Physics48 (Fall20l1)
Anr-4:l
--_:_==:_
Chapter 29: Magnetic Fields Due to Carrents
_,\v.
"Nothing can bring you peace but yourself. " - Ralph lValdo Emerson
"Thefoolish man seekshappinessin the distance,the wise man grows it under hisfeet. "
JamesOppenheim
"Happinessis not a stcrtion),ou arrive at, but a mannerof traveling." - MargaretB. Runbeck
Reading: sections764- 781
Outline:
= Biot-SavartLaw
magneticfie1dfrom a long straightwire
magneticfield from a infinite straightwire
magneticfield at the centerofa circulararc
+ forcebetweentwo parallel cunents
= Ampere'sLaw
usingAqtp"t"'. tu*
= solenoids
andtoroids
+ currentcarryingcoil asa magneticdipole (readon your own)
ProblemSolvingTechniques
A few of the eariyproblemsdealwith the fieid of a long shaightwire. You shouldbe ableto find the
magnitudeanddirection of the field at anypoint in space,given the currentin the wire. Use
B : /hi/21v to find the magnitudeandthe right-handrule to find the direction.You may be askedfor
the total fie1dof two or more long straightwires. You will then needto carry out vector addition.
Onceyou havefoundthemagneticfield you maybe askedfor the forceit exertson a moving
charge.Use qi x E . Don't forgetto includethe sineof theanglebetweeni and .D whenyou
calculatethemagnitude.Know how to usethe right-handrule to find thedirectionofthe force.Be
sureto takeinto accountthe signofthe charge.
Someproblemsaskyou to usetheBiot-savartlaw to computethe magneticfield of a current.Divide
thecurrentinto infinitesimaleiements,
write the expression
for the field ofan element,thenintegrate
eachcomponent
overthe current.You will needto writetheintegrandin termsof a singlevariable.
If the wire is straightplaceit alongthex-axis,say,andusethe coordinate
x ofpoints alongthewire
asthe variableofintegration.If thewire is a circularloopusethe anglemadeby a radialline with a
coordinateaxisasfhe variableofintepration.
ln many casesyou may think of an electricalcircuit as composedof finite straight-1ineand circulararc segments,eachof which producesa magneticfield. You can then calculatethe field producedby
and vectorially sum the individual fields to find the total field. Use the result given in
eachs"egment
Probleri 11 if you need to frnd the field on the perpendicularbisector of a finite straightwire. To
frnd the field at someother point you sill needto integlatethe BiofSavart 1aw.Use the procedure
given in Section 1 for an infinite wire but replacethe limits of integrationwith finite values.lJse
B : t4id l4nR for the field of a circular arc at its centerof curvature'
Someproblemsask you to usewhat you leamedin the last chapterto calculatethe force that one
wire exefison another.
you to
Arnpere'slaw problems take three forms. The most straightforwardgive the currentsand ask
find the value of {f . ai arounda given path. You must pay attentionto the directionsof the
observewhich currents are encircledby
currentsasthey pierce the plane of the path and carefu11y
the path and w-hichare not. Other problemsask you to useAmpere'slaw to calculatethe magnetic
fieli. Carefully choosethe Amperian path you will use andpay attentionto the evaluationof the
Ampere'slaw integral in terms of the unknown field. Still other problems give you the magnetrc
heli as a function ofposition and ask for the currentthrough a given region. Carry out the line
integralofthe tangentialcomponentofthe field aroundthe boundary of the region and equatethe
result to /41, then solve for l.
(in
You shouldknow how to computethe magneticfields of somespecialcurrent configurations
ideal
of
a.n
field
The
additionto a long straightwire): a solenoid,a toroid, and a magnetic dipole.
=
solenoidis given by B : ltnni inside the solenoid andby B 0 outside.The fieid inside is aiong the
:
cylinder axis. The field of a toroid is given by B pnNl2tn at a point inside, a distancer fiom the
center.The field inside the hole and outsidethe toroid is zero.The field lines inside are circlesthat at
B = po F lZtz3 at a point on the
concentricwith the toroid. The field of a magneticdipole is given by
axis definedby the direction ofthe dipole moment,a distancez from the dipole. You shouldreca11
from the last ciapter how to find the dipole moment of a curent loop (both magnitudeand direction)
and how to computethe torque of a uniform magneticfield on a dipole'
QuestionsandExampleProblemsfrom Chapter29
QuestionI
in whichlongparallelwirescarryequalcurrentsdirectly
Tie figurebelowshowsfour arrangements
to the
accolding
Rankthearrangements
intooiout ollhe pageat thecomersofidenticalsquares.
