1. Ch. 21: Magnetic Induction & Faraday’s Law of Induction
Chapter 29 opener. One of the great laws of physics is Faraday’s law of induction, which says that a changing magnetic flux produces an induced emf. This photo shows a bar magnet moving inside a coil of wire, and the galvanometer registers an induced current. This phenomenon of electromagnetic induction is the basis for many practical devices, including generators, alternators, transformers, tape recording, and computer memory.

2. Topics Outline Faraday’s Law
Induced EMF
Faraday’s Law
of Induction
Lenz’s Law
EMF Induced in a Moving Conductor
Electric Generators
Back EMF & Counter Torque

3. A Changing Magnetic Flux Produces an Electric Field!
More Topics
Eddy Currents
Transformers & Transmission of Power
Faraday’s Law:
A Changing Magnetic Flux
Produces an Electric Field!
Some Applications of Induction:
Sound Systems
Computer Memory
Seismograph,….

4. Magnetic Induction
Electric & magnetic forces both act only on particles carrying an electric charge
Moving electric charges create a magnetic field
A changing magnetic field creates an electric field
This effect is called magnetic induction
This links electricity and magnetism in a fundamental way
Magnetic induction is also the key to many practical applications. See next slides!

5. Speedometers & Odometers
Some Applications
Magnetic
Resonance
Imaging (MRI)
Speedometers & Odometers

6. Electric Guitars & Other Instruments
The solid-body electric guitar was invented by Les Paul (1915–2009; shown here). He is the only person who has been inducted into both the Rock and Roll Hall of Fame and the Inventor’s Hall of Fame.

7. Hybrid Automobiles
Figure A hybrid car such as this Toyota Prius contains a gasoline-powered motor along with an electric motor, a generator, and batteries.

8. An electric field can produce Can a magnetic field produce
Electromagnetism
Electric and magnetic phenomena were first connected by Ørsted in 1820
He discovered that an electric current in a wire can exert a force on a compass needle. This indicates that an electric field can lead to a force on a magnet. He concluded that
An electric field can produce
a magnetic field
So, people wondered
Can a magnetic field produce
an electric field?
Experiments by Faraday found that the answer is yes!

9. Michael Faraday
1791 – 1867
British physicist & Chemist.
Great experimental Scientist.
A “hands” on Experimental
Scientist. Strong on experiment
design. Less strong on math!
2 of his Major Contributions to Electricity: 1. Electromagnetic induction
2. Laws of electrolysis

10. Michael Faraday A Productive Inventor!!!
Some Major Inventions
1. Motor
2. Generator
3. Transformer

11. Faraday’s Discoveries: An electric current could be
1. Whenever the magnetic
field about an electromagnet was made to grow or collapse by closing or
opening the electric circuit of which it was a part,
An electric current could be
detected in a separate
conductor nearby.

12. Faraday’s Discoveries:
2. Moving a permanent magnet into & out of a coil of wire also
induces a current
in the wire
while the magnet is in motion.
3. Moving a conductor near a stationary permanent magnet causes a current to flow in the wire also, as long as it is moving.

13. Faraday’s Experiments
Faraday attempted to observe a B field induced E field
He used an ammeter instead of a light bulb
If the bar magnet was in motion, a current was observed
If the magnet was stationary, the current & the electric field were both zero

14. Another Faraday Experiment
A solenoid is positioned near a loop of wire with the light bulb. Current passes through the solenoid by connecting it to a battery
When the current through the solenoid is constant, there is no current in the wire
When the switch is opened or closed, the bulb does light up

15. Induced EMF Michael Faraday looked for evidence
that a magnetic field would induce an
electric current with this apparatus:
Figure Faraday’s experiment to induce an emf.

16. when the switch was turned on or off.
He found no evidence when the current was
steady. But, he saw an induced current
when the switch was turned on or off.
Figure (a) A current is induced when a magnet is moved toward a coil, momentarily increasing the magnetic field through the coil. (b) The induced current is opposite when the magnet is moved away from the coil ( decreases). Note that the galvanometer zero is at the center of the scale and the needle deflects left or right, depending on the direction of the current. In (c), no current is induced if the magnet does not move relative to the coil. It is the relative motion that counts here: the magnet can be held steady and the coil moved, which also induces an emf.

