {TEXT -1 30 "University of Nebraska-Lincoln" }}{PARA 19 "" 0 "" 
{TEXT -1 23 "Lincoln, NE, USA  68588" }}{PARA 19 "" 0 "" {TEXT -1 20 "
sdunbar@math.unl.edu" }}{PARA 0 "" 0 "" {TEXT -1 32 "http://www.math.u
nl.edu/~sdunbar" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 417 "\n8-1:  Biot-
Savart law, Ampere's Law, \n\nKEYWORDS: Biot-Savart Law, Ampere's Law,
 magnetic field, current, \n\nOBJECTIVE: Write the Biot-Savart law and
 employ it to find the magnitude and direction of  the differential ma
gnetic field dB at a point P1 caused by a differential current element
 at another point P2 and find the the magnetic field B at the center o
f a circular or semicircular loop of current-carrying wire." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 496 "Write Ampere's
 law \\int \\mathbold\{B\} \\dot d\\mathbold\{l\} = \\mu_0 I, define a
ll symbols appearing in it, and apply the sign convention relating the
 positive directions of d\\mathbold\{l\} and I.  Use Ampere's law to d
etermine the magnitude and direction of the magentic field B caused by
 currents in appropriately symmetric conductors, such as a long straig
ht wire, a long solenoid, or a toroid, apply the principle of superpos
ition to find the resultant when these fields are combined in simple w
ays." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "
\nPREREQUISITE: Know the principle of superposition, know basic vector
 calculus, including cross product and line integrals." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "LENGTH:  One class p
eriod, about 50 minutes." }}{PARA 0 "" 0 "" {TEXT -1 59 "AUDIENCE:  In
structor, student use in computer lab setting." }}{PARA 0 "" 0 "" 
{TEXT -1 21 "SOFTWARE: Maple V R 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 105 "FUTURE:  Draw the figure, similar to Fig
ure 32-3, page 451 in Sears-Zemansky, also Figure 32-7, page 454." }}
{PARA 0 "" 0 "" {TEXT -1 146 "Add a derivation of the nearly uniform m
agnetic field due to Helmholtz coils.  See Williams and Spangler, Chap
ter 26, problem 40 for a derivation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 69 "HISTORY:  Created October 15, 1997, sli
ghtly revised October 18, 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Suggested InfoMall Reading:  Dudl
ey Williams and John Spangler, Physics for Science and Engineering, Su
ggested Keywords:  Biot-Savart law, Ampere's Law." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 46 "Magnetic Field Ve
ctors Surrounding a Long Wire" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "
Fact 0:  A moving charge (current) creates a magnetic field." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "Fact 1:  The m
agnetic field set up at any point by the current in a circuit is the v
ector sum of all the fields due to all the moving charges in the circu
it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 243 "F
act 2: Ampere's Law (also called Biot law, Biot-Savart law):  The Fren
ch physicist Jean Biot in 1820 deduced that the vector of the magnetic
 force at a point P distance s away  caused by a current I flowing in \+
a short segment of circuit dl is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 1 "d" }{TEXT 257 1 "B" }{TEXT -1 19 " = (mu0/
(4*Pi)) I d" }{TEXT 256 2 "l " }{TEXT -1 2 "x " }{TEXT 259 1 "s" }
{TEXT -1 5 "/s^2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 67 "d\\vector\{B\} =   (mu0/(4*Pi))* I d \\vector\{l\} \\cros
s \\vector\{s\}\}/s^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 41 "Fact 3:  Teh total magentic flux density " }{TEXT 260 
2 "B " }{TEXT -1 67 "at P due to the entire entire circuit is the vect
or sum of all the " }{TEXT 261 2 "dB" }{TEXT -1 41 "'s due to the curr
ent flowing in all the " }{TEXT 262 2 "dl" }{TEXT -1 22 "'s around the
 circuit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 90 "Now consider  a long current-carrying wire along the y-ax
is.   The current is set to be I." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 63 "We will consider the contribution of the \+
small charge element d" }{TEXT 258 1 "l" }{TEXT -1 54 " located along \+
the segment from  (0,y,0) to (0,y+dy,0)" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 124 "Consider a point  a distance r from
 the current-carrying wire. We will derive the magnetic force field ve
ctor at this point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 71 "Consider (without loss of generality) the point  (0,0,r
) on the z-axis." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 67 "Explain why you can consider this point without lo
ss of generality:" }}{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 22 "restart; with(lin
alg);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 62 "Note the following use of ordered-triple notation for vec
tors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "
The vector r from (0,y,0) to (0,0,r)  is (0,-y,r)." }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 94 "Write (in Maple notation) the unit vector in the dir
ection from current segment dl to point P:" }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "bolds := vector(
 [   ,    ,    ] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 64 "Write the distance s to from the current elemen
t to the point P." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "s :=        ;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Written Answer (Not Maple
 Input):  Write a unit vector in the direction of the current:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
dboldl := vector( [0, i, 0]); # dy" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "dboldB := scalarmul( crossp
rod(dboldl, bolds), ((mu0)/(4*Pi))/s^2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "Written Answer:  What is
 the direction of the element of magnetic force?  Why does the directi
on of this element of the  magnetic force vector point in this directi
on?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xcomponent := dboldB[1];" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(xcomponent, y = -infini
ty .. infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 91 "What other electromagnetic field fact that we have
 earlier derived does this remind you of?" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 62 "Magnetic Field of a Circular Tur
n of Wire Containing a Current" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 93 "Consider a circular turn of wire of r
adius R carrying a current I and lying in the x-y plane." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Point P is on the \+
axis of the turn (the z-axis) at a distance r  from the center." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "restart; with(linalg);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 82 "What is the parametric representation o
f of a circle of radius R in the x-y plane?" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "current_element_position := vector([ R*cos(t), R*sin(
t), 0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dboldl := vecto
r( [i*(-sin(t)), i*cos(t), 0]); # dt" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "boldP := vector( [0,0,r]);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 50 "dbolds := evalm(boldP - current_element_position);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dbold_unit_s := evalm( \+
dbolds/sqrt(R^2 + r^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"s := sqrt(R^2 + r^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "d
B := scalarmul( crossprod( dboldl, dbold_unit_s), (mu0/(4*Pi))/s^2 );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dB_x := dB[1];\ndB_y :=
 dB[2];\ndB_z := dB[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "
B_x := int(dB_x * R, t = 0..2*Pi);  # factor R from polar coord integr
ation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B_y := int(dB_y * \+
R, t = 0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B_z := \+
int(dB_z * R, t = 0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "No
w calculate the field  at the centrer of the circular loop" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(subs( , ));  # erase" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "26 0 0" 33 }
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