Chapter 19
F a r a da y's La w Ch a pt er Review
E QU ATI ON S :
•
Φm =
∫
B • dS
[Th is is t h e m ost gen er a l expr ession for t h e m a gn et ic flu x Φ m gen er a t ed by
S
a m a gn et ic field B pa ssin g t h r ou gh t h e fa ce of a closed pa t h . It in st r u ct s u s t o defin e a
differ en t ia l su r fa ce a r ea vect or dS per pen dicu la r t o t h e fa ce of t h e closed pa t h , m u lt iply
t h a t differ en t ia l a r ea by t h e com pon en t of B pa r a llel t o dS (t h a t is wh a t t h e dot pr odu ct
does), t h en su m a ll su ch pr odu ct s over t h e en t ir e fa ce via in t egr a t ion . TH IS
E XP RE SSION IS ONLY USE D WH E N B VARIE S over t h e fa ce. Th e u n it s of m a gn et ic
flu x a r e t es la s . m 2 , com m on ly ca lled weber s .]
• Φm = B • A
[Th is m a gn et ic flu x expr ession is u sed wh en B is con st a n t over a pa t h 's fa ce.
Not e t h a t t h e expr ession is equ a l t o B A cos θ , wh er e B is t h e m a gn it u de of t h e m a gn et ic
field, A is t h e t ot a l a r ea of t h e fa ce, a n d θ is t h e a n gle bet ween B a n d A. USE TH IS
E XP RE SSION WH E NE VE R YOU CAN. Th er e is n o r ea son t o m a ke a pr oblem h a r der
t h a n it h a s t o be by u n n ecessa r ily m essin g wit h t h e in t egr a l expr ession pr esen t ed ju st
a bove.]
• E M F [An elect r om ot ive for ce (E MF ), sym bolized by a n ε in a cir cu it , h a s t h e u n it of volt s
a n d is t h e fa ct or wit h in a n elect r ica l cir cu it t h a t m ot iva t es ch a r ge t o m ove a s a cu r r en t .
In a t ypica l DC cir cu it , t h e E MF is pr ovided by a ba t t er y or power su pply.]
• ε = −N
d Φm
[Th is is F a r a da y's La w. It st a t es t h e followin g: If you t a ke a wir e coil t h r ou gh
dt
wh ich a n ext er n a l m a gn et ic field B pa sses, t h er e will be a m a gn et ic flu x Φ m through the
fa ce of t h a t coil (i.e., t h r ou gh t h e a r ea bou n ded by t h e loops). If t h e m a gn et ic flu x t h r ou gh
t h a t fa ce ch a n ges, a n in du ced E MF will be set u p t h a t will m ot iva t e ch a r ge in t h e coil t o
d Φm
flow. Th e size of t h e in du ced E MF is pr opor t ion a l t o t h e r a t e of ch a n ge of flu x
dt
t h r ou gh t h e fa ce, wit h t h e pr opor t ion a lit y con st a n t bein g t h e opposit e of t h e n u m ber of
win ds in t h e coil, or -N. Not e t h a t TH IS E XP RE SSION IS USE D WH E N Φ m VARIE S IN
A NON-LINE AR WAY. Th e expr ession t h a t follows is ea sier t o u se if t h e ch a n ge is
l i n e a r .]
∆Φ m
∆ ( B A cos θ )
= −N
[Th is is F a r a da y's La w for a sit u a t ion in wh ich t h e ch a n ge
∆t
∆t
of m a gn et ic flu x is lin ea r . Not e t h a t t h er e a r e a lot of wa ys in wh ich t h e m a gn et ic flu x
ca n ch a n ge. If, sa y, t h e m a gn et ic field pr odu ces t h e ch a n ge, t h e expr ession becom es
 ∆B 
 ∆B 
ε = −N 
 A cos θ . Not e a lso t h a t t h e in for m a t ion r equ ir ed t o det er m in e t h e 

 ∆t 
 ∆t 
t er m ca n be given in a t lea st t wo wa ys. Specifica lly, you cou ld be given t wo m a gn et ic
field va lu es cor r espon din g t o t wo specified t im es (exa m ple: B = 2 t esla s a t t = 3 secon ds
a n d B = 14 t esla s a t t = 5 secon ds), or you cou ld be given t h e r a t e a t wh ich B ch a n ges wit h
• ε = −N
370
F a r a da y's La w
t im e (exa m ple: 6 t esla s/secon d). In a n y ca se, t h is for m of F a r a da y's La w is ea sier t o u se
t h a n t h e m or e for m a l, der iva t ive-la den ver sion .]