17. Faraday concluded that: A Changing Magnetic Field Induces an EMF.
His experiments used a magnetic field that
was changing because the current producing it
was changing; the picture shows a magnetic
field that changes because the magnet is moving.
Figure (a) A current is induced when a magnet is moved toward a coil, momentarily increasing the magnetic field through the coil. (b) The induced current is opposite when the magnet is moved away from the coil ( decreases). Note that the galvanometer zero is at the center of the scale and the needle deflects left or right, depending on the direction of the current. In (c), no current is induced if the magnet does not move relative to the coil. It is the relative motion that counts here: the magnet can be held steady and the coil moved, which also induces an emf.

18. An EMF is Produced by a Changing Magnetic Field
A loop of wire is connected
to a sensitive ammeter.
When a magnet is moved
toward the loop, the
ammeter deflects.
The direction was arbitrarily chosen to be negative.

19. held stationary, there is no deflection no induced current.
When the magnet is
held stationary, there
is no deflection
of the ammeter.
Therefore, there is
no induced current.
Even though the magnet is in the loop

20. in the opposite direction!
If the magnet is moved
away from the loop.
The ammeter deflects
in the opposite direction!

21. Induced Current, Summary

22. 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.
No battery is present in
the secondary coil.
The secondary coil is not directly
connected to the primary coil.

23. Close the switch & observe the current readings on the ammeter.

24. Faraday’s Findings
At the instant the switch is closed,
the ammeter changes from zero in one direction, then returns to zero.
When the switch is opened,
the ammeter changes in the opposite direction, then returns to zero.
The ammeter reads zero when there is a
steady current or when there is no
current in the primary circuit.

25. Faraday’s Experiments: Conclusions
An electric current can be induced in
a loop by a changing magnetic field.
This is the current in the secondary
circuit of this experimental set-up.
The induced current exists only while the magnetic field through
the loop is changing.

26. Faraday’s Experiments: Conclusions
An electric current is induced in a secondary circuit during the time when the current through the solenoid is changing. Faraday’s experiments show that an electric current is produced in the wire loop only when the magnetic field at the loop is changing
A changing magnetic field
produces an electric field

27. Faraday’s Experiments: Conclusions
A changing magnetic field
produces an electric field
An electric field produced in this way is called
an induced electric field.
The phenomena is called
electromagnetic induction

28. Faraday’s Experiment: Conclusions
All of this is usually expressed as:
An induced emf is produced in the loop by the changing magnetic field.
Just the existence of the magnetic
field is not sufficient to produce the
induced emf, the field must be
changing.

29. Magnetic Flux ΦB  B A cos θ
Faraday developed a quantitative theory of induction that is now called Faraday’s Law
The law shows how to calculate the induced electric field in different situations
Faraday’s Law uses the concept of magnetic flux
Magnetic flux is similar to the concept of electric flux
Let A be an area of a surface with a
magnetic field passing through it
The flux is defined as
ΦB  B A cos θ

30. Magnetic Flux
If the field is perpendicular to the surface, ΦB = B A
If the field makes an angle θ with the normal to the surface, ΦB = B A cos θ
If the field is parallel to the surface, ΦB = 0

31. 1 Wb = 1 T . m2 The magnetic flux can be defined for any surface
A complicated surface can be broken into small regions and the definition of flux applied
The total flux is the sum of the fluxes through all the individual pieces of the surface
The surfaces of interest are open surfaces
With electric flux, closed surfaces were used
SI unit of magnetic flux = the Weber (Wb)
1 Wb = 1 T . m2

32. Faraday’s Law of Induction: Lenz’s Law
Faraday found that the induced emf in a wire loop is
Proportional to the time Rate of Change of the Magnetic Flux Through the Loop.
Magnetic Flux is defined similarly to electric
flux: If B is constant over a surface area A,
then the magnetic flux passing through A is
ΦB  BA = BA cosθ
(The scalar or dot product of vectors B & A)
The SI Unit of Magnetic flux = Weber (Wb):
1 Wb = 1 T·m2.

33. This figure shows the variables in the flux equation:
ΦB = BA = BA cosθ
Figure Determining the flux through a flat loop of wire. This loop is square, of side l and area A = l2.

34. Magnetic Flux is analogous to electric flux: It
is proportional to the total number of
magnetic field lines passing through the loop.
Figure Magnetic flux ΦB is proportional to the number of lines of B that pass through the loop.

35. Conceptual Example: Determining Flux
A square loop of wire encloses area A1. A uniform
magnetic field B perpendicular to the loop extends
over the area A2.
What is the magnetic flux through the loop A1?
Solution: Assuming the field is zero outside A2, the flux is BA2.

