hrb==.-trn't
Pltysics48 (Fail 201I)
Chupter30: Induction and fnductunce
-7--
"For everyminuteyou are angry, you lose 60 secondsofhappiness." - Ralph Waldo Emerson
"Considerhow much more oftenyou sufferf'om your anger and grieJ than.from lhose very things
.for which you are angry and grieved." Marcus Aurelius
"Holding on to anger is like grasping a hol coal with the intent of throwing it at someoneelse;you
are the one who gets burned. " - Buddha
791- 813
Reading:pages
O utline:
= introduction
Lawof induction
Faraday's
Lenz'sLaw
\-/
= emf inducedin a moving conductor
= inducedelectric frelds
Faraday'sLaw
= inductorsand inductance
inductanceof a soienoid
self-induction
= RL circuits
+ energystoredin a magnetic field
energydensity of a magneticfield
+ mutual induction (read on your own)
Problem Solving Techniques
You should know how to computethe flux through a given area.In some cases,the magneticfield is
uniform and you simpiy multiply the perpendicularcomponentby the area.In other cases,the fteld
is not uniform and you must cany out an integrationofthe perpendicularcomponentover the area.
In someproblems,the magneticfield is given while in othersthe currentthat producesthe field is
given and you must use what you learnedin the last chapterto write an expressionfor it as a
function ofposition. For example,the field may be producedby current in a long straightwire.
Once an expressionfor the magneticflux is found, you differentiate it with respectto time to find the
emf aroundthe boundary of tlre region. Look carefully at the expressionfor the flux and decidewhat
is changingwith time. It might be the magnitudeof the magnetic field, the magnitudeof the areaof
the region, or the orientation of the areawith respectto the fie1d.
'-
In someproblems,the boundary ofthe region is a conductorand you are askedto find the currentin
it. First, note al1the emfs aroundthe boundary.One of them is the emf associatedwith the changing
magneticflux but there may be others,producedby batteriesor generators,for example.Add them
with their correctsigns and divide the total by tJreresistancein the loop. Obviously, you must be
very careful aboutthe signs ofthe various emfs here.
Someproblemsask for the electric field associatedwith a chilnging magnetic field. Use Faraday's
a
law in theforrn P E 'a i = -Aolat.all of theproblemshavecylindricalsymmetry,with the
electricfield linesformingcirclesaroundthe cylinderaxis.For them,you integratethetangential
oneof rhefieldlines.withresultI f '6i :2rrE.where r
of theelectriclieldaround
component
is theradiusofthe circle.Find therateofchangeofthe magneticflux throughthe circleandsolve
1awequationfor E.
the Faraday's
(L : N@/i)to computeoneof the
Someproblemsaskyou to usethedefinitionof self-inductance
quantitiesthatappearin it. To cany out this task,you mayneedto computethe magneticflux (Dof
problemsdealwith theemf
the magneticfield, perhapsby carying out an integration.-Other
-L
di/dt.
Use B:
generated
by an inductorwhenthe currentchanges.
Someproblemsdealwith RZ seriescircuits.If a sourceof emf, suchasa battery,is in the circuitand
for example,a switchis closedthen),the cunentis givenby
the cunentis 0 at time I : 0 (because,
t() = (E/x) (t - n-'f
:
:
") ,wheretheinductivetime constantis 11 L/R.If the currentis lo au 0
and no sourceof emf is in the circuit, calculate
. -llT,
t\e ''
You may be askedto computethe potential differenceacrossthe resistor(iR) or acrossthe inductor
(L dildt). Be sureyou can tell which end is at the higher potential. You may also be askedfor the rate
with which the sourceof emf is supplying energy(l'€ ), the rate at which the resistoris dissipating
energy(;2R),or the rate at which the inductor is storingenergy (Li dildt).lf you are askedfor the
I
energydensitiesuse ur : i so* andua : El2po.
Someproblemsdeal with the calculationof mutual inductance.Here you assumea curent ln one
circuit and computethe magnetic flux it producesin anothercircuit.