•
ε = iR [In a coil in wh ich a n in du ced cu r r en t is gen er a t ed by a n in du ced E MF , t h e
r ela t ion sh ip bet ween t h e cu r r en t a n d t h e E MF is st ill su m m a r ized by Oh m 's La w.]
di
[If you ch a n ge t h e m a gn et ic flu x down t h e a xis of a coil, you will get a n in du ced
dt
E MF a cr oss t h e coil's lea ds. At som e poin t , som eon e r ea lized t h a t a ch a n ge in t h e
ext er n a lly dr iven cu r r en t in t h e coil ca u ses a m a gn et ic field ch a n ge t h a t ca u ses a
m a gn et ic flu x ch a n ge t h a t ca u ses a n in du ced E MF in t h e coil. As su ch , on e cou ld wr it e
t h e in du ced E MF in t er m s of t h e r a t e of ch a n ge of cu r r en t wit h t im e, or di/dt . Th is
expr ession r equ ir es a pr opor t ion a lit y con st a n t t h a t is ca lled t h e coil's in du ct a n ce. Th e
sym bol for in du ct a n ce is L, a n d it s MKS u n it is h en r ys, t h ou gh m ost coils h a ve a n
in du ct a n ce in t h e m illih en r y (i.e., m H ) r a n ge. As is t h e ca se wit h a r esist or 's
r esist a n ce R or a ca pa cit or 's ca pa cit a n ce C, a n in du ct or 's in du ct a n ce L is t h e pa r a m et er
t h a t iden t ifies t h e size of t h e en t it y. Not e t h a t du e t o t h is, coils a r e oft en r efer r ed t o a s
in du ct or s wh en in a n elect r ica l cir cu it . Ot h er com m on colloqu ia l t er m s a r e ch ok e a n d
s ole n oid .]
• ε = −L

−( R L ) t 
• i ( t ) = io  1 − e
 [Th is is t h e solu t ion t o Kir ch off's La ws wr it t en for a n RL cir cu it .


YOU WILL NE VE R USE TH IS E XP RE SSION. It h a s been in clu ded h er e beca u se it
m a t h em a t ica lly h igh ligh t s t h e fa ct t h a t cu r r en t s in RL cir cu it s r ise slowly F ROM ZE RO
(ver su s im m edia t ely ju m pin g t o m a xim u m cu r r en t a s in a st r a igh t r esist or or R C
cir cu it ).]
 L
• τ =   [On e t im e con st a n t τ for a n RL cir cu it is t h e a m ou n t of t im e it t a kes a ft er power is
 R
su pplied for t h e cu r r en t in t h e RL cir cu it t o r ise t o 63% of it s m a xim u m . J u st a s wa s t h e
ca se wit h ca pa cit or s, t wo t im e con st a n t s (i.e., 2 τ ) is a ssocia t ed wit h 87%. Not e t h a t t h is
fu n ct ion is r ecipr oca l. Th a t is, it will t a ke on e t im e con st a n t for a n est a blish ed cu r r en t
in a n RL cir cu it t o dr op 63% of it s or igin a l va lu e (i.e., down t o 37% cu r r en t ) wh en power
is r em oved fr om t h e cir cu it .]
• E n ergy stored =
1 2
Li
2
[A cu r r en t ca r r yin g in du ct or st or es en er gy in t h e m a gn et ic field t h a t
1 2
L i . Note that this is also the
2
a m ou n t of en er gy r equ ir ed t o get i's wor t h of cu r r en t flowin g t h r ou gh t h e coil in t h e fir st
pla ce, a ssu m in g n o en er gy loss occu r s in t h e pr ocess.]