36. Faraday’s Law of Induction:
“The emf induced in a circuit is equal to the time rate of change of magnetic flux through the circuit.”
For a coil of N turns: N

37. Faraday’s Law, Summary Only changes in the magnetic flux matter
Rapid changes in the flux produce larger values of emf than do slow changes
This dependence on frequency means the induced emf plays an important role in AC circuits
The magnitude of the emf is proportional to the rate of change of the flux

38. Faraday’s Law, Summary
The magnitude of the emf is proportional to the rate of change of the flux
If the rate is constant, then the emf is constant
In most cases, this isn’t possible and AC currents result
The induced emf is present even if there is no current in the path enclosing an area of changing magnetic flux

39. Flux Though a Changing Area
A magnetic field is constant and in a direction perpendicular to the plane of the rails and the bar.
Assume the bar moves at a constant speed.
The magnitude of the induced emf is ε = B L v
The current leads to power dissipation in the circuit

40. Conservation of Energy
The mechanical power put into the bar by the external agent is equal to the electrical power delivered to the resistor
Energy is converted from mechanical to electrical, but the total energy remains the same
Conservation of energy is obeyed by electromagnetic phenomena

41. tends to restore the changed field.
The minus sign gives the direction of
the induced emf.
 Lenz’s Law:
A current produced by an induced emf moves in a direction so that the magnetic field
it produces
tends to restore the changed field.

42.  Lenz’s Law: An induced emf is always in a direction that OPPOSES
The minus sign gives the direction of
the induced emf.
 Lenz’s Law:
Alternative Statement:
An induced emf is always
in a direction that OPPOSES
the original change in flux
that caused it.

43. Lenz’s Law
Lenz’s Law gives a way to determine the sign of the induced emf
Lenz’s Law states that the magnetic field produced by an induced current always opposes any changes in the magnetic flux

44. Lenz’s Law, Example 1
Assume a metal loop in which the magnetic field passes upward through it
Assume the magnetic flux increases with time
The magnetic field produced by the induced emf must oppose the change in flux
Therefore, the induced magnetic field must be downward and the induced current will be clockwise

45. Lenz’s Law, Example 2
Assume a metal loop in which the magnetic field passes upward through it
Assume the magnetic flux decreases with time
The magnetic field produced by the induced emf must oppose the change in flux
Therefore, the induced magnetic field must be upward and the induced current will be counterclockwise

46.  = - ([BAcos(θ)]/t) Example The induced emf is ΦB = BA = BAcos(θ)
Assume a loop enclosing
an area A that lies in a
uniform magnetic field.
The magnetic flux
through the loop is
ΦB = BA = BAcos(θ)
The induced emf is
 = - ([BAcos(θ)]/t)

47. Methods of Inducing an EMF Using Faraday’s Law
The magnitude of the magnetic field can
change with time.
The area enclosed by the loop can
The angle between the magnetic field
& the normal to the loop can change
with time.
Any combination of the above can occur.

48. Changing a Magnetic Flux, Summary

49. A Loop of Wire in a Magnetic Field
Example
A Loop of Wire in a Magnetic Field
A square loop of wire of side l = 5.0 cm is in a
uniform magnetic field B = 0.16 T.
Calculate
(a) The magnetic flux in the loop when B is
perpendicular to the face of the loop.
(b) The magnetic flux in the loop when B is at an
angle of 30° to the area A of the loop,
(c) The magnitude of the average current in the loop
if it has a resistance of R = Ω and it is
rotated from position (b) to position (a) in 0.14 s.
Solution: a. The flux is BA = 4.0 x 10-4 Wb.
b. The flux is BA cos θ = 3.5 x 10-4 Wb.
c. The emf is ΔΦB/Δt = 3.6 x 10-4 V; then I = emf/R = 30 mA.

50. So, I = (/R) = (3.6  10-4 V)/(0.14) = 2.6  10-3 A
Example A Loop of Wire in a Magnetic Field
Square loop l = 5.0 cm, uniform magnetic field B = 0.16 T.
A = l2 = 2.5  10-3 m2 In general, ΦB = BA = BAcos(θ)
(a) Magnetic flux when B is perpendicular to face of loop.
 = 90°, cos (90°) = 1
ΦB = (0.16)(2.5  10-3) = 4.0  10-4 Wb
(b) The magnetic flux in the loop when B is at an angle of
30° to the area A of the loop cos(30°) = 0.866
ΦB = (0.16)(2.5  10-3)(0.866) = 3.5 10-4 Wb
(c) The magnitude of the average current in the loop of
resistance R = Ω & it is rotated from
position (b) to position (a) in t = 0.14 s.
B = ( )  10-4 Wb = -5  10-5 Wb
 = - (B/t) = (5  10-5)/(0.14) = 3.6  10-4 V
So, I = (/R) = (3.6  10-4 V)/(0.14) = 2.6  10-3 A
Solution: a. The flux is BA = 4.0 x 10-4 Wb.
b. The flux is BA cos θ = 3.5 x 10-4 Wb.
c. The emf is ΔΦB/Δt = 3.6 x 10-4 V; then I = emf/R = 30 mA.