Questions and Exahple Problems from Chapter 30
QuestionI
In thefigurebelow,a long straightwire with currenti passes(withouttouching)threerectangular
wire loopswith edgelengthsL, 1.5L,and2L. The loopsarewidely spaced(so asto not affectone
another).Loops1 and3 aresymmetricaboutthe longwire.Rankthe loopsaccordingto thesizeof
greatestfirst.
the currentinducedin themif currenti is (a) constantand(b) increasing,
(qJ
(b)
W
Question 2
If the circulat conductorin the figure below undergoesthermal expansionwhile it is in a uniform
magneticfield, a currentwiil be induced clockwise aroundit. Is the magnetic field directedinto the
pageor out of it?
t
o,^tA'AA
A;""*
)
Question3
R in the left-handcircuitofthe figurebelowis increasedat a steadyrate,is
If thevariableresistance
the cunentinducedin the right-handloop clockwiseor counterclockwise?
a j-A d.{.D.4e,,s",'-g.-}o
-)
I
dJ^r^".d
&^^r^S
$'rr^
7.tt-o-i
,c-a-o-c-&.'l*.,r"
"t
c^r-t"'r"g
",*r.'$r'r*
B.-*
Problem1
accordingto the reiation
ln the figurebelow,the magneticflux throughthe loop increases
(De:6.0f + 7.01,where@eis in milliwebersandt is in seconds.
(a) What is the magnitudeof the
emf lnducedin the loop when t = 2.0 s? (b) Is the directionof the currentthrough R to the right or
left?
do = 6.o L\+?.ot
^lr"r*1" ,Jt
l/r
yJ-
^ 6r= (6'0ry) f- ( r'o *$e ) t
x) = - laZ-?.D
= -JAt (6.or*+?.0
lF-] = l),t,"1.o--z l€(*'a-ou)tt -I
t-------
llat=
t)
l}'(e.o)r?.o =31
|
3l.V=3t"to-3V I
42,,*.*,Bo*,^^r*a.6Y*f^?" ""A ;*-.""'V
Qrr**r"- $^^,*;"
fro'NrJt-t+a a""t't /r,.,'$ fv*'*" " 3 {.,.,}J ;h itr gr'4r. )
y;
Problem 2
A squarewire loop with 2.00 m sidesis perpendicularto a uniform magneticfield, with half the area
of the loop in the field as shown in the figure below. The loop containsan ideal battery with emf
e.= 20.0 V. If the magnitudeof the field varies with time accordingto B : 0.0420 - 0.8701,with B in
teslasand t in seconds,what are (a) the net emf in the circuit and (b) the direction of the current
throush the batterv?
=) -Ar;".r
a"*L Et;,r...,
,.,-,-^a}-{A r-r} 14 n^qx
sa rcx^'''^Ita".)^zl"-rrv;g
t. ea"t
"l
st
.,,t-ax,nr.*+l^.t; d,^" of-ro..r^-nrg
B''u
N=1 €=- %t(vns
t=-rta6"/at
t/a't^
k =
€ = -(dBAr) P
B = tl. o'lloT-(O.t ?oTlr)r
lU/lf = - 6,x1or/s
1.
-
t-
t
rrr,
l.
,t!
g n^)l ^n ,r,^c'.Octv*
/1
t---------------Problem 3
|/1( ].o o. a
;
-- - (- o"r?or/,)
€
cl*tL-uFatt
oa .!-taffi"a1
t = at.rV I
I rr'.d-
4.0 O.
In the figurebelow,a wire formsa closedcircularloop,with radiusR:2.0 m andresistance
wire is
on a long straightwire; at time t:0, the currentin the long^straight
Thecircleis centered
:
(2.0
(The
Ns")f
.
straight
according
to
i
5.0
A
5.0A rightward.Thereafter,the currentchanges
wire is insulated,sothereis no electricalcontactbetweenit andthe wire of the loop.)What arethe
masnitudeanddirectionof the currentinducedin the loop at timest > 0?
)
-a;^r*
fr\,
dElr-t/En1 la.,ta',tt
/o\ )&\p ,P-*"5 tfi,*"g'*.
