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Physics48 (Fail 2011)
Chapter28: Magnetic Fields
"It is neverto late to be whatyou might havebeen." - GeorgeEliot
"Only thosewhorisk going toofar canpossiblyfind oul howfar theycango." -T.S.Eiiot
Reading: pages735- 757;skip section28-5
Outline:
\-/
+ magneticfields
introduction(PowerPoint)
magneticfield lines (PowerPoint)
+ magneticforce on a chargedparticle
crossproductand right-handrule (RHR)
+ crossedelectricandmagneticfieids
= a circulatingchargedparticle
helicai paths
auroraborealis(PowerPoint)
+ cyclotronsand synchrotrons(readon your own)
:+ magneticforce on a currentcarryingwire
= torquean a current loop
magneticdipoie moment
Problem SolvingTechniques
You shouldknow how to computethe magneticforce on a moving charge.UseF = qvBsin?to
computethe magnitudeandusethe right-handrule to find the direction.Draw a diagramto help find
B . In^othercasegthe velocity andfie1dmight be given in
the angle d betweenthe v^e"tgtr
:*f
unit vectornotation.
Use i xJ= *, i xk: i,a16 & x i= J. You alsoneedto knowthatthesignof
'i
: -i ,for example.Remembertlat
the vectorproductreversesif the factorsareinterchanged
,
";
thevectorproductoftwo parallelvectorsis 0: i x i: 0, for example.
Someproblemsdealwith the simultaneousinfluenceof electricandmagneticfields. Use
F : q(E + r' x 6 ). ln many cases,the chargeis not acceleratedbut continuesin a straightline with
constanlspeed.Then, E + v x B = 0. This equationis oftenusedto find the value of one of three
quantiliesthatappearin it.
Someproblemsdeal with a chargedparticle that movesarounda circular orbit, underthe influence
ofa uniform magneticfield. You shouldknow the relationshipbetweenthe particle speed,the orbit
radius,andthe magneticfield. The period of the motion is given by T = 2nm/qB,whereneis the
massof the particle,4 is its charge,andB is the magnitudeof the magneticfield. The radiusof the
orbit is given by r: mv/qB,whereu is the particle speed.
You shouldknow how to calculatethe force of a magneticfield on a current-carryingwire andthe
torque ofa uniform magneticfield on a current-carryingloop. Oftenthe easiestway to computethe
torque is by using r = px B . You will needto know how to computethe dipole moment F, both
magnitudeanddirection,ofa currentloop
Questionsand ExampleProblemsfrom Chapter 28
Question I
For four situations,hereis the velocity i ofa Drotonat a certaininstantasit movestlroueh a
unifonn maBneticfield .E:
q= l=c_,d (
@)n =2i -3j na E =+i
@)n 4i +2j nd E=aE
@)r =tj-zic nd E=ai
@)n =zli andE=-+i
F= F--iFiFr'
)
Without written calculation,rank the situations4ccordingto the m/gnitude ofthe uragrreticforce on
\J
geatest
fitst.
tbeproton,
/r\
,-, - .1r -,3\
= -sJ-Be
(a) s(?,P)-ru(],0)
=2-\r
-t
Quesrion2
\
/\
(b)-lr(li[7-at1^t)= '4
-'t
(c) lr(i-?)-?((1) =
"
i
Bb
The figure showsthreesituationsin which a chargedparticle with vei6city i travelsthrougha
uniform magnaic field -8. h eachsituation,what is tle direction of the magneticforce I on the
oarticle?
\o)
1.7
I "t
(.0*r4{)"
-
)
ProblemI
Arr electrotrin a TV cameratube is moving at 7.20 x 10om/s in a magneticfield of sfength
83.0mT. (a) Without knowing the direction of ttrefield, what canyou say aboutthe grcatestaxrd
leastmagnitudesofthe forceacting on the electrondueto tle field? (b) At onepoint the electronhas
an accelerationofmagnitude4.90 x 1Ot4ntJs2.What is the anglebetweenthe electron'svelocity and
the masneticfield?
(q)
^tr'du
^ r . ' lu
r ^ 6$'/t/
n - 8 3 . b , -l t 3 T
- i . 6 o e ,i f ' e c
r,,t -
*-*
(b)
-
-j
.r:*,;
,!
,,"r*.r,,r.. . . . i
F u = l X l v B = ( l . 6 r A , y o ' e ) { z . r o * 1 o 6n j s ' ) ( D . o , i t i a t 2
D
rv\
:
t'te
l o o l v l ) * , , , . , 4 ' - - - - ? + ' . a v "r r J r a .. ' . : , g
4-
'l"5brlb
Q,r1,1g,"'l'c
f)=
'^
h B - - _ l ?, 1v" "r)a^r',:J
=ts
rn
,-_-+
_A/hr B :
N)ta
=- 4.G^l^ Io-=
Ji,lvB
n
e= O.au?"