magnitudeof the net magnetic field at the centerof the square,greatestfirst.
larar?l\
q-----o
o-----o
tqlq-l
!, K
. - ' - / 2' tr
l\',D
:
xal
i
O *"6\'l
--*"O
q
t,tJ
3
I
; z/tx ;
0-*---a
4th\3
9- _---6
i.,Yi
d**---o
Ll
(,)
3
+-----9
i\i3,,X
|
1
.,/,/, \
t,'/'\|
:. t
|
o"-*--o
4
(,!j
Question2
The Figurebeiow showsthree cirouits, eachconsistingof two radial lengths and two concentricarcs,
one ofradius r and the other ofradius R > r. The circuits have the samecurrent through them and the
sameanglebetweenthe two radial lengths.Rank the circuits accordingto the magnitudeofthe net
magneticfield at the center,greatestfirst.
,--
(?
B=
R
('
)
[.]l
I
itua
I *8 -+ ,c'"R &"4^; "" d'
"6
Y1
(0)
(e)
q.--n A,\&,
E 4wJTI ;2!AI a)''v\'.r'J )JL)t\4iJ
n
p'aih
A,n""ta "7 B-rr *;tl 4 ar\c
(c,
C-+ cr* "d Ps,
A-
A--1 ),,t6
<9^'b, o'L a^.aLLl'o'\ ot"\'r
-.I-
( - r ntagaAa
tik Vry A --r ,u^.t,p-go
c, o, b
Problem1
d: 16.0cm carry current\:3.67 mA and
Ia the{igurebelow,two longstraightwiresat separation
iz 3.00hout of thepage.(a) At whatpoint on thex axisshownis thenet magreticfield dueto the
currentsequalto zero?(b) If thetwo currentsaredoubled,is thepointofzeto magneticfield shift
towardwire 1, shiftedtowardwire 2, or unchanged?
d-x
.1, c
B.*r -r, ^%yny',o
;e"
l[ro
Ml,il4j
-et*uea
B, pr"di, v1\u)'Atdn)
F-"i-J
K=l-(
l"
JJo
L r
?..rr(-r
A, L,
4..-Tra
_
t-
a.
L\o La
2..ir C^
= ad3';J
----+ I=
8,ri.(a-.)
f= f,/,t = q*
(b)
I
(-
?
l
{-c
Jr*= d
-(.
-- z/.oer,
arQz',g ^a Lx= 3->at,, )gv -?arfi,';'
"B
D^l .o Aoeuii rf*y^
Problem2
In thefigurebe1ow,a currenti: 10A is setup in a long hairpinconductorformedby bendinga wire
into a semicircleof radiusR = 5.0mm. Pointb is midwaybetweenthe straightsectionsandso
asbeinganinfinitewire. Whatare distantfrom the semicirclethateachsectioncanbe approximated
the (a)magnitudeand(b) direction(into or out ofthe page)ofB at point a ard the (c) magnitudeand
(d) directionofB at point b.
1-
,*$ll,.,&
t'
;ggt*
afi"a,'4t,i;
gtnni,lwi 6,n1
zu"kd
p,= n,;r1
"3
7;F,
i,o^rtTdil
= Xt *)
E-..
Ir N\Q- "'
etifi
\
I
= dlf
r1 A
- (.ln"r;.'I',/n)(134: i t.o'.w) atery"4ni
-rr (
>.;
5oo^ 16
Problem3
In thefigurebelow,a wire formsa semicircleof radiusR: 9.26cm andtwo (radial)straight
.-_/
=
are
the
i
mA.
What
cm.
wire
carries
current
34.8
13.1
The
eachof lengthL:
segments
(a)magriitudeand(b) direction(into or out of thepage)ofthe netmagneticfield at the semicircle's
centerof curvatureC?
-0eNt"J^a"{r,f^rfl'c,n ;
I
?r
t_l
t-'l
I
A,4tl* rt-'Artf ,Ory'r',,atr}
os
--lEria
o
c
:::--(-
4r
dt
t-t
F\
B=o
t-
dB=&
qf
JB = ,^t:- {jiiad=o
i Jr a,,"*tta' --o ----) B =o
6?.
