Part IIMagnetostatics
8: Magnetic Phenomena8.1 History
8.1.1 Relation to Electrostatics
Magnetic phenomena are a lot less common in nature than electrostatice!ects. There are lots of ways to accidentally build up a static electric chargeand it isn’t hard to notice that something strange is happening. By contrast,it takes deliberate e!ort to produce magnetism where it didn’t already exist,and there aren’t a lot of naturally occuring magnets. Think about yourown experience: when was the last time you happened to notice somethingbehaving in a strange way that could have been a magnet, other than a man-made object? Probably never. On the other hand, electrostatic shocks canbe a daily experience. Of course, these are also made more frequent by theproliferation of di!erent fabrics in modern life, but with furs and such lyingaround for all of human history, someone was bound to notice the buildupof charge pretty quickly. Despite all of this, it turns out that electricty andmagnetism were first studied seriously (at least as far as we know) aroundthe same time in Greece. Electricity was named by the Greeks after amberbecause amber would readily build up electrostatic charge, while magnetismwas named after the province (Magnesia) where magnetite (an inherrentlymagnetised iron ore) was found.
We study electricity and magnetism together because we now know thatthe two are intimitely connected phenomena and even understand (just wait!)what that connection is. But even in the ~600s BCE, it was clear that the twowere similar and attempts were made to relate the two (unsuccessfully). It
91
was only after the development of controlled electrical currents that scientistswere able to understand the connection between the two.
8.1.2 Permanent Magnets
In any event, there were magnetised objects occurring naturally. Lodestonesand magnetite are obvious examples, but it is also possible to produce amagnet accidentally in the course of working with metal. Rubbing two piecesof iron together in the same orientation for a very long time will slowlymagnetise them both. Apparently you can use this to build a compass in anemergency, but I’d suggest just packing one for your trip instead.
It was realized pretty pretty quickly that magnets have the special prop-erty of reliably preferring to point (roughly) north. This is of course theentire basis for the functioning of a compass. This is also the origin for muchof the nomenclature associated with magnetism.
8.2 Behavior
8.2.1 Permanent Magnets
a-\?-* 0v'- ' -
Le r lu ' l - f ,?v:r t>
'J
ffi 4ffiffi-->
F?
L
g 0i fd l r ,
-wm/V" h flM
O&t'A\o,.o
yol* rf
0;(dl (
I
:
t
- l '
( tH-{
D:p, t&
E"r V\ ' , /\"t- \
@gr. l \
Figure 8.1: Opposites Attract
Every magnet has 2 di!erent “ends”, known as poles because of their ten-dency to orient towards the earth’s poles. This tendency led to the obviouschoice of names: the north pole of a magnet is the end that points towardthe north pole of the earth, the south pole... points the other way. However,this choice leads to a confusing fact. If you take two magnets and identifythe north pole of both (by testing where they point when allowed to rotate
92
freely), you will find that when you bring the two together, the north polesrepell, while the north poles attract the southern poles. Like repells like andopposites attract (as with electric charge). Following this logic, then, we findthat what we have cleverled labeled the “north” pole of our magnet must beattracted to a “south” pole: the North Pole is magnetically south! If youthought it was a little odd that electrons ended up with a negative charge,screwing up our definition of current, this should put it into perspective. Ofcourse, nobody refers to the north pole of the earth as the south magneticpole. We use words like “geomagnetic north”, which means “we know its thenorth pole but the magnet actually isn’t north and we’re just calling it northanyway”.
a- lL- ovt
-
Le "1u,. Yi nv{t} ' )
Di(dl *"
I
I
+vDloale
ffiilh
rye1,&i**nq,'.we.w6isrtur*_,r"
--_-- _._.
g 0 i fd l t t
-wn/V" r f lB
O&tt1Y\
e*n*o y olt ,"
--">
F*?
-->r?
F?
L
E"r V\"' /\"t\
Figure 8.2: Magnetic North is... Magnetic South
Monopoles
There is no such thing as a magnetic monopole. If you separate a north and asouth pole, you just end up with 2 of each. This entirely unlike electrostatics.It is possible to construct an electric dipole, but it is made of 2 oppositecharges which can exist and act independently. There simply isn’t an analogin magnetism, despite extensive searches by very serious people for a verylong time and some pretty interesting ideas about why they should exist.This is an idea that I imagine won’t ever die, and could even be correct, buthas no real support at the current time.
93
a- lL- ovt
-
Le "1u,. Yi nv{t} ' )
Di(dl *"
I
I
+vDloale
ffiilh
rye1,&i**nq,'.we.w6isrtur*_,r"
--_-- _._.
g 0 i fd l t t
-wn/V" r f lB
O&tt1Y\
e*n*o y olt ,"
--">
F*?