Problem 2
An electrontlrathasvelocityg =(2.0x106m/s)i+(3.0x1062/s)j movesthroughthe magnetic
field ,F= (0.300ni - Q.15Di . (a) Find theforceon the electron.(b) Repeatyour calculationfor a
proton havingthe samevelocity,
---
v-^B-=(v-e.v,r)^ ( B-a*Br3)
-1 ?
/\
CJK
--l
VXD.*
- V*B*(?,1?)
* V-B,( l^3)ov) B^(3-r)
Vx \6
o
B^ 8., o
V',o | ^l
=?
9,"/-tl
u. "/* R
B*ol
* [rg, (r^s) -t
t"l
B^brl
V* = A,o^161.7r V, . 3.ox 16617,
$* = o.3oo1
B.i= -0.15 T
(3,o ^rou.l)(o.3oT)]
V^D:
:
F_
(b)
(-re"lo' T'1,) f
%f.B = (-r.uoa*td,Jc)G\.e,
tou:-,1)Q = (t.re'.to''x)R
-lJ^^o,.'|*\)
%=
ol,{,oa*;d'qc-4o
= - ( l"g?..ld't t,) 0
a\
Problem3
A proton travelstbroughuniform magneticandelectricfields. The magneticfield is E = -Z.Si mt .
At oneinstantthe velocity of the proton is t = 2000jm/s. At that instantandin unit vector notation,
whatis thenetforceactingon theprotonif the electricfield is (a) 4.00iYlm,(b) 4.00iy/m,nd
(c)4.oorv/m?
-
l7-= (aooo"7r)J
._.\
F =%E-qi* B'=g ( i- il"F')
t _a
B'=a a.s'r6-;r)
*lo-'r)q1-'l)l
(o) tr = (L"6ol.^ld''c)
fUloov1fiil+ (ao*"1r)(-r-s
.
=
s'orm75
P
(6.qx1d,1N) f*
<a.o.1o',oru)f
b +c -,
-+J
a..;+
f
.B
Oo,"'DjjW E )-*
P)(-t)
,*' ,o-'o
l-<
t
Problem4
An electronwith kinetic energy1.20keV circlesin a planeperpendicularto a uniform magnetic
field. The orbil radiusis 25.0 cm. Find (a) the speedof the electron,(b) the magneticfreld, (c) the
fiequencyof circling, and(d) the period of the motion.
(o) KF=Ynu*- v=FF
(F= l.ao"to?"v
( '4?|u"")
=
1.q3.*ld'osq"11'.le-.,
l 5 , O c . r r :; e , ) S o r n
p:
I f ] e = Q " 1 1* l o
''Kg
/t\
(bl
f=D!_--->
xb
(c)
{=
i
,=
(a)
?F
_-
\/
/ ,''
/ :--
N. V
-:--l|r
B = 4.t"'l* lD-'T
aT$
?..n( 9'rt*16u'Ee
)
'|
f-)
6=
=
.-...-I
1.3t"Idnr
- --)
(&"os
q.tt'to''trE\
(\----=:J-L-----1,
"lJo
(r.roa.,t
o'"c)(o.a50,.)
I
Problen 5
An electronhasa velocity of (32i + qOh hds as it entersa unifomr magneticfieid of magnitude
field B -- 6OipT. What are (a) the radiusoftle helical pathtakenby the electronand (b) ttre pitch of
the path?(c) To an observerlooking into the magneticfield region from the entrancepoint of the
electron,doesthe electon spiral clockwiseor counterclockwiseasit moves?
i* ltor'7.)i
f = (:rKm/.)
Vr= loYt/s = 1-o^Plr1,
d= (torppbT)a
Y,
f = nvt
(4)
= 3) Kr/5 = 3.3.'1orm19
(1.o^to"n/,
3.9,.10
%b
(b)
P=v,,r=
P= l.g^l0-'r.,
^1 rlQ&a h ,^^"fud!!ua^*"6e
zV
_r
,/
4A
Problem6
A 13.0g wire of lengthL:62.0 cm is suspended
by a pair of flexible leadsin a uniform magnetic
field of magnitude0.440T (seethe figure below). What arethe (a) magnitudeand (b) direction (left
or rigbt) ofthe currentrequiredto removethe tensionin the supportingleads?
,ll.n-t
"v\.t ^i
,!r. tt Pw N49,f
f,4
^n ilp'-^A -lr"rr*
Pfu
--'l
-i
I
tr--tL*b'
XXX
l=----
-.-.:,
L
F, = teo, =o
/
/-
R
'
v\J
+
II -
aLB)r\
t=
trrq
Lb.
M \ ^ = r\)- '
lrr:l
4
?o
= fft4
?0" -= fqg
o.?b'l h
= L9:"!3rgll:€r&) ---> L = jr,o
^;etd
');
(o'ba^)(o.sg"r)
Yg*
Problem7
A wire 50.0.cm long lying along the x axis carriesa cunentof 0.500A in the positive x direction,
tluough a magneti"6"16 fr = 1t.00ml j + (l}.OnT)fr . In unit vectornotation,what is the magnetic
force on the wire?