F
o)
-)
"1
lfi, u,^bt 4
fl=
4::j_-_A,;
tcl
tanA
q r;^tvc^-t'a-t
o'vL "A
= @
4R
b= &
|
d
= n a",*,"'vt
--l"tt*tdnl4(o.oqa.6m)
Ilr B'= l'lrr16-zTr"hJtuo'.+I
Problem 4
In the figure below, point P2is at perpendiculardistanceR:25.1 cm from one end of straightwire
of lengthL = 13.6 cm canytng current l= 0.693 A. (Note that the wire is not long.) What is the
magnitudeof the magretic freld atPz?
rd=* t#
?','4t
i
I
dg t"-" 'ie/ln",@fu4*
l!
a^r*"tr pa.ag
JB=
j*- +:: ---)B-+? (
Jsi"g
\
)nt
x
A"t ( anL
a---.) p.=
4';A (
P
.r
(Fril'u
4t
)
"t;- )
ds--. Jx
4i^e, = -B-.v"
CA\x")
rl".n
A +
l#-,-=;fiju" g: a";
i?6n'^/
fx= l**x*
V'A
B=
dx
6;t*
-L
-l'
l-]a* lO-T ru* a6
Problem.5
form a squareof edgelengtha : 8.50cm.Eachwire carries15.0A, andall thecurrentsareout of
thepage.In unit-vectornotation,what is the net magneticfield forceper meterof wire lengthon
wire 1?
rr $a'*' .b&a"een )' 6a-""'-0{at ouuvrz,da'-topu.r-*,.1
t
.1.
L
rl
Fl\
t,
XJ'U
a--*S
^
tt
rr \
3
r
xF,=
+#"*ffi,t
)Ar,lL
p*fM
cu^",f4 "fr-,}
t\?41&
= 4' Lt i r"'r*;
q
l\
a
I
.
X L = F^r,E ,*"e
tFr= -trr-tr"-^e
!l-
flf-
,"
O.45"
I
@-.. n-.*@
.'
LtL^
atrd
a.
.l
!
J4n,
^l
l
!!
r{
=
I
'-
d
e
l'-t
:
1t],,.L
aj rq
:
z(Ltn r l Ido
tl Yt1
I
(l 5.oAl
.a
L
w
.].
J P-L
q 7-q
J-T
,
=
( o.ots.,)
'
" Ni,,)1
t - ('t,z.l''to
(1,.t\,1c li/,",)
Problem 6
The figure beiow showstwo very long straightwires (in crosssection)that eachcarry a currentof
4.00 A tlirectly out of the page.Distanced1: 6.00 m and distanced2:4.00 m. What is the
magnitudeofthe net magnetic field at point P, which lies on a perpendicularbisectorofthe wires?
I
34 : 1e* ---r o= 3t".1"
l,oo r'\
d a.
i= S'ootl
tr--
i ",
/
-- qO-g = 9o-3r".1' -- S3.1.
d
U./
-;::".{(
l
Z=:-
= 4ui?
P -AD\'x".t
I'tT I-
PJ
3,
1\t/u'
-
A
PF4I
do = 8,4^d* B.d^,d
-- a B.o"nfl
. -- 4
I
a\L
B^"*
)A^t
1
f
af
l - o m 6/
=
J
't
'
: -Tf
9-z:-nNn$
I
f
= (4'i. ro'to/n) (q.oa*) _o,v\53-1.
- r 1 .( 5 . o o q )
r
= 5. ll,* ld tT ,pr""""\
Problem 7
Showthat a ruriformmagneticfield cannotdrop abruptiyto zero(asis suggestedby the lack of fie1d
to E , sayalongthe
linesto theright ofpoint a in the figurebelow)asonemovesperpendicular
pathshownby the
horizontalarrowin thefigure.(Hint: Apply Ampere'slaw to therectangular
dashedlines.)In actualmagnets"fringing" of themagneticfield linesalwaysoccurs,whichmeans
zeroin a gradualmanner.Modify the held iinesin the figureto indicatea more
that .E approaches
realisticsituation.
r't'
I r--l--
I r, I
|:
t
'
(
f
\, t \ B,'d s, + \
'^ s a ' d| --'s e*
O
( nr"o)
Dq'd>"
r\l
f-!
( t>\
.-{ 3
A-
-
)
.+1. r ,i
I r*!i r
o
( B ,r l u , ;
-=(
(r
n'
J.--f-'
Jr'9 t
q b'.i5
I B"ls'
J-,
t_, ,
L,-
,/'
(
L
:\,.