-->r?
F?
L
E"r V\"' /\"t\Figure 8.3: No Magnetic Monopoles
Magnetic Fields
We aren’t yet ready to define a magnetic field, but even casual experimen-tation with magnets make a field seem even more real than in the case ofelectric fields. Holding two reasonably strong magnets and trying to pushthe like poles together gives a definite sense that something is in the way.You can even feel out the shape of the field lines to a certain extent.
Magnetic field lines are similar to electric field lines but also quite di!er-ent. They represent a vector field, as do electric field lines, but the meaningof their direction is less straightforward and we must wait to define it. Theyalso, unlike electric field lines, have no starting and stopping points. Mag-netic field lines form continuous loops that exit from a magnet at the northpole and enter at the south pole, but continue inside and in this regiontravel from south to north. This is actually the same as saying that thereare no magnetic monopoles. You can split an electric dipole by separatingthe sources of field lines from the sinks, but with a magnet there are no ends,so sources and no sinks. We denote the magnetic field as !B. I actually haveno idea why we use the letter B for the magnetic field, but everyone does.
Magnetic field lines, like electric field lines, cannot cross, and will tryand evenly distribute themselves. The field of a bar magnet looks much likethe field of an electric dipole for good reasons, even tho they have di!erent
94
Figure 8.4: Field of a Bar Magnet
behavior, as we will see. Since magnetic field lines emerge from N poles andenter into S poles, it is possible to construct a region of uniform magneticfield in much the same way as we can use two charged plates to construct auniform electric field. The positively charged plate will emit lines which thenegative plate receives: likewise, lines (externally) will flow from N to S.
,|-
ebdoCIl: Fa]
& der.ol ts ( Fa lJ J i r**dsJ
',^ L= *,L* f d? f' O J!""{ r >'-tr-E
*?sfJ fs,,n' i^p l:}**.ff;,*, &'t-f*{P#.
C"rr tu & u' ! t \ t * \ cr {*c !-
lLlg A* f^?e,
\i
r\ ' l - iIl ' ,t, l
l
g
\ .4Fr
*I7x
TJ-Bs, G?)
1^l\ere e \,st
o,utd..ffB'
r t lu t - .L l ' r fq ic,
+*,-,,l*-#d!;Jdd+!*:#i:r##d*v{*is'.$@4@@dMr,w6#R't1i:}r&##r'Ft*'l'l*J4
,N
tl(s .
-.--, Y,'-' '')*
7 T r ] rSBr
/['A
elr^*L
Ll^*:
&1 . l-rcn'&
i t FA,*L
a Ji.. ar A it k.\
e/) t f dt o^e
pr- dvr( *
cu((e^ b
? o',n t av!f
Figure 8.5: Uniform Magnetic Field
95
Figure 8.6: Magnetic Field from a Straight Current
8.2.2 Electromagnets
With the invention and relatively easy access to controllable and knownelectrical currents, it was possible to discover and study the e!ect by whichmoving electrical charges (currents) produce magnetic fields. The resultingfields are simple, but unlike what we have seen so far. Rather than in anyway emitting from the wire, the field lines will form perfect concentric circlesaround a straight wire. If you bend the wire (into a loop, for instance), thenthese circles will distort against one another and change a bit, but close tothe wire you always have perfect little circles.
So there are a few strange things about these fields already. First, theyaren’t produced by a charge of any sort, but the motion of a charge. This isstranger than the di!erence between gravitation and electrostatics in a sense,because there all we did was change what the “charge” was, and we were inbusiness. The force formula was even the same! Now we have to worry aboutcharges movement. Not only that, however. Notice that the direction of flowof (positive) charge is along the current carrying wire, while the field is goingaround the wire. Not towards, or away, or with, but around it. This means
96
that the field is at a right angle to the direction of flow of the current. Ifit had been radially outward, we could have convinced ourselves that thecharge for magnetic fields is just moving charge, and then everything elsewould be the same. But alas, things are not so straightforward. Hang inthere, however, because there is a charge analogy to be made later down theroad.
I specified that the field goes around the wire. Around which way? Mag-netic fields obey various right-hand-rules, and the first is that if you havea current, you can find the directon of the associated field by aligning your(right) thumb along the current and curling your fingers. The direction ofcurl will indicate which way the field goes. At this point you should askyourself where else you’ve seenthe RHR used, and the answer is: anywherea vector (cross) product was lurking (such as torque). Sure enough, we arezeroing in on a cross product, but we aren’t there yet.
8.3 Forces
8.3.1 Force Exerted by a Magnetic Field on a Length of Cur-rent
We saw that magnets exert a force on one another (tho we didn’t fully definewhat it was), and we’ve seen that electric currents produce magnetic fields.It should not then be a surprise to find that magnetic field exert a force oncurrents. What may be a surprise, however, is that the force applied is notattractive or repulsive but rather orthgonal. A current bearing wire passingthrough a magnetic field will feel a force which is perpendicular to both thedirection of current flow, and the magnetic field direction. Three mutuallyorthogonal related vectors is another good sign that the three are connectedby a cross product, and in fact once again we use the right hand rule toidentify the relationship between the directions of the problem. There isa special memnonic for the right hand rule when used for force, magneticfield, and current based on the acronym FBI. Put your thumb, pointer, andmiddle finger in a mutually orthogonal position. Now in order your middlefinger is the force (!F ), pointer is magnetic field ( !B) and your thumb is in the
97
direction of current (!I). The acronym is in the order of your fingers, startingin the middle with F. Alternatively (and resulting in an alternative set ofassignments, so pick one way and stick with it!) we can use the standardtechnique based on the order of a cross product once we define the forceequation. While these 2 methods will assign a di!erent quantity to di!erentfingers, the resulting relationship is the same, so either one works but youare probably better o! sticking with one way to remember.
,|-
ebdoCIl: Fa]
& der.ol ts ( Fa lJ J i r**dsJ
',^ L= *,L* f d? f' O J!""{ r >'-tr-E
*?sfJ fs,,n' i^p l:}**.ff;,*, &'t-f*{P#.
C"rr tu & u' ! t \ t * \ cr {*c !-
lLlg A* f^?e,
\i
r\ ' l - iIl ' ,t, l
l
g
\ .4Fr
*I7x
TJ-Bs, G?)
1^l\ere e \,st
o,utd..ffB'
r t lu t - .L l ' r fq ic,
+*,-,,l*-#d!;Jdd+!*:#i:r##d*v{*is'.$@4@@dMr,w6#R't1i:}r&##r'Ft*'l'l*J4
,N
tl(s .
-.--, Y,'-' '')*
7 T r ] rSBr
/['A
elr^*L
Ll^*:
&1 . l-rcn'&
i t FA,*L
a Ji.. ar A it k.\
e/) t f dt o^e
pr- dvr( *
cu((e^ b
? o',n t av!f
Figure 8.7: Force on current due to magnet
Experimentation allows us to fully identify the form of the force felt bythe wire, which isfound to vary with the angle between the current and fieldas
!F = I!"! !B = I"B sin #!B[F ] = Cs mT where T = N s
mC = NA·m is a Tesla.
,|-
ebdoCIl: Fa]
& der.ol ts ( Fa lJ J i r**dsJ
',^ L= *,L* f d? f' O J!""{ r >'-tr-E
*?sfJ fs,,n' i^p l:}**.ff;,*, &'t-f*{P#.
C"rr tu & u' ! t \ t * \ cr {*c !-
lLlg A* f^?e,
\i
r\ ' l - iIl ' ,t, l
l
g
\ .4Fr
*I7x
TJ-Bs, G?)
1^l\ere e \,st
o,utd..ffB'
r t lu t - .L l ' r fq ic,
+*,-,,l*-#d!;Jdd+!*:#i:r##d*v{*is'.$@4@@dMr,w6#R't1i:}r&##r'Ft*'l'l*J4
,N
tl(s .
-.--, Y,'-' '')*
7 T r ] rSBr
/['A
elr^*L
Ll^*:
&1 . l-rcn'&
i t FA,*L
a Ji.. ar A it k.\
e/) t f dt o^e
pr- dvr( *
cu((e^ b
? o',n t av!f
Figure 8.8: Magnetic Force on a Current
where !" is the wire segment within the field. The field was initiallydefined by this relation, tho we find a more fundamental definition later.The strength of the force is proportional to the length of the wire, and
98
this formula can be applied for curved or non-uniform currents by, as usual,turning things inter di!erential elements:
d!F = Id!"! !B.This is di!erential force element due to a small piece of wire. We could
also find the force element on a wire by a di!erential element of field:d!F = I!"! d !B.This is actually more analogous to what we would use for Coulomb’s law
but isn’t useful until we define a way of calculating the field !B. We won’tdo this for a little while yet, but there is plenty we can do without such adefnition.