r esides down it s a xis. Th e a m ou n t of en er gy is equ a l t o
• If N s > N p , then ε > ε a n d i s < i p [If t h e t u r n s-r a t io of a t r a n sfor m er is su ch t h a t t h e
s
p
n u m ber of t u r n s N s in t h e secon da r y coil is gr ea t er t h a n t h e n u m ber of t u r n s N p in the
pr im a r y coil, t h en in du ced E MF
ε s in t h e secon da r y will be gr ea t er t h a n t h e E MF ε p in
t h e pr im a r y, a n d t h e cu r r en t i s in t h e secon da r y will be less t h a n t h e cu r r en t ip in the
pr im a r y. As t h e volt a ge in cr ea ses bet ween t h e pr im a r y a n d t h e secon da r y in t h is ca se,
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su ch t r a n sfor m er s a r e ca lled st ep u p t r a n sfor m er s. If t h e t u r n s r a t io h a d been t h e ot h er
wa y a r ou n d, t h e t r a n sfor m er wou ld h a ve been ca lled a st ep down t r a n sfor m er .]
• ε in d u ced = N B av [DO NOT ME MORIZE TH IS RE LATIONSH IP . It is t h e der ived
expr ession for t h e E MF gen er a t ed in a coil of widt h a a n d win ds N a s t h e coil is pu lled
fr om a con st a n t m a gn et ic field B wit h velocit y v. It h a s been in clu ded h er e t o r em in d you
t h a t m ot ion a l E MF 's a r e im por t a n t , a n d t h a t you sh ou ld u n der st a n d t h e t h eor y well
en ou gh t o be a ble t o DE RIVE t h e a bove st a t ed expr ession . You sh ou ld a lso be a ble t o der ive
a n expr ession for t h e cu r r en t gen er a t ed in t h e coil, a n d a n expr ession for t h e for ce on t h e
coil du e t o t h e in du ced cu r r en t 's in t er a ct ion wit h t h e ext er n a l m a gn et ic field.]
• N
d Φm
= − ∫ E • d l [An in du ced E MF t h a t is gen er a t ed in a coil m u st be a ssocia t ed wit h a n
dt
elect r ic field (a ft er a ll, a n elect r ic field is wh a t m a k es ch a r ge m ove in a wir e). Th e
pr oblem is t h a t pr eviou sly, we h a ve r ela t ed elect r ic fields a n d elect r ica l pot en t ia l
differ en ces by t h e r ela t ion sh ip ∆ V = − ∫ E • d l . A difficu lt y sh ows it self wh en we t r y t o
u se t h is r ela t ion sh ip in con ju n ct ion wit h t h e closed pa t h F a r a da y's La w r equ ir es
(r em em ber , F a r a da y's La w dea ls wit h t h e in du ced E MF set u p a r ou n d a closed pa t h du e t o
t h e pr esen ce of a ch a n gin g m a gn et ic flu x). In su ch ca ses, ∆ V should be zero as the
elect r ica l pot en t ia l differ en ce bet ween a poin t a n d it self is zer o. As t h a t ca n 't be t h e ca se
(ot h er wise, − ∫ E • d l wou ld a lso be zer o), t h er e m u st be som et h in g wr on g wit h ou r
r ea son in g. Th e pr oblem goes a wa y wh en we r ea lize t h a t a ch a n gin g m a gn et ic flu x does
n ot pr odu ce a con ser va t ive for ce field. As su ch , t h er e ca n be n o pot en t ia l en er gy fu n ct ion
a ssign ed t o t h e for ce field, a n d t h e con cept of ∆ V becom es n on sen se. Th e r ela t ion sh ip
t h a t does exist bet ween a n elect r ic field E t h a t h a s been in du ced a r ou n d a closed pa t h du e
t o t h e pr esen ce of a n E MF in du cin g, ch a n gin g m a gn et ic flu x, is
N
d Φm
dt
= −∫ E • dl .]
COMME NTS, H INTS, a n d TH INGS t o be a wa r e of:
• Th e con cept of a n E MF is n ot n ew. Wh en you pu t a volt m et er a cr oss t h e t er m in a ls of a power
su pply, t h e volt a ge m ea su r ed is ca lled t h e t er m in a l volt a ge. Beca u se t h e power su pply
will h a ve a n in t er n a l r esist a n ce t h a t ca u ses a volt a ge dr op wh en cu r r en t is dr a wn fr om
t h e sou r ce, t h e t er m in a l volt a ge is n ot equ a l t o t h e power su pply's E MF . It is, in st ea d,
equ a l t o ε - ir i , wh er e ε is t h e sou r ce's E MF , i is t h e cu r r en t bein g dr a wn fr om t h e sou r ce,
a n d r i is t h e in t er n a l r esist a n ce wit h in t h e sou r ce.