--
Bu(,At,. VL
,/ -A
r
-)
\B"ds"+\Bu"dr,
o
( B * - !d s " )
..-.r
j
i>
r
Problem8
A solenoid1.30m long and2.
insidethesolenoidis 23.0mT
,
esa currentof 18.0A. Themagneticfie1d
wire formingthe solenoid.
l --1"3a^
\, -I"3ocr)
C
= d./d ^\
|
-I,jo ^lo-tm
?
L
--
r<
^
$'ot "'"t1Ata!
e
t-\
h= E'3'o*lo"r
-= 8,.- = 6'3'o"lo-37
(+r^lotTr/n) ( tr'ai+;
4;
tl
.,
i:=
:
B =A"; n
n
| o'<z 1-1
a*o" i)'
t\l
D= t:!l9s-=!Y=t
t_.
ge,d
,We -t-nfr
h= l"o)..x|o3fu,n/n
+ F,,ru U%"P4)x )sv\etrP4/a'"
(ne)x(?'rrr)
:
^)Yrtl
(r.oa^o.w,.7^) ( l"ao';( A-r) ( l'3oxrotr)
=l-loe;l
Problem9
A long solenoidhas 100tums/cmandcarriescunentL An electronmoveswithin the solenoidin a
to thesolenoidaxis.The speedof the electronis 0.0460c
circleofradius2.30cm perpendicular
(c: speedoflight). Find the currenti in thesolenoid.
----t F =-DV*,/'\ eA.r?
t
%B
k"B
tri
-
=Ioi
I
I
t
tt
)
I
J
?.lt* lo*?'t(S
o
O.D'l6f (:.oa16r*/s)
|-3t^ foern/s
q= l -i"o'a*lo t*c
= 4 r^ to " rnr/n
ri
\-1
l-
-
(* = \v
3 ('u'; n)
I
lootP^]#=1oq;Eonu/.,
O . oa,3or.,,
r\V
qr"n {(t.tl' td'' K+)t l'3t- lD?r'i/,
(r,oor*td'*e)(lri-r5?r"6)1
Qrlfr)
ll:;il]
t"
b. ollon)
Problem10
The figurebelow showsan arrangementknown asa Helmholtz coi1.It consistsof two circular
by a distances : R. Thetwo coiis
coaxialcoils,eachof 200 tumsandradiusR = 25.0cm,separated
carryequalcurrentsi: 12.2mA in the samedirection.Find the magnitudeof thenet magneticfield
at P, midwaybetweenthe coils.
J]* l"'ae.,&'/
)
)#G u^#n'r -f,VsJr,. )4
'.
g*94 t
At-r*,an* z aLxg
+*n o rcc"L 4, tt l*r*
3Gi
_
i"
Nl ,r,1.( .r,
2-l4"*z)"/''
\
=)
ai
poa^,*P o d,ut*t'*n z= Alr. 4"E"' a^4
O& t fu nog..-fui yA; $*n Qle\ /od^l'P*ift 'nr #v + x-t
^ f N l^"t8",
lJ^.d= l[;;;l]!12.
- 4
+L'vr/
LK
J
I
1
=
, na
N,UoLll
r-
^2t\7/^
l.n
/4)
>n ( o.r s.) \
( re.?..1o
( ryn- |o " -rr,r
(r.oo)
=
/*
)
)
B
I s (o.er.)'/q7Y"
.t
( B.rr"td'r)i
I
JJ
-\)
Problem11
In the figwe below, cunent t : 56.2mA is setup in a loop having two radial le,ngthsandtwo
of radii a : 5.72cm andb:9.36 cm with a commoncenterP. What arethe (a)
semicircles
magritudeand(b) direction (into or out of the page)of the magrreticfield at P andthe (c) magnitude
and(d) directionof the loop's maeneticdipole moment?
(a) d= o .{.-'L fi,ar4ntat4.*A a* V
B-
D^*= 4iL
qlfb
+U='
tlta
U-L
4nR
d,t'c/^'trt o5r't^cuu+ o^+
1 d'h-* b,*a"'t@
"l^n q".3-
(t.*)
+d
I
B^ur=@
Ll
E^* = 4 ,1r7*lo-?t:
'1
I{L*-,r..n5Ja-*v;'.J