,|-
ebdoCIl: Fa]
& der.ol ts ( Fa lJ J i r**dsJ
',^ L= *,L* f d? f' O J!""{ r >'-tr-E
*?sfJ fs,,n' i^p l:}**.ff;,*, &'t-f*{P#.
C"rr tu & u' ! t \ t * \ cr {*c !-
lLlg A* f^?e,
\i
r\ ' l - iIl ' ,t, l
l
g
\ .4Fr
*I7x
TJ-Bs, G?)
1^l\ere e \,st
o,utd..ffB'
r t lu t - .L l ' r fq ic,
+*,-,,l*-#d!;Jdd+!*:#i:r##d*v{*is'.$@4@@dMr,w6#R't1i:}r&##r'Ft*'l'l*J4
,N
tl(s .
-.--, Y,'-' '')*
7 T r ] rSBr
/['A
elr^*L
Ll^*:
&1 . l-rcn'&
i t FA,*L
a Ji.. ar A it k.\
e/) t f dt o^e
pr- dvr( *
cu((e^ b
? o',n t av!f
Figure 8.9: Force on current in uniform magnetic field
8.3.2 Force on a Single Moving Charge
We know from our study of circuits that a current is just a collection ofmoving electric charges. If this leads you to expect that individual movingcharges also feel a force due to a magnetic field, you are correct. We canfind the force on a single charge simply by dividing the force on a currentup amongst all of its constituent charges. Or, equivalently, we can define asingle moving charge as a tiny current !Iq = q!v.
!F = q!v ! !B
The force on a wire with current flow is actually the sum of the forceson all of the moving electrons inside the wire. Because the electrons cannotescape the wire, they keep getting pushed against the sides in the directionof the force applied by the magnetic field, and this pushing translates intoa force on the wire as a whole. When a particle passes through an electric
99
field on its own, however, there is nothing to push back and keep it going ina straight line. Because of this, a particle passing through a magnetic fieldwill be deflected in a direction determined by the right-hand-rule. However,once the particle is deflected, the force it will experience is in a di!erentdirection because of the cross product. This will cause the particle to bedeflected further, which results in a new direction of force, and so on andso forth. It is in fact not possible to construct a magnetic field which willonly exert a force on a particle in a single direction, as it is with the electricfield. The dependence upon !v means that the force will always depend onhow the particle is moving, and since the force will change how that particleis moving, charged particles never travel in straight lines through a magneticfield. This is manifestly di!erent from electric or gravitational forces. It ispossible to have curved trajectories in either of those fields, but not necessary.It is quite possible to fall straight down, and two opposite charges at rest willattract one directly together in a straight line. This simply cannot happento a free charged particle in a magnetic field. By restricting a charge to awire, we can force it into a straight line, but even the wire will try to bendand move too accomodate the flowing electrons need to travel in a curve.
t
l " il r I
v-11- av*88,f,''t-F-#"
)(
{
*
{
{.
X
XX K
^A..d v?\t
^q;-+
Jl\.x \ r . (
'rFl l t, r / t
X- x4-I{_>F>
F ' 'K N r ,y ' t
tq- {.F-t **?t
t ' x"-x x
x
x
A
\4\
.i4r!6l*F ji@hw@]*q;,e!i*4$L
vn,,.e4kaq14{
X
X
r
X
/-X
KX
t { i B lnL
y,{
Y a?e ( "ni fo^)
1{ l lil,l Jj We Arr i
' , t t -Se?rv\-a^ t
-\( Gr-JBr hb-
. l - r
L\,t (*,- or\ c tt 3
\,^''1 rs1 ' Ab fs(-, L0,
}Jts i
TJ8ft
{v(wrr e^
bht 1
6og*^1" ) ^^"1 l l
t - f fou*t I r
c a^ 'e lo[ ]*a
r sia tts) /-l)
6)n r*) Lr)
t'r,,$) co)
-\rGFe,bL
-->
1 ;S
4
1
F co'?Js
!I?"
t -e
d c + T16F A'+ hu*n l - 6o - ,
\6re 4un s!e &U*
&rl cf t tb ' ,F* {*rrs
nf!'q?le J l t ' re F' t r^\ ,
W '{'l L'''u pf r^,
TT\
k t'h"--
b ors wle
t # t*#
,+b. nrJ
tI^*
knsw
Figure 8.10: Magnetic force on a moving charge
In the special case of a constant magnetic field, a charged particle willtravel in a closed circle. This is because it will feel a constant force perpen-dicular to its direction of motion, exactly as a ball on a wire or any otherforced circular motion in mechanics.
