1. Lect. 15: Faraday’s Law and Induction
Chapter 30
HW 5 (problems): 28.5, 28.10, 28.15, 28.27, 28.36, 28.45, 28.50, 28.63, 29.15, 29.36, 29.48, 29.54, 30.14, 30.34, 30.42, 30.48
Due Thursday, July 17.

2. 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

3. 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 direction was arbitrarily chosen to be negative

4. 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 in the loop

5. EMF Produced by a Changing Magnetic Field, 3
The magnet is moved away from the loop
The ammeter deflects in the opposite direction
Use the active figure to move the wires and observe the deflection on the meter

6. Faraday’s Experiment – Conclusions
An electric current can be induced in a loop by a changing magnetic field
This would be 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
This is generally expressed as: an induced emf is produced in the loop by the changing magnetic field
The actual existence of the magnetic flux is not sufficient to produce the induced emf, the flux must be changing

7. Faraday’s Law – Statements
Faraday’s law of induction states that “the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit”
Mathematically,

8. Faraday’s Law – Statements, cont
Remember FB is the magnetic flux through the circuit and is found by
If the circuit consists of N loops, all of the same area, and if FB is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

9. Faraday’s Law – Example
Assume a loop enclosing an area A lies in a uniform magnetic field
The magnetic flux through the loop is FB = BA cos q
The induced emf is
e = - d/dt (BA cos q)

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

11. 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 some frequency, the magnetized segment produces a changing flux through the coil
The induced emf is fed to an amplifier

12. Motional emf
A motional emf is the emf induced in a conductor moving through a constant magnetic field
The electrons in the conductor experience a force, that is directed along ℓ

13. Motional emf, final For equilibrium, qE = qvB or E = vB
The electric field is related to the potential difference across the ends of the conductor: DV = E ℓ =B ℓ v
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

14. Sliding Conducting Bar
A bar moving through a uniform field and the equivalent circuit diagram
Assume the bar has zero resistance
The stationary part of the circuit has a resistance R

15. Sliding Conducting Bar, cont.
The induced emf is
Since the resistance in the circuit is R, the current is

16. Sliding Conducting Bar, Energy Considerations
The applied force does work on the conducting bar
This moves the charges through a magnetic field and establishes a current
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

17. Lenz’s Law
Faraday’s law indicates that 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’s law
Developed by German physicist Heinrich Lenz

18. Lenz’s Law, cont.
Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop
The induced current tends to keep the original magnetic flux through the circuit from changing

19. Lenz’ Law, Example
The conducting bar slides on the two fixed conducting rails
The magnetic flux due to the external magnetic field through the enclosed area increases with time
The induced current must produce a magnetic field out of the page
The induced current must be counterclockwise
If the bar moves in the opposite direction, the direction of the induced current will also be reversed

20. 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 is nonconservative
Unlike the electric field produced by stationary charges

21. 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:

22. 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 of would be zero and it isn’t

23. 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
Use the active figure to adjust the speed of rotation and observe the effect on the emf generated

24. Rotating Loop
Assume a loop with N turns, all of the same area rotating in a magnetic field
The flux through the loop at any time t is FB = BA cos q =
BA cos wt

25. Induced emf in a Rotating Loop
The induced emf in the loop is
This is sinusoidal, with emax = NABw

26. Induced emf in a Rotating Loop, cont.
emax occurs when wt = 90o or 270o
This occurs when the magnetic field is in the plane of the coil and the time rate of change of flux is a maximum
e = 0 when wt = 0o or 180o
This occurs when the magnetic field is perpendicular to the plane of the coil and the time rate of change of flux is zero

27. Motors
Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work
A motor is a generator operating in reverse
A current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotate

28. Motors, final
The current in the rotating coil is limited by the back emf
The term back emf is commonly used to indicate an emf that tends to reduce the supplied current
The induced emf explains why the power requirements for starting a motor and for running it are greater for heavy loads than for light ones

29. Some Terminology
Use emf and current when they are caused by batteries or other sources
Use induced emf and induced current when they are caused by changing magnetic fields
When dealing with problems in electromagnetism, it is important to distinguish between the two situations

30. Self-Inductance
When the switch is closed, the current does not immediately reach its maximum value
Faraday’s law can be used to describe the effect

31. Self-Inductance, 2
As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time
This increasing flux creates an induced emf in the circuit
The direction of the induced emf is opposite the direction of the emf of the battery

32. Self-Inductance, 3 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 εL is called a self-induced emf

33. Self-Inductance, Equations
An induced emf is always proportional to the time rate of change of the current
The emf is proportional to the flux, which is proportional to the field and the field is proportional to the current
L is a constant of proportionality called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics

34. 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; The SI unit of inductance is the henry (H)

35. 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 flux through each turn of area A is

36. 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

37. Effect of an Inductor in a Circuit
The inductance results in a back emf
Therefore, the inductor in a circuit opposes changes in current in that circuit
The inductor attempts to keep the current the same way it was before the change occurred
The inductor can cause the circuit to be “sluggish” as it reacts to changes in the voltage

38. RL Circuit, Analysis An RL circuit contains an inductor and a resistor
Assume S2 is connected to a
When switch S1 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

39. RL Circuit, Analysis, cont.
Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives
Looking at the current, we find

40. 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

41. 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

42. RL Circuit Without A Battery
Now set S2 to position b
The circuit now contains just the right hand loop
The battery has been eliminated
The expression for the current becomes

43. 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

44. Energy in a Magnetic Field, cont.
Looking at this energy (in terms of rate)
Ie 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 stored in the magnetic field

45. 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

46. Energy Density of a Magnetic Field
Given U = ½ L I2 and assume (for simplicity) a solenoid with L = mo n2 V
Since V 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)

47. Energy Storage Summary
A resistor, inductor and capacitor all store energy through different mechanisms
Charged capacitor
Stores energy as electric potential energy
Inductor
When it carries a current, stores energy as magnetic potential energy
Resistor
Energy delivered is transformed into internal energy

48. Mutual Inductance
The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits
This process is known as mutual induction because it depends on the interaction of two circuits

49. Mutual Inductance, 2 The current in coil 1 sets up a magnetic field
Some of the magnetic field lines pass through coil 2
Coil 1 has a current I1 and N1 turns
Coil 2 has N2 turns

50. Mutual Inductance, 3
The mutual inductance M12 of coil 2 with respect to coil 1 is
Mutual inductance depends on the geometry of both circuits and on their orientation with respect to each other

51. Induced emf in Mutual Inductance
If current I1 varies with time, the emf induced by coil 1 in coil 2 is
If the current is in coil 2, there is a mutual inductance M21
If current 2 varies with time, the emf induced by coil 2 in coil 1 is

52. Mutual Inductance, Final
In mutual induction, the emf induced in one coil is always proportional to the rate at which the current in the other coil is changing
The mutual inductance in one coil is equal to the mutual inductance in the other coil
M12 = M21 = M
The induced emf’s can be expressed as