,"n"rq**}Ie
p-"fo ^t ,6 e Vn
,roriffo !*^1
M$"
|fu^K
^/olt9\, ,,u-f.{.,*r
Ml
SAta s6ul!-8* t*'
eu,^e"'J a -iin JJ14fu^r:f**
tt'"*\a
Q"=o
-r,tr, !-, ttu
\
,D-
))4.o8/-u!e .t>
o*"Jd
r"
,ea.*r*A
i9* q-€r
n'"*.*& 4'$9*j )Vu
t = (Lt,soA/r') t'- (ro'oA/s)t
-
Probfemn
Zn
For thesituationshownin tl6 figurebelow,a: 12.0cm andb : 16.0cm. The currentin the long
(a) Find the emf
straightwire is givenby i = 4.50t'- 10.01,wherei is in amperesandt is in seconds.
in thesquareloop at t: 3.00s. (b) What is thedirectionofthe inducedcurrentin theloop?
->
J,,rr,* j-
(-C
a
,{-f -\
B..JA = (
\,
Fl
^ dir-
b-o.
/ d't
a
o\-q
BdAc-ao" = ("(n"')(Ul.)
.arfc//
'
-- 4'j)
xtt
L-o
q
't
,U". b 9^. /
2\rT
t_"
!?=
92 +t*, +^,w"1
P'oth;\ Ptr-I-.F
= - Jd",,
l-a
/
A )*-P aL-p ,v')^i4',a
g'"ib^
S'^
#*
d'
(
r-
b-q
-) d=
- a,6
€=- d6/dt = 2,-r '^t
@
/ f.,\ ,at t= 3""t
d%tro u
B v,""t-^.g
6 )^^-'^;^g
Problem 5
condu&ifrlT66fr-,61ViiltfildsistanceR, andmassm, is
In thefigurebelow,a longrectangular
magnetic
field thatis directedinto thepageandthatexistsonly above
hungin a horizontal,uniform
until it reachesa certainterminal
line aa.Theloop is thendropped;duringits fall, it accelerates
find
that
terminal
speed.
v1.
Ignoring
air
drag,
speed
Fe
t_
F"-^Z= o ---) 4-m3
tl-
J
|ni
t LI3 -^^- 9o" = Ifrg
,
L=':'l--
foq
Lt3
,"=t'/n =k l'6111- * lTtu^tI
:
BLv
=
:,---A
nnq
Lt3
--->
+ a(r%t)=*a(t
fno, A
--J4--
Btl'
dv/w) = VY
/2
,\
.--
Problem6
of a close-packed
coil of 400tumsis 8.0mH. Calculatethe magneticflux though
Theinductancq
the coil whenthe currentis 5.0 mA.
N--foo
L= g "o"to-afr
l'= 5"o' td?A
d=tt./^
/=N6/t--@
= (f E . o , t 1 - t l i ) ' ( S . o " r o - : n )
0"-?
Problem7
A batteryis connectedto a seriesRL oircuit at time t: 0. At what multiple of t1 will the cuuent be
0.100%lessthanits equilibriumvalue?
- t.1
= ''/
^
/ ) ,
7A
Ar.o4rte --;. L
-,- 72" )
,'- - ?,/|
l
/
A
\
t
L
v
"-q,t^I^-0r,.-t*
o.gcq?/,1" e/n(l_"r".) __- o.ggq= 1- f"/''
- t/2,
-- -'o'1(o"oor)
---. n
o/a=
o"ool
t = e"9t.z.
---J
Problem8
Considerthe RL circuit of the figure below. In termsof the batteryemf e, (a) what is the selfinducedemf er when the switch hasjust beenclosedon a, and (b) what is es whent:2.0ey? (c) ln
termsof €s,when will es bejust one-halfthe batteryemf e?
'T'=Lh
;= €7o(l-.t/'')
€"= -tdAt = -t %rLt/n(l-e-"'/r)7
€,=
- g qt/z, --) ) t,) = t e*t/"
t= 6, t:a
a*
a-o
ce.
=)
la
= 2 c^, =
(b) I e,) = € e--^'o'"/"'
cc) lt,) = t;t/'gr, ('lr) - -t/-r,-
t
tr,l'-r
Jta
€r=
€, w&("'a;*'?"n&
(n&'"
,")t
e")
"&.,1
r'
u" tJ5 a
--n 9/a= €[*/'"
f = -(n"'a)1.