100
8.3.3 Examples
We can calculate forces on currents and moving charges using the equationsabove. As with electric fields, the principal of superposition still applies: ifwe have multiple magnetic fields, the resulting total force will just be thesum of the forces due to each field. Also, in the case of currents we needto consider forces on di!erent pieces of a current carrying wire. In our firstexample, there are 3 sections of wire in a uniform magnetic field. Eachsegment feels a di!erent force and must be calculated separately and thenadded together.
Wire loop inserted into a uniform magnetic field
t
l " il r I
v-11- av*88,f,''t-F-#"
)(
{
*
{
{.
X
XX K
^A..d v?\t
^q;-+
Jl\.x \ r . (
'rFl l t, r / t
X- x4-I{_>F>
F ' 'K N r ,y ' t
tq- {.F-t **?t
t ' x"-x x
x
x
A
\4\
.i4r!6l*F ji@hw@]*q;,e!i*4$L
vn,,.e4kaq14{
X
X
r
X
/-X
KX
t { i B lnL
y,{
Y a?e ( "ni fo^)
1{ l lil,l Jj We Arr i
' , t t -Se?rv\-a^ t
-\( Gr-JBr hb-
. l - r
L\,t (*,- or\ c tt 3
\,^''1 rs1 ' Ab fs(-, L0,
}Jts i
TJ8ft
{v(wrr e^
bht 1
6og*^1" ) ^^"1 l l
t - f fou*t I r
c a^ 'e lo[ ]*a
r sia tts) /-l)
6)n r*) Lr)
t'r,,$) co)
-\rGFe,bL
-->
1 ;S
4
1
F co'?Js
!I?"
t -e
d c + T16F A'+ hu*n l - 6o - ,
\6re 4un s!e &U*
&rl cf t tb ' ,F* {*rrs
nf!'q?le J l t ' re F' t r^\ ,
W '{'l L'''u pf r^,
TT\
k t'h"--
b ors wle
t # t*#
,+b. nrJ
tI^*
knsw
Figure 8.11: Force on curent loop partially inserted into magnetic field
We can measure the strength of a magnetic field by inserting a knowncurrent of known length and direction into the field. Note that in this partic-
101
ular example we allow 2 forces which act in di!erent places to cancel. Thisis correct, the total force does become 0, but if the loop were not alreadyperpendicular to the field, this would create a torque (as there is whenevertwo forces act in opposition from di!erent points on the same object).
Curved wire in uniform magnetic field
o-1*
fJ
()
a
0
6
oT
og
-eF-..-o
a
o
oo
GE
c)#G
6J
oao
IJ
t
o
&o
oo
-\d
/t --\t- a\
t t.T \
;
("[.,<)L
o
(6A
o
o
o
&*l
o
8o
G)
o
Tha roJ' \n5 nl &[* L ' t ( r t*
)s . lu>l Lc ] l t ,
A, i ' r tL Io b ] 'groi l .-m,
wa L.ar* f 9e 2Y^* b oL*
wir , ( / $nJ Ag r"r i l t o?al^
(aac!l Cf i , (3^b r1,dvs vve,
h(r l { r" }a\r7r* lc [o 2!t
be.
T.. o,)* , to eluai J ,e-*{ , , 'o !' , th1
i,n b*?t* t,* )n Lh* di"of l,,r o t(aau!ra t l oqr \ L 'ur(&A L,
f * t r{,
-5 'L"f - \ - rF-b
f - -w2-
Ji f ixd= " 1* ad Bq@ g,:sin 6\+ca(\
/) I F cotg(d) q] ad
(t^a^\ "
\^/ tt cnn ( (\ V ? $1'^n*'al '1 tTl* f
T,u!*" $,"{ l} { e'n d[) E z- ryL'-o {- o* i 9
+I-1
Pr-
Figure 8.12: Measuring !B with a curved current loop
This is the same a the last situation, except we are insering a curvedsection of wire instead of a straight one. This requires us to do an integralover the angle of the semicircle of the current loop. Notice, however, that the
102
force is unchanged by this modification. The force is acting on componentsof charge from left to right to produce the y component of force, and there isjust as much of this in one case as the other. The x component just cancelsout by symmetry.
