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Edward C. Jordan Memorial Offering of the First
Course under the Indo-US Inter-University
Collaborative Initiative in Higher Education and
Research: Electromagnetics for Electrical and
Computer Engineering
by
Nannapaneni Narayana Rao
Edward C. Jordan Professor of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
Urbana, Illinois, USA
Amrita Viswa Vidya Peetham, Coimbatore
July 10 – August 11, 2006
2.3
Faraday’s Law
2.3-2
Faraday’s Law
d
C E • dl  – dt S B • dS
B
S
C
dS
2.3-3
C E • dl = Voltage around C, also known as
electromotive force (emf) around C
(but not really a force),
 V m  m, or V.
S B • dS = Magnetic flux crossing S,
2
2
Wb
m

m

 , or Wb.
d
– S B • dS = Time rate of decrease of
dt
magnetic flux crossing S,
Wb s, or V.
2.3-4
Important Considerations
(1) Right-hand screw (R.H.S.) Rule.
The magnetic flux crossing the
surface S is to be evaluated toward
that side of S a right-hand screw
advances as it is turned in the sense of C.
C
2.3-5
(2) Any surface S bounded by C.
The surface S can be any surface bounded by
C. For example:
z
R
z
R
C
O
C
Q
O
y
P
x
x
Q
y
P
This means that, for a given C, the values of
magnetic flux crossing all possible surfaces
bounded by it is the same, or the magnetic flux
bounded by C is unique.
2.3-6
(3) Imaginary contour C versus loop of wire.
There is an emf induced around C in either
case by the setting up of an electric field. A
loop of wire will result in a current flowing in
the wire.
(4) Lenz’s Law.
States that the sense of the induced emf is
such that any current it produces, if the closed
path were a loop of wire, tends to oppose the
change in the magnetic flux that produces it.
2.3-7
Thus the magnetic flux produced by the
induced current and that is bounded by C must
be such that it opposes the change in the
magnetic flux producing the induced emf.
(5) N-turn coil.
For an N-turn coil, the induced emf is N times
that induced in one turn, since the surface
bounded by one turn is bounded N times by
the N-turn coil. Thus
d
emf  – N
dt
2.3-8
where  is the magnetic flux linked by one turn

D2.5 B  B0 sin t ax  cos t a y
B
S

d S = B0 sin t
d
C E d l   dt  B0 sin t 
  B0 cos t V
z
1
C
1
x
y
2.3-9

B0
0
 dec.

2
3
t
 inc.
–B0
emf
B0
0
–B0
emf < 0

2
3
emf > 0
Lenz’s law is verified.
t
2.3-10
(b)
S B • dS
1
1
 B0 sin t – B0 cos t
2
2
1
 


B0 sin t –

2
4 
C E • dl
z
1
C
1
x
d  1


–
B0 sin t –

dt 
4 
 2

B0


–
cos t –
V


2
4
1 y
2.3-11
(c)
z
S B • dS
1
 B0 sin t  B0 cos t


 2 B0 sin t 

4 
C
E • dl
d 
–
2 B0 sin t

dt 

 – 2 B0 cos t 

C
1 y
1
x
 

4 


V
4 
2.3-12
Motional emf concept
B
C
l
S
x
z
S
d S = B0ly
 B0l  y0  v0t 
v0 ay
conducting
rails
y
B  B0az
B
dS
conducting bar
y  y 0  v 0t
2.3-13
d
C E • dl  – dt S B • dS
d
   B0l  y0  v0t  
dt
  B0lv0
This can be interpreted as due to an electric field
F
E   v0 B0 a x
Q
induced in the moving bar, as viewed by an observer
moving with the bar, since
l
v0 B0 l  x0 v0 B0a x • dx a x
l
 x0 E • dl
2.3-14
where
F  Qv  B
 Qv0 a y  B0 a z
 Qv0 B0 a x
is the magnetic force on a charge Q in the bar.
Hence, the emf is known as motional emf.