! t =;"b'lt::
-,
€
17^'=6t/z'
-- $L"t)2.
Problem9
In the figurebelow,thebatteryis ideal,e : 10V, Rr = 5.0 O, Rz: 10C2,andL = 5.0H. SwitchS is
what are(a) ;1,@) lz, (c) the currentisthroughthe switch,(d) the
closedat time t = 0. Justafterwards,
potentialdifferenceVz acrossresistorR2,(e) thepotentialdifferenceVs acrossthe inductorL, and
(f) the rateof changedi2/dfl A long time later, what are(g) 11,(h) t , (D r., (i) Vz, ft) V1, and (I) dir/dt?
kr^-
t"dr-*4
(o.-r--.1rU""6 ) :
c- EA Q- d'/'")
I = - Z d'/2.
()
t.= - t;t/z'
(r) lZ= ,-ol
(,4tv
t-
v
)
t= pt
" - F'/
.-74
---1 t.= -L J f=oe
, \€..\-@
( i, .\4"4
-''-,ry')
=
( h) i^= €/A,(t-;ttz'S ,o*t-u > t^= €1A^
#
:l@
( i) t, =t,, i^ =.Fo{f
:
(i) V^-- = 1r,o*)(to.or)
IT-r;4
"8,
()<) tt = - 2 6t/2" --E;;l ,,+t= "'
( l) d,/r* =
/0, Lc/q(t-;'/'')]
-- t:
t-i.
o'?'lt=
-g/F
-, *
.'/.?1".)
o[.^*e="-
=
Td*o'
Problem10
to an idealbattery
of 10O is suddenlyconnected
of 2.0 H anda resistance
A coil with aninductance
with e: i00 V. 41 t = 0.10s afterthe connectionis made,whatis therateat which (a) energyis
beingstoredin the magneticfield, (b) thermal.energyis appearingin the resistance,and (c) energyis
beingdeliveredby the battery?
(a) P=;€,:tLd4"
L=\.oH
4 = tolt
e = loDv
(
I
= L; t,Zt ) \
Av"/at=d7417trat;')
\)
"t ?=
p =L letott-;'h,)f(F-/,d"')
= e74
0-;+t.,
(d'ft)
,
.+h,
; =Z(t-;'e)
-th"
, .' .
t"z
o6t=
,=9
7" 6w'
----->o) {: b.LoS
( b) +ol q'\aa).lfq, P=itR = LtlnTtit/ttl'R
-.:
P = E7n( l-.oz*'ra
a* t'o'ios
L.) &*a 2,tu)At P =f t = L€/u(\'d''4r7Z
= E7A(l:*/2") -- a* t= o,ro5JP = :.q,"to"lv
P
f)f)
/@'" Yum2 = Y"y^x^*?,;-r^",a
Problem11
Provethat, after switch S in the figure to the right hasbeenthrown from a to b,
all the energystoredin the inductor will ultimately appearasthermalenergyin
theresistor.
*tL)"rAt',ht-
= ; (d"/at)
-JE/at L; J|AL ----) 2- = r/^ti\
?=;gL
l\^
----.
-b^.L^Jyl
.M+.1,f4) ^h\
tl.l-
^J{.r^rJWL
)
^u,u.,"r- nllMr"rt )"1\*'Y\ 6xl"*
t/at;,')
4r-^riilA ,'.4 oJ a (E=
I t= o)a\]'fr,'ffi;
.a
a
/:= (Flt
z-))J
=(t^AJt-(
*
-i^!''l f
tr L:A l'='/^
o
r.
E
t/-;^27 t ' L - - l / ^ i " ^ A
/^uo,/)
^[
ta i"'hh!fu
L. w!^t\
i=Lo;z/z' ,.= /a
&
(
-=
A
i.:
r,;t'-l^A J,
- \x/zL
o
o
[%1 =
I/1L"'"^
- -gr\t\q4
---a:,.
/fi."A
en@
wr anr'ol"u;tl7r