Single Charge in a uniform magnetic fieldZ:,!kfI
o
o
{=u
s('l'r
o
o
fro,q- n!altctni* : , r^r& bna"- Lh"t d
C,t r t *\ ar rr4 o [i* h* * cl ( ettl '" Pr I o
A,- V}
drsa6,r d l* l ctu*. [*u1 f*rcLr,? y
)
o
o
Q;ro
e)
6)
B*C eru ><r \ lr 4
Pro U [e,*- "- I
FrLat iu^- ip
J l r"*-* [- ]*^ o b
dre. d. i^2 tht :
cln e [. 6- f "o "r, LX*
' * bLry{ae{tit [*
\q *ut{8a b"-
fat t l .k rn
I a cc4* aL'ta\
r *FFr- -h^
/, - l_ \S'rYt(e i e i )
, le le c t t O*r, t ,aa oV ,^
)
n - ; rb - '&
D*,pe"J i,u2 o^
ve 4o.* golvc
ryn ele*[run i :
& o$- { , r t D
LL.* h^ s? **\ic f,* *a \caeP't,) LL
ir^ rr L't ( l* lr ,
y{a l'^. F-a\ /
et'Yy
vtc* sfis.- lttr*a*' *"*{' d sloe
'*
{T'
{r. ,^r}n ', c\e v er i s n"n Ln et*A ' i t ar
L*u.-v\ Fhls l cal t6 l1t t* , 'U, Sc ^raC
,*rh o t-
LV,,5
d
o
o
o
bo S,^}, tt{- 3 .),
At any given moment, the force on a point charge is simply given by theformula above,
!F = q!v ! !B.There isn’t much more to it. However, if we wish to calculate the trajec-
tory a charged particle takes, we would use the equationd!F = q!v ! d !B "# d!a = d"F
m
and use it in the expression for the arbitrary position of a particle undera force,
!r (q, T ) =ˆ T
0
!!v0 (q) +
ˆ t
0d!a (q)
"dt.
However, there really isn’t anything particularly interesting to be gainedby wading through the application of this bit of nested calculus. I mention itjust to point out that it is doable and to give a sense of the added complexityof dealing with magnetic forces in general. The electrostatic analog wouldlook the same, except that the second integral would just be some number,
103
Z:,!kfI
o
o
{=u
s('l'r
o
o
fro,q- n!altctni* : , r^r& bna"- Lh"t d
C,t r t *\ ar rr4 o [i* h* * cl ( ettl '" Pr I o
A,- V}
drsa6,r d l* l ctu*. [*u1 f*rcLr,? y
)
o
o
Q;ro
e)
6)
B*C eru ><r \ lr 4
Pro U [e,*- "- I
FrLat iu^- ip
J l r"*-* [- ]*^ o b
dre. d. i^2 tht :
cln e [. 6- f "o "r, LX*
' * bLry{ae{tit [*
\q *ut{8a b"-
fat t l .k rn
I a cc4* aL'ta\
r *FFr- -h^
/, - l_ \S'rYt(e i e i )
, le le c t t O*r, t ,aa oV ,^
)
n - ; rb - '&
D*,pe"J i,u2 o^
ve 4o.* golvc
ryn ele*[run i :
& o$- { , r t D
LL.* h^ s? **\ic f,* *a \caeP't,) LL
ir^ rr L't ( l* lr ,
y{a l'^. F-a\ /
et'Yy
vtc* sfis.- lttr*a*' *"*{' d sloe
'*
{T'
{r. ,^r}n ', c\e v er i s n"n Ln et*A ' i t ar
L*u.-v\ Fhls l cal t6 l1t t* , 'U, Sc ^raC
,*rh o t-
LV,,5
d
o
o
o
bo S,^}, tt{- 3 .),
or a very simple function, rather than an integral.The only case of single particle motion we want to consider carefully at
this point is a charge in a uniform field. In this case, we have already arguedthat the motion will be circular. Using what we already know, we can getmore specific.
Helical motion of a charged particleA- lb ov
I f o t', \Juc LL\r ?e
o$ v!loctkY / t t
d3 c ' r Ln, tc ' r f t
w lLh C 6mf d^e", It'l
fo -b qb w.\ l
l"ra t l rn.
E V'-f *u,
{n (!
n*e J L, F',^ JAs we
!ir\
t .-{
A
I
?x&
T,Ib
fo. c.e> a'f'\j t> d
.->
tcg
- -r J0{5-; e
ci)w
g*
q)
c: et" h
fu*?- 1,f$ s'rt .9 i
T1
}JB
rf G
l- Psrg -
-)t-
r (sr=.Jrfcv-
F-\f
bac t t , of
.-\a-!(--
-.->l="Ab
_\.?1f :,- ATL
blq**&.->c+-> VAU
t ...+'.
fxFntt-.