[:
\\---l
o"5o*;
[ = A.5a'i-.
*)
B - = ( o . o " ' = o T -j,* { o . c r o T . , f '
*)
F
= ( D"5i A) i o.so,",)
( t. S;i * I o.go*){o,eo
(o.o<,=,oT)
n) {c.oo3or) (t, t)
.
==_"___--___
r-
-
I
---=-
, -3--\
-
(--.5'.
0
--F.-
\.^
-^
.
! l i *' \ r '(" ^?r v_ q * t n - ? - r )- lt '(
tJ
J
lO
Problem o8 '
rroDtcm
;
I
A single-tumctment loop, carryiag a cunent of 4.00 A, is in the shapeof a rigbt trianglewith sides
50.0,120,and 130cm. Theloop is in a unifonnmagreticfield of magnitude75.0mT whose
direction is parallelto the cunent in the 130cm sideof the ioop. What is the magnitudeofthe
magneticforceon (a) the 130cm side,(b) the 50.0cm side,and(c) the 120cm side?(d) Whatis the
magnitudeof the net forceon the loop?
..\
\\
l?Ocm
zU-r^ngri
,G^Af^|rr\e
B
-
-r
P = A,tn
Y
6-qo-o:
-)
i L"b
$O cm
/ l f , o c m\
(-,.*l=
,-t-
--:
-lA
\
|J-""/
qx:-6"
or*'*
=;
r
befr!@'tl
B)''"r.'O
tr
7r€
-ld
(4) F= ; LB^^^oo
-+
/r\
\--_,1
C.)
6?.1.'. -.
r -.
L <.----{r
1r>.,6"\ g?.1"
\
J,
t 5 ? , qo
f
l:i
lF=
= (t,oaA) ( o.5DA)(0 5 ^15=T),ailql 1a..6"
I
/
o"l36N ,"u*4P'++ ,l
-- ( 1.oo*)- \ 1r.lo^1( ? 5 ^ lo 3T) -4;nn15?. I '
tr
\
/
-/-
6"1,10
b
(r)
=.N
DN*(o.rzn)1t- (0.{3tN)0
Problen 9
A circular yire loop vihoseradiusis 15.0cm carriesa currentof 2.60 A. It is placedso that the
normal to its planemakesan angleof 41.0owith a uniform magneticfield of 12.0T. (a) Calculate
the magneticdipole momentof the loop. (b) What torque actson the loop?
N=t
c = o "l 5 t . ,
t - A.boA
O = ?i.o"
B = le.or
(A) /,,\ = bjr= = Ni
4
f , -
/,\
Lrl(
4-
i'.rtr')
/.t
+ . ', p-o ah\, i -T r /( au,- l :1( 5,
i:i
I
=- ; ; ;L; r; l( ' ( -
- - - a- ' ;, /; 4
(b) z- =i "8. ---- z =)^$ti>
rl Irr r
= t"+S
= (o, l$'/Aro')(ta"o'r)-o,r,,.tl"o'
--i
Z
lZ
Problem 10
t
A circular coil of 160tums has a radiusof 1.90 cm. (a) calculatethe currentthat resultsin a
magneticdipole momentof 230 Afri2. (b) Find tle maximumtorquethat the coi1,carryingthis
cunent,can experiencein a miform 35.0 mT magretic field.
N--lr"o
(a)
)=
F = D ' O l 9 Or ,
c=
/tr = X.:oAn'*
$:
Js,o'1[31
NiA = N;(r.)
l,\
Nrrr-
=
Q,.34A rn}
l tuo;rr(o.orto^)\
1'= lt.? A
(b) ?-= ,il.d ---->2 =AB-a*'g
't^^
rv'"a4z = 4 B
: (1.=offtq')(ss^lot3T)
=
7.o5^ lo-) N,q
Problem11
Two concentric,circular wire loops, of radii 11= 20.0 cm and 12= 30.0 cm, arelocatedin the xy
plane;eachcarriesa clockwisecurrentof 7.00 A (seethe figure below). (a) Find the net magnetic
dipole momentof this system.(b) Repeatfor reversedcurrentin the inner loop.
J4= l.J i A = N i (tn") "u*pu ;,1-'**, Jroo"t", d.g"r1"{
po"'f ;u-,k e^44
*An LL","ta
@) d'aA" A^-'","d &^
"
v
( U) /^)ii|
.,i1,."t.NlTfc',t* i'l ;rt':
c' n- ( c,:'nc.a)
l.l^"* = (?.oo6; -rt f {o.ao";^*
(o.ao^;'] =
@'. ffv
a-rr,rvr.o.",*
p'v"v',!'n'?zw'p'a*A
= LT(r"'*o')
)J.,d.- irr.l-ircl
(1'aaft) n f(o.:",)a*(o.eo".;l
t'AW
=
{.8t A nrr
C,;afW)
6rr'"2"*"er';b
i*t"i,-
Wvilt'
**T
l.loAn^(,*r.wJ