1. Induced Voltages and Inductance
Chapter 20
Induced Voltages and Inductance
Conceptual questions: 1,2,4,6,12,13
Quick Quizzes: 1,3,5
Problems: 26, 28, 34, 39,56

2. Induced emf
A current can be produced by a changing magnetic flux

3. Magnetic Flux A loop of wire is in a uniform magnetic field B
The loop has an area A
The flux is defined as
ΦB = BA = B A cos θ
θ is the angle between B and the normal to the plane
SI units of flux are
T m² = Wb (Weber)

4. Magnetic Flux
The value of the magnetic flux is proportional to the total number of lines passing through the loop
When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum
When the area is parallel to the lines, no lines pass through the area and the flux is 0

5. Faraday’s Law
For a single loop
For N tightly wound up loops
Since ΦB = B A cos θ, the change in the flux, ΔΦ, can be produced by a change in B, A or θ.
Thus, the induced electromotive force can be produced by changing B, A or θ, or their combinations.

6. emf induced by changing magnetic field

7. QUICK QUIZ 20.1
The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

8. Motional emf, changing A
A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field
The electrons in the conductor experience a magnetic force
F = q v B
The electrons tend to move to the lower end of the conductor

9. Motional emf, cont
The potential difference between the ends of the conductor can be found by
ΔV = B ℓ v
A potential difference is maintained across the conductor as long as there is motion through the field
If the motion is reversed, the polarity of the potential difference is also reversed

10. Motional emf in a Circuit
The induced, motional emf, acts like a battery in the circuit

11. QUICK QUIZ 20.3
You wish to move a rectangular loop of wire into a region of uniform magnetic field at a given speed so as to induce an emf in the loop. The plane of the loop must remain perpendicular to the magnetic field lines. In which orientation should you hold the loop while you move it into the region of magnetic field in order to generate the largest emf? (a) With the long dimension of the loop parallel to the velocity vector; (b) With the short dimension of the loop parallel to the velocity vector. (c) Either way—the emf is the same regardless of orientation.

12. Faraday’s Law and Lenz’ Law
The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law
The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop
The induced current tends to maintain the original flux through the circuit

13. Lenz’ Law Revisited – Moving Bar Example
As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases
The induced current must in a direction such that it opposes the change in the external magnetic flux

14. Lenz’ Law, Bar Example The bar is moving toward the left
The magnetic flux through the loop is decreasing with time
The induced current must be clockwise to to produce its own flux into the page

15. Lenz’ Law, Moving Magnet Example
A bar magnet is moved to the right toward a stationary loop of wire (a)
As the magnet moves, the magnetic flux increases with time
The induced current produces a flux to the left, so the current is in the direction shown (b)

16. Lenz’ Law, Final Note
When applying Lenz’ Law, there are two magnetic fields to consider
The external changing magnetic field that induces the current in the loop
The magnetic field produced by the current in the loop

17. Conceptual questions
A circular loop is located in a uniform and constant magnetic field. Describe how an emf can be induced in the loop.
Does dropping a magnet down a copper tube produce a current in the tube?
12. A bar magnet is dropped toward a conducting ring lying on a floor. As the magnet falls toward the ring, does it move as a freely falling body?
4. A loop of wire is placed in a uniform magnetic field. For what orientation of the loop is the magnetic flux a maximum? For what orientation is it zero?
6. A bar moves perpendicularly to the
magnetic field. Is an external force
required to keep it moving with
a constant velocity?

18. What is the direction of the current induced in the resistor at the instant the switch is closed?
Problem I B I B
Induced
current
Induced B

19. 20-28. Find the direction of the current in R the instant the switch is closed.

20. Applications of Faraday’s Law – Electric Guitar
A vibrating string induces an emf in a coil
A permanent magnet inside the coil magnetizes a portion of the string nearest the coil
As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil
The changing flux produces an induced emf that is fed to an amplifier

21. Applications of Faraday’s Law – Ground Fault Interrupters

22. Tape Recorder A magnetic tape moves past a recording and playback head
The tape is a plastic ribbon coated with iron oxide or chromium oxide
To record, the sound is converted to an electrical signal which passes to an electromagnet that magnetizes the tape in a particular pattern
To playback, the magnetized pattern is converted back into an induced current driving a speaker

23. Recording

24. Tape playing

25. AC Generators
As the loop rotates, the magnetic flux through it changes with time
This induces an emf and a current in the external circuit
The ends of the loop are connected to slip rings that rotate with the loop
Connections to the external circuit are made by stationary brushed in contact with the slip rings

26. AC Generators
If the loop rotates with a constant angular speed, ω, