1. Michael Faraday discovered the principles of electromagnetic induction in the early 1830s through experiments showing that a changing magnetic field can induce an electromotive force (emf) in a nearby conductor. 2. According to Faraday's law, an emf is induced in a closed loop of conductor when there is a change in the magnetic flux through the loop. The magnitude of the induced emf is proportional to the rate of change of the magnetic flux. 3. Self-induction describes the phenomenon of induction within a single circuit - as the current in a circuit changes, it induces its own opposing emf through its own magnetic field according to Lenz's law. This effect is quantified by the circuit's inductance

1. UNIDAD V
Inducción
Electromagnética

2. Michael Faraday
 1791 – 1867
 Great experimental
physicist
 Contributions to early
electricity include
 Invention of motor,
generator, and
transformer
 Electromagnetic
induction
 Laws of electrolysis

3. Induction
 An induced current is produced by a
changing magnetic field
 There is an induced emf associated with
the induced current
 A current can be produced without a
battery present in the circuit
 Faraday’s Law of Induction describes
the induced emf

4. EMF Produced by a
Changing Magnetic Field, 1
 A loop of wire is connected to a sensitive ammeter
 When a magnet is moved toward the loop, the
ammeter deflects
 The deflection indicates a current induced in the wire

5. EMF Produced by a
Changing Magnetic Field, 2
 When the magnet is held stationary, there is
no deflection of the ammeter
 Therefore, there is no induced current
 Even though the magnet is inside the loop

6. EMF Produced by a
Changing Magnetic Field, 3
 The magnet is moved away from the loop
 The ammeter deflects in the opposite
direction

7. EMF Produced by a Changing
Magnetic Field, Summary
 The ammeter deflects when the magnet is
moving toward or away from the loop
 The ammeter also deflects when the loop is
moved toward or away from the magnet
 An electric current is set up in the coil as long
as relative motion occurs between the
magnet and the coil
 This is the induced current that is produced by an
induced emf

8. Faraday’s Experiment –
Set Up
 A primary coil is connected to
a switch and a battery
 The wire is wrapped around
an iron ring
 A secondary coil is also
wrapped around the iron ring
 There is no battery present in
the secondary coil
 The secondary coil is not
electrically connected to the
primary coil

9. Faraday’s Experiment –
Findings
 At the instant the switch is closed, the
galvanometer (ammeter) needle deflects in
one direction and then returns to zero
 When the switch is opened, the galvanometer
needle deflects in the opposite direction and
then returns to zero
 The galvanometer reads zero when there is a
steady current or when there is no current in
the primary circuit

10. Faraday’s Experiment –
Conclusions
 An electric current can be produced by a time-
varying magnetic field
 This would be the current in the secondary circuit of
this experimental set-up
 The induced current exists only for a short time
while the magnetic field is changing
 This is generally expressed as: an induced
emf is produced in the secondary circuit by
the changing magnetic field
 The actual existence of the magnetic field is not
sufficient to produce the induced emf, the field must
be changing

11. Magnetic Flux
 To express
Faraday’s finding
mathematically, the
magnetic flux is
used
 The flux depends on
the magnetic field
and the area:
B d
  
B A

12. Flux and Induced emf
 An emf is induced in a circuit when the
magnetic flux through the surface
bounded by the circuit changes with
time
 This summarizes Faraday’s experimental
results

13. Faraday’s Law – Statements
 Faraday’s Law of Induction states that
the emf induced in a circuit is equal
to the time rate of change of the
magnetic flux through the circuit
 Mathematically,
B
d
dt


 

14. Faraday’s Law –
Statements, cont
 If the circuit consists of N identical and
concentric loops, and if the field lines
pass through all loops, the induced emf
becomes
 The loops are in series, so the emfs in the
individual loops add to give the total emf
B
d
N
dt


 

15. Faraday’s Law – Example
 Assume a loop
enclosing an area A
lies in a uniform
magnetic field
 The magnetic flux
through the loop is
B = B A cos q
 The induced emf is
 
cos
d
BA
dt
 q
 

16. Ways of Inducing an emf
 The magnitude of the field can change
with time
 The area enclosed by the loop can
change with time
 The angle q between the field and the
normal to the loop can change with time
 Any combination of the above can occur

17. Applications of
Faraday’s Law – Pickup Coil
 The pickup coil of an electric
guitar uses Faraday’s Law
 The coil is placed near the
vibrating string and causes a
portion of the string to
become magnetized
 When the string vibrates at
the some frequency, the
magnetized segment
produces a changing flux
through the coil
 The induced emf is fed to an
amplifier

18. Motional emf
 A motional emf is one
induced in a conductor
moving through a
magnetic field
 The electrons in the
conductor experience
a force that is directed
along l
B q
 