(i^*a) =
Lo" L- el 'rhi. L- .t J I'
- W6ine (0)a
+u cY) --
-...>+?1*ew
I496,,'roa-
?' -s '+x F.o =
ryshs
(i"*a),
bL*I
e,4ff\ (-tJ
co\@) (g);al
s/Ft (t)
-J{b - -TJBe' Ats >
-J/\
F (*)** IJ 6 'a
f* wi l\ f'oJn** &L s*r\! 0*'* qs
b*fore, on\1 hotr )b Con 6* ot- o^ ' '
orn ?\ *' f[' L* &t o !"- #u n *LB' cc's ( *)
wlt \ P'rJo"e t tc- n 'c*JeJ t ' r7n Lt
F*u!*r5& &L J" '**h; u l d(e8( j""1 L"
bt* r l thL h"n*- rv+l c* )
g!7,r*^ L-
( i^Lo p oort )
( st . : i lc- ""e)
(o*L ol P. .?a)
: }$$lat9 (-$)Gt(
Figure 8.13: Helical trajectory
If a particle has components of motion both parallel to and perpendic-
104
ular to an external magnetic field, its motion will be circular in the planeperpendicular to the magnetic field, while traveling una!ected in the paralleldirection. The path traced out by such motion is a single helix.
Aurora Borealis
Figure 8.14: Aurora Borealis: Helical motion at work!
The Aurora Borealis (and their southern counterpart the Aurora Aus-tralis) are perhaps the most naturally beautiful examples of electric or mag-netic phenomena. Charged particles from the sun (“Solar Wind”) get caughtby the earth’s magnetic field and then spiral down to the surface (much likethe helical example above) until they reach the atmosphere. Once thesecharged particles (which have been accelerated by the magnetic field) hitatoms in the atmosphere, they heat it up and cause it to glow in a varietyof colors and patterns.
105
8.3.4 Torque
If we repeat the example of inserting a current loop into a uniform magneticfield, but this time insert the loop rotated 90 such that the end segment#!"$
is parallel to the magnetic field, we find that the forces on the othertwo lengths exert a torque on the loop. This is analogous to the case of anelectric dipole in a constant electric field feeling a torque around its center.
A* 13- oV
B P os',\rc LL\'rge-
o$ v!loc)12 / t r
d5 c'r L uior qt
w l t t r C om( d^e^ |
! . , 6 qb w.\ l
l"ra h ? rn,
B ta $*un
fr* t#As
&1."
fe&: T* &
' ' T, lblgr=.J
Y*- I-Ar3
-.)( ,F hB-
*TJ D
\irf e
&
T1
I
s,+ff\.(-tJcos?) (0)
3l
t*!) (t)t .*_{
* S )n orrf cr>'e
fuL=Vi'Y
E;" }^" o{ t\* s!->
E ' , L,*u+t ?r ' -
Js x Fns^/\ '
fu*?- J'tY3 Los$11
t\z-Jy' .
F oa= IJB 'A
.-...- \4n'<4e#^er#!t8
446?#' t :=d:r
--)
Trv wi t\ P'oJn*t !L sos^e Pqrc* n*
b*fore, on\2 horr )k cun 6o ot- o^) '
dn ?\* ' fS' L* &toF [u' otr-E' cc's( e)
wl t \ proJ* 'e bt* A,eeJeJ t ' r7n Lr
l'* u **r* !. &l^* J'o'** h.lut cx { eQ( J "^
} L"
-..>f="Ab
/lr
U ATL: TIJ {'
r--- 5ln I?
?' --\+xFco =
rys)qb
f.. ce> "''PP\; t> cs
.->
tcO
-rya
, A\
L-? )
WTc-
r a\L-? )
Lo, L^ el ,^rh. I o J J
Wsin(o) {*E)a
4 AI al I i -
99.,"[s)er(
(luxa) z
[, $),^. \
xz l :,i+
UCY) ?
!*o--
(i"
n*e J &,"' F',n Jr/
Ct fie h g!") l ' .a.t L*
( i^ Lo ? ol t )
( st . : i J e v"t)
( o* f- o l, P..1a)
Just like in the case of an electric dipole, a magnetic dipole in a field hasan associated potential energy given by the torque and angle through whichit must be rotated:
U =´
$d# = NIAB´
sin #d# = µB (" cos #) + U0.If we chose the 0 of our potential to be at # = #
2 , this becomesU = "!µ · !B
which we can readily compare to the potential energy of an electric dipolein a constant electric field !E:
UE = "!p · !E.