• If, AT A GIVE N INSTANT, t h e m a gn et ic field is t h e sa m e ever ywh er e a cr oss t h e fa ce of a
bou n ded a r ea , even if t h e m a gn et ic field fu n ct ion is ch a n gin g wit h t im e, you ca n u se B . A
t o det er m in e t h e gen er a l expr ession for t h e m a gn et ic flu x AT TH AT P OIN T IN TIME .
Use t h e in t egr a l expr ession ONLY if t h e m a gn et ic field is differ en t fr om poin t t o poin t
a cr oss t h e fa ce AT A GIVE N INSTANT IN TIME . In sh or t , don 't u se t h e in t egr a l for m of
t h e m a gn et ic flu x expr ession if it isn 't a bsolu t ely n ecessa r y.
• Don 't u se t h e in t egr a l for m of F a r a da y's La w if it isn 't a bsolu t ely n ecessa r y. Th e
∆ ver sion wor k s ju st fin e, a ssu m in g t h e m a gn et ic flu x is ch a n gin g lin ea r ly.
372
F a r a da y's La w
• An in du ced E MF gen er a t ed by a coil lea vin g a m a gn et ic field (a m ot ion a l E MF pr oblem )
ALWAYS gen er a t es a for ce t h a t OP P OSE S t h e m ot ion . Th a t is, if t h e coil is m a de t o
lea ve t h e field, t h e in du ced cu r r en t in t h e coil will in t er a ct wit h t h e ext er n a l m a gn et ic
field (F = iLxB) pr odu cin g a for ce on t h e wir e t h a t is opposit e t o t h e dir ect ion of t h e m ot ion .
Th e sa m e is t r u e if t h e coil en t er s a m a gn et ic field.
• E ddy cu r r en t s r efer t o t h e swir lin g m ot ion of in du ced ch a r ge flow in a solid, m et a llic pla t e
(t h e pla t e doesn 't h a ve t o be m a gn et iza ble--a lu m in u m wor ks ju st fin e) a s pa r t of t h e pla t e
is for ced in t o or ou t of a m a gn et ic field. Th e in t er a ct ion of t h ese in du ced cu r r en t s a n d t h e
ext er n a l m a gn et ic field pr odu ces a for ce t h a t ALWAYS slows t h e pla t e's m ot ion . Th is is
t h e ba sis of a n eddy cu r r en t br a ke.
• Wh en ever a n y m a t er ia l exper ien ces a ch a n ge of m a gn et ic flu x, wh et h er t h e m a t er ia l be a
con du ct or or in su la t or , t h er e will ALWAYS be a n in du ced elect r ic field set u p in t h e
d Φm
m a t er ia l su ch t h a t
= − ∫ E • d l . If t h e m a t er ia l h a ppen s t o be a con du ct or , ch a r ge
dt
will m ove a n d you will get cu r r en t . If t h e m a t er ia l is a n in su la t or , n o ch a r ge will m ove
(a h sa y, it 's a n in su la t or ), t h ou gh t h e field will n ever t h eless be t h er e.
• If you a r e a sked t o u se Kir ch off's La ws on a cir cu it in wh ich t h er e exist s a t lea st on e
in du ct or , don 't be pu t off. J u st a s t h e volt a ge dr op a cr oss a r esist or is iR a n d t h e volt a ge
di
dr op a cr oss a ca pa cit or is q/C, t h e volt a ge dr op a cr oss a n in du ct or is L
. Not e, t h ou gh ,
dt
t h a t beca u se a n in du ct or is essen t ia lly wir e wr a pped a s a coil, t h er e is a lso r esist or -like
r esist a n ce a ssocia t ed wit h it . Th a t m ea n s t h er e ca n a lso be a n on -zer o volt a ge dr op
a cr oss a n in du ct or equ a l t o ir L , wh er e i is t h e cu r r en t t h r ou gh t h e in du ct or a n d r L is t h e
r esist or -lik e r esist a n ce in h er en t wit h in t h e wir e it self. Th is volt a ge dr op is som et im es
ign or ed beca u se it is oft en sm a ll, bu t don 't be t h r own off if you r u n in t o a pr oblem t h a t
in clu des it .
373