51. if the area of the loop changes.
The Magnetic Flux will change
if the area of the loop changes.
Figure A current can be induced by changing the area of the coil, even though B doesn’t change. Here the area is reduced by pulling on its sides: the flux through the coil is reduced as we go from (a) to (b). Here the brief induced current acts in the direction shown so as to try to maintain the original flux (Φ = BA) by producing its own magnetic field into the page. That is, as the area A decreases, the current acts to increase B in the original (inward) direction.

52. Magnetic Flux will change if the angle between the loop & the field changes.
Figure A current can be induced by rotating a coil in a magnetic field. The flux through the coil changes from (a) to (b) because θ (in Eq. 29–1a, Φ = BA cos θ) went from 0° (cos θ = 1) to 90° (cos θ = 0).

53. Application: Induction stove.
In an induction stove, an ac current exists in a coil that is the “burner” (a burner that never gets hot). Why will it heat a metal pan but not a glass container?
Solution: The magnetic field created by the current induces a current in the metal pan, which heats due to resistance. Very little current is induced in a glass pan (or in your hand, which is why it does not feel hot).

54. Problem Solving: Lenz’s Law
Determine whether the magnetic flux is increasing, decreasing, or unchanged.
The magnetic field due to the induced current points in the opposite direction to the original field if the flux is increasing; in the same direction if it is decreasing; and is zero if the flux is not changing.
Use the right-hand rule to determine the direction of the current.
Remember that the external field and the field due to the induced current are different.

55. Practice with Lenz’s Law
Conceptual Example:
Practice with Lenz’s Law
In which direction is the current induced in the circular loop for each situation?
Solution: a. Pulling the loop to the right out of a magnetic field which points out of the page. The flux through the loop is outward and decreasing; the induced current will be counterclockwise.
b. Shrinking a loop in a magnetic field pointing into the page. The flux through the loop is inward and decreasing; the induced current will be clockwise.
c. N magnetic pole moving toward the loop into the page. The flux through the loop is inward and increasing; the induced current will be counterclockwise.
d. N magnetic pole moving toward loop in the plane of the page. There is no flux through the loop, and no induced current.
e. Rotating the loop by pulling the left side toward us and pushing the right side in; the magnetic field points from right to left. The flux through the loop is to the left and increasing; the induced current will be counterclockwise.

56. Practice with Lenz’s Law
Conceptual Example:
Practice with Lenz’s Law
In which direction is the current induced in the circular loop for each situation?
Solution: a. Pulling the loop to the right out of a magnetic field which points out of the page. The flux through the loop is outward and decreasing; the induced current will be counterclockwise.
b. Shrinking a loop in a magnetic field pointing into the page. The flux through the loop is inward and decreasing; the induced current will be clockwise.
c. N magnetic pole moving toward the loop into the page. The flux through the loop is inward and increasing; the induced current will be counterclockwise.
d. N magnetic pole moving toward loop in the plane of the page. There is no flux through the loop, and no induced current.
e. Rotating the loop by pulling the left side toward us and pushing the right side in; the magnetic field points from right to left. The flux through the loop is to the left and increasing; the induced current will be counterclockwise.

57. Pulling a coil from a magnetic field.
Example (Giancoli, p 589)
Pulling a coil from a magnetic field.
A 100-loop square coil of wire, with
side l = 5.00 cm & total resistance
100 Ω, is positioned perpendicular to
a uniform T magnetic field. It
is quickly pulled from the field at
constant speed (moving perpendicular to
B) to a region where B drops to zero.
At t = 0, the right edge of the coil is at the edge of the field. It
takes s for the whole coil to reach the field-free region.
Find: (a) the rate of change in flux through the coil, and
(b) the emf and current induced.
(c) the energy dissipated in the coil.
(d) the average force required (Fext).
Solution: a. The flux goes from BA to zero in s, so  ΦB/ t = BA/t = x 10-2 Wb/s.
b. The emf is –N  ΦB/ dt = 1.50 V. The current is emf/R = 15.0 mA.
c. E = Pt = I2Rt = 2.25 x 10-3 J.
d. F = W/d = N.