F v B

19. Motional emf, cont
 Under the influence of the force, the electrons
move to the lower end of the conductor and
accumulate there
 As a result of the charge separation, an
electric field is produced inside the conductor
 The charges accumulate at both ends of the
conductor until they are in equilibrium with
regard to the electric and magnetic forces

20. Motional emf, final
 For equilibrium, q E = q v B or E = v B
 A potential difference is maintained
between the ends of the conductor as
long as the conductor continues to
move through the uniform magnetic field
 If the direction of the motion is reversed,
the polarity of the potential difference is
also reversed

21. Sliding Conducting Bar
 A bar moving through a uniform field and the
equivalent circuit diagram
 Assume the bar has zero resistance
 The work done by the applied force appears as
internal energy in the resistor R

22. Sliding Conducting Bar, cont
 The induced emf is
 Since the resistance in the circuit is R,
the current is
B
d dx
B B v
dt dt


     
B v
I
R R

 

23. Sliding Conducting Bar,
Energy Considerations
 The applied force does work on the
conducting bar
 This moves the charges through a magnetic
field
 The change in energy of the system during
some time interval must be equal to the
transfer of energy into the system by work
 The power input is equal to the rate at which
energy is delivered to the resistor
 
2
app
F v I B v
R

  

24. AC Generators
 Electric generators take
in energy by work and
transfer it out by
electrical transmission
 The AC generator
consists of a loop of
wire rotated by some
external means in a
magnetic field

25. Induced emf
In an AC Generator
 The induced emf in
the loops is
 This is sinusoidal,
with max = N A B w
sin
B
d
N
dt
NAB t

w w

 


26. Lenz’ Law
 Faraday’s Law indicates the induced
emf and the change in flux have
opposite algebraic signs
 This has a physical interpretation that
has come to be known as Lenz’ Law
 It was developed by a German
physicist, Heinrich Lenz

27. Lenz’ Law, cont
 Lenz’ Law states the polarity of the
induced emf in a loop is such that it
produces a current whose magnetic
field opposes the change in magnetic
flux through the loop
 The induced current tends to keep the
original magnetic flux through the circuit
from changing

28. Lenz’ Law – Example 1
 When the magnet is moved toward the stationary
loop, a current is induced as shown in a
 This induced current produces its own magnetic field
that is directed as shown in b to counteract the
increasing external flux

29. Lenz’ Law – Example 2
 When the magnet is moved away the stationary loop,
a current is induced as shown in c
 This induced current produces its own magnetic field
that is directed as shown in d to counteract the
decreasing external flux

30. Induced emf
and Electric Fields
 An electric field is created in the conductor
as a result of the changing magnetic flux
 Even in the absence of a conducting loop,
a changing magnetic field will generate an
electric field in empty space
 This induced electric field has different
properties than a field produced by
stationary charges

31. Induced emf
and Electric Fields, cont
 The emf for any closed path can be
expressed as the line integral of
over the path
 Faraday’s Law can be written in a
general form
B
d
d
dt


   
E s
d

E s

32. Induced emf
and Electric Fields, final
 The induced electric field is a
nonconservative field that is generated
by a changing magnetic field
 The field cannot be an electrostatic field
because if the field were electrostatic,
and hence conservative, the line
integral would be zero and it isn’t

33. Self-Induction
 When the switch is
closed, the current
does not
immediately reach
its maximum value
 Faraday’s Law can
be used to describe
the effect

34. Self-Induction, 2
 As the current increases with time, the
magnetic flux through the circuit loop
due to this current also increases with
time
 The corresponding flux due to this
current also increases
 This increasing flux creates an induced
emf in the circuit

35. Self-Inductance, 3
 The direction of the induced emf is such that
it would cause an induced current in the loop
which would establish a magnetic field
opposing the change in the original magnetic
field
 The direction of the induced emf is opposite
the direction of the emf of the battery
 Sometimes called a back emf
 This results in a gradual increase in the
current to its final equilibrium value

36. Self-Induction, 4
 This effect is called self-inductance
 Because the changing flux through the
circuit and the resultant induced emf arise
from the circuit itself
 The emf L is called a self-induced
emf

37. Self-Inductance, Equations
 An induced emf is always proportional
to the time rate of change of the current
 L is a constant of proportionality called
the inductance of the coil
 It depends on the geometry of the coil and
other physical characteristics
L
dI
L
dt
  

38. Inductance of a Coil
 A closely spaced coil of N turns carrying
current I has an inductance of
 The inductance is a measure of the
opposition to a change in current
 Compared to resistance which was
opposition to the current
B
N
L
I dI dt


  

39. Inductance Units
 The SI unit of inductance is a Henry (H)
 Named for Joseph Henry
A
s
V
1
H
1



40. Joseph Henry
 1797 – 1878
 Improved the design of
the electromagnet
 Constructed one of the
first motors
 Discovered the
phenomena of self-
inductance

41. Inductance of a Solenoid
 Assume a uniformly wound solenoid
having N turns and length l
 Assume l is much greater than the radius
of the solenoid
 The interior magnetic field is
o o
N
B nI I
 
 

42. Inductance of a Solenoid, cont
 The magnetic flux through each turn is
 Therefore, the inductance is
 This shows that L depends on the
geometry of the object
B o
NA
BA I

  
2
o
B N A
N
L
I


 

43. RL Circuit, Introduction
 A circuit element that has a large self-
inductance is called an inductor
 The circuit symbol is
 We assume the self-inductance of the
rest of the circuit is negligible compared
to the inductor
 However, even without a coil, a circuit will
have some self-inductance

44. RL Circuit, Analysis
 An RL circuit contains
an inductor and a
resistor
 When the switch is
closed (at time t=0), the
current begins to
increase
 At the same time, a
back emf is induced in
the inductor that
opposes the original
increasing current

45. RL Circuit, Analysis, cont
 Applying Kirchhoff’s Loop Rule to the
previous circuit gives
 Looking at the current, we find
0
dI
IR L
dt
   
 
1 Rt L
I e
R
 
 

46. RL Circuit, Analysis, Final
 The inductor affects the current
exponentially
 The current does not instantly increase
to its final equilibrium value
 If there is no inductor, the exponential
term goes to zero and the current would
instantaneously reach its maximum
value as expected

47. RL Circuit, Time Constant
 The expression for the current can also be
expressed in terms of the time constant, t, of
the circuit
 where t = L / R
 Physically, t is the time required for the
current to reach 63.2% of its maximum value
   
1 t
I t e
R
t
 
 

48. RL Circuit,
Current-Time Graph, 1
 The equilibrium value
of the current is /R
and is reached as t
approaches infinity
 The current initially
increases very rapidly
 The current then
gradually approaches
the equilibrium value

49. RL Circuit,
Current-Time Graph, 2
 The time rate of
change of the
current is a
maximum at t = 0
 It falls off
exponentially as t
approaches infinity
 In general,
t
dI
e
dt L
t
 


50. Energy in a Magnetic Field
 In a circuit with an inductor, the battery
must supply more energy than in a
circuit without an inductor
 Part of the energy supplied by the
battery appears as internal energy in
the resistor
 The remaining energy is stored in the
magnetic field of the inductor

51. Energy in a
Magnetic Field, cont
 Looking at this energy (in terms of rate)
 I is the rate at which energy is being supplied by
the battery
 I2R is the rate at which the energy is being
delivered to the resistor
 Therefore, LI dI/dt must be the rate at which the
energy is being delivered to the inductor
2 dI
I I R LI
dt
  

52. Energy in a
Magnetic Field, final
 Let U denote the energy stored in the
inductor at any time
 The rate at which the energy is stored is
 To find the total energy, integrate and
UB = ½ L I2
B
dU dI
LI
dt dt


53. Energy Density
of a Magnetic Field
 Given U = ½ L I2,
 Since Al is the volume of the solenoid, the
magnetic energy density, uB is
 This applies to any region in which a
magnetic field exists
 not just the solenoid
2
2
2
1
2 2
o
o o
B B
U n A A
n

 
 
 
 
 
2
2
B
o
U B
u
V 
 

54. Inductance Example –
Coaxial Cable
 Calculate L and
energy for the cable
 The total flux is
 Therefore, L is
 The total energy is
ln
2 2
b
o o
B a
I I b
BdA dr
r a
 
 
 
     
 
 
ln
2
o
B b
L
I a


  
   
 
2
2
1
ln
2 4
o I b
U LI
a


 
   
 

55. Magnetic Levitation –
Repulsive Model
 A second major model for magnetic
levitation is the EDS (electrodynamic
system) model
 The system uses superconducting
magnets
 This results in improved energy
effieciency

56. Magnetic Levitation –
Repulsive Model, 2
 The vehicle carries a magnet
 As the magnet passes over a metal plate that
runs along the center of the track, currents
are induced in the plate
 The result is a repulsive force
 This force tends to lift the vehicle
 There is a large amount of metal required
 Makes it very expensive

57. Japan’s Maglev Vehicle
 The current is
induced by magnets
passing by coils
located on the side
of the railway
chamber

58. EDS Advantages
 Includes a natural stabilizing feature
 If the vehicle drops, the repulsion becomes
stronger, pushing the vehicle back up
 If the vehicle rises, the force decreases
and it drops back down
 Larger separation than EMS
 About 10 cm compared to 10 mm

59. EDS Disadvantages
 Levitation only exists while the train is in
motion
 Depends on a change in the magnetic flux
 Must include landing wheels for stopping and
starting
 The induced currents produce a drag force as
well as a lift force
 High speeds minimize the drag
 Significant drag at low speeds must be overcome
every time the vehicle starts up