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Electromagnetic
Induction
EMF Induced in a Moving Conductor
v
EMF Induced in a Moving Conductor
L
q
F
v
Work done by force
W  FL  qvB L
Force on charge
F  qvB
W  qvBL
EMF
Work
qBLv
ε

Ch arg e
q
 BLv
Motional
Emf
Magnetic Flux
Φ  BA cos θ
Φ  B A
Magnetic flux  Weber (Wb)
1 Wb  1 T  m 2
Magnetic Flux
Φ  BA cos θ
Faraday’s Law of Induction
Dt
At t = 0
Fo = BA = BLx
B
v
After Dt
F = Fo + DF
F = BLx + BLDx
DF = BLDx
DF = BL(vDt)
x
Dx
ΔΦ
 BLv
Δt
ε
ΔΦ

Δt
L
Faraday’s Law of Induction
Magnetic flux will change if the area of the loop changes:
Faraday’s Law of Induction
Magnetic flux will change if the angle between the loop and
the field changes:
Faraday’s Law of Induction
Induced
emf
Faraday’s Law
Change
in Flux
DF
  N
Dt
Lenz’s
Law
Number
of Turns
Time for
Change
Lenz’s Law
Magnetic flux in the loop increases
Counterclockwise
Lenz’s Law
Magnetic flux in the loop increases
Clockwise
Lenz’s Law
Magnetic flux in the loop decreases
Clockwise
Lenz’s Law
Magnetic flux in the loop decreases
Counterclockwise
Transformers
A transformer consists of two coils, either interwoven or
linked by an iron core. A changing emf in one induces an
emf in the other.
NP
NS
Primary
Secondary
VS
VP
ΔΦ
ΔΦ
VP   N P
VS   N S
Δt
Δt
ΔΦ VP


Δt N P
VP VS

N P NS
VS NS

VP N P
ΔΦ VS


Δt N S
Transformers
IP
NP
NS
IS
VS
VP
Power in primary = Power in secondary
I P VP  I S VS
VS I P

VP I S
VS NS I P


VP N P I S
Field Energy Density
Energy density:
uE  1 εoE2
2

1 ε E2
2 o
uM 
2
1B
2μ
o
2
1B
2μ
o
E
1

B
ε oμ o
E

B
1
8.85x1012 C2 / N  m 2 4π x 10 7 T  m/A




Faraday’s Law of Induction
Problem 21-04
A 9.6-cm-diameter circular loop of wire is in a 1.10-T magnetic
field. The loop is removed from the field in 0.15 s.
What is the average induced emf?
ε
ΔΦ
AΔB


Δt
Δt


π 0.048 m 2 0  1.1 T 
ε
0.15 T
ε  5.3 x 10 2 V
Faraday’s Law of Induction
Problem 21-06
A 10.2-cm-diameter wire coil is initially oriented so that its plane
is perpendicular to a magnetic field of 0.63 T pointing up. During
the course of 0.15 s, the field is changed to one of 0.25 T pointing
down. What is the average induced emf in the coil?
ε
ΔΦ
AΔB


Δt
Δt


π 0.051 m 2  0.25 T  0.63 T 
ε
0.15 T
ε  4.8 x 10 2 V
Faraday’s Law of Induction
Problem 21-12
The moving rod is 12.0 cm long and is
pulled at a speed of 15.0 cm/s. If the
magnetic field is 0.800 T, calculate
(a) the emf developed in the rod.
ε  Blv  0.800 T0.120m 0.150 m/s 
 1.44 x 10 2 V
(b) the electric field felt by electrons in the rod.
ε  Ed
ε
1.44 x 10 2 V
E

d
0.120 m
 0.120 V/m, down
Faraday’s Law of Induction
Problem 21-14
The moving rod is 13.2 cm long and
generates an emf of 120 mV while
moving in a 0.90-T magnetic field.
(a) What is its speed?
ε  Blv
ε
v

Bl
0.12V
0.90 T 0.132 m 
 1.0 m/s
(b) What is the electric field in the rod?
ε  Ed
ε
E
0.12 V

0.132 m
d
 0.91 V/m, down
Faraday’s Law of Induction
Problem 21-16
A 500-turn solenoid, 25 cm long, has a diameter of 2.5 cm.
A 10-turn coil is wound tightly around the center of the solenoid.
If the current in the solenoid increases uniformly from 0 to 5.0 A
in 0.60 s, what will be the induced emf in the short coil during
this time?
ΔΦ
NAΔB   NA  μ o ΔIN sol 


 L

Δ
t


Δt
Δt
sol
ε

10  π 0.0125 m 2  4p x 10  7 T  m/A  5A 500 



0.60 s
ε  1.0 x 104 V


0.25 m


Faraday’s Law of Induction
Problem 21-17
The rod moves with a speed of 1.6 m/s
is 30.0 cm long, and has a resistance
of 2.5 W. The magnetic field is 0.35 T, and
the resistance of the U-shaped conductor
is 25 W at a given instant.
Calculate (a) the induced emf,
ε  Blv
 0.35 T 0.300 m 1.6 m/s 
 0.168 V
(b) the current in the U-shaped conductor
ε
I
R
0.168 V

27.5 Ω
 6.1 x 10 3 A
(c) the external force needed to keep the rod’s velocity constant at
that instant.
4
F  ILB  6.1 x 10 3 A 0.300 m 0.35 T   6.4 x 10 N


Faraday’s Law of Induction
Problem 21-73
A square loop 24.0 cm on a side has a resistance of 5.20 W. It is
initially in a 0.665-T magnetic field, with its plane perpendicular
to B but is removed from the field in 40.0 ms. Calculate the
electric energy dissipated in this process.
ΔΦ
AΔB


Δt
Δt
ε
ε
I
AΔB

R
RΔt
A 2 ΔB 
A
Δ
B


U E  PΔt  I 2 RΔt  
 RΔt 
RΔt
 RΔt 
2
2

0.240 m 2 0  0665 T 2
 7.05 x 10 3 J
UE 
5.20 Ω 0.0400 s 
Lenz’s Law
Problem 21-02
The rectangular loop shown is pushed into the magnetic
field which points inward. In what direction is the induced
current?
counterclockwise
Transformers
Problem 21-30
A transformer is designed to change 120 V into 10,000 V, and
there are 164 turns in the primary coil. How many turns are in
the secondary coil?
VS NS

VP N P
 VS 
 10,000 V 
  164 turns

N S  N P 
 120 V 
 VP 
 13,700 turns
Transformers
Problem 21-32
A step-up transformer increases 25 V to 120 V. What is the
current in the secondary coil as compared to the primary coil?
I S VS  I P VP
 VP 
I P   25 V I P
I S  
 120 V 
 VS 
 0.21 I P
Transformers
Problem 21-36
A transformer has 330 primary turns and 1340 secondary turns.
The input voltage is 120 V and the output current is 15.0 A. What
are the output voltage and input current?
Output voltage:
VS NS
NS
1340 turns
 120 V 

 VS  VP
330 turns
VP N P
NP
 487 V
Input current:
NS
I P NS
1340 turns

 I P  IS
 15 A 
IS N P
NP
330 turns
 60.9 A