106
A* 13- oV
B P os',\rc LL\'rge-
o$ v!loc)12 / t r
d5 c'r L uior qt
w l t t r C om( d^e^ |
! . , 6 qb w.\ l
l"ra h ? rn,
B ta $*un
fr* t#As
&1."
fe&: T* &
' ' T, lblgr=.J
Y*- I-Ar3
-.)( ,F hB-
*TJ D
\irf e
&
T1
I
s,+ff\.(-tJcos?) (0)
3l
t*!) (t)t .*_{
* S )n orrf cr>'e
fuL=Vi'Y
E;" }^" o{ t\* s!->
E ' , L,*u+t ?r ' -
Js x Fns^/\ '
fu*?- J'tY3 Los$11
t\z-Jy' .
F oa= IJB 'A
.-...- \4n'<4e#^er#!t8
446?#' t :=d:r
--)
Trv wi t\ P'oJn*t !L sos^e Pqrc* n*
b*fore, on\2 horr )k cun 6o ot- o^) '
dn ?\* ' fS' L* &toF [u' otr-E' cc's( e)
wl t \ proJ* 'e bt* A,eeJeJ t ' r7n Lr
l'* u **r* !. &l^* J'o'** h.lut cx { eQ( J "^
} L"
-..>f="Ab
/lr
U ATL: TIJ {'
r--- 5ln I?
?' --\+xFco =
rys)qb
f.. ce> "''PP\; t> cs
.->
tcO
-rya
, A\
L-? )
WTc-
r a\L-? )
Lo, L^ el ,^rh. I o J J
Wsin(o) {*E)a
4 AI al I i -
99.,"[s)er(
(luxa) z
[, $),^. \
xz l :,i+
UCY) ?
!*o--
(i"
n*e J &,"' F',n Jr/
Ct fie h g!") l ' .a.t L*
( i^ Lo ? ol t )
( st . : i J e v"t)
( o* f- o l, P..1a)
4= 4^b
7' l j - ov
rJ,{o s i t g q-A.fuGl- LeD r
^L+ w4
Wc arr ' l {e
r^cltza b; ' ,J"nk
ob Ll* rno(c
l',, o Ll on bAo
7!"n&rc ' l
r j
ue ol*
D* Fuu
7; A,
\,{ o_
A;,t*
a t t o
W!,', c I
J*[ntu
J r { ' ,^e
lc I',nLs
l* ls
C on
A {-s)Itui l f ,
T l ' . r : u, l l I onl?
of d $tach
Ll. V e c[ 'n
e{)
fAj (11' 'v
On Orfq-
out l" aF
t/ t .> vy,r t , ,
" l
wa (b)
bho t
pe{ Al'*
i $ we dt [', n- LL d',rtoh'^
bh* t*,vrt ( *\ t f \o *-J c*,r orrr J
r't 7\ t l^ * A r't [e , lte ah'tn7 )
ofl.^, \^/ e
L[^ls C o'9e/
F^r.r\ t'11 la ftrt 3, I^
Oi
*,1\ f A u/ rr''e
,J, ^
I oo
-s"l\
F;F
*L Wrtr c ln* l , ?^ '> drd 3 !)
t^.!- Js $!n*
9o
le ?* Nt d xE-.
Breo LSe wi{& lo*Ps
!-xLrc*\2 Prvualrn L,
*'A
NAn
*kn
/tr or^e hc 0t prl* ,frrrnn'k
107
4= 4^b
7' l j - ov
rJ,{o s i t g q-A.fuGl- LeD r
^L+ w4
Wc arr ' l {e
r^cltza b; ' ,J"nk
ob Ll* rno(c
l',, o Ll on bAo
7!"n&rc ' l
r j
ue ol*
D* Fuu
7; A,
\,{ o_
A;,t*
a t t o
W!,', c I
J*[ntu
J r { ' ,^e
lc I',nLs
l* ls
C on
A {-s)Itui l f ,
T l ' . r : u, l l I onl?
of d $tach
Ll. V e c[ 'n
e{)
fAj (11' 'v
On Orfq-
out l" aF
t/ t .> vy,r t , ,
" l
wa (b)
bho t
pe{ Al'*
i $ we dt [', n- LL d',rtoh'^
bh* t*,vrt ( *\ t f \o *-J c*,r orrr J
r't 7\ t l^ * A r't [e , lte ah'tn7 )
ofl.^, \^/ e
L[^ls C o'9e/
F^r.r\ t'11 la ftrt 3, I^
Oi
*,1\ f A u/ rr''e
,J, ^
I oo
-s"l\
F;F
*L Wrtr c ln* l , ?^ '> drd 3 !)
t^.!- Js $!n*
9o
le ?* Nt d xE-.
Breo LSe wi{& lo*Ps
!-xLrc*\2 Prvualrn L,
*'A
NAn
*kn
/tr or^e hc 0t prl* ,frrrnn'k
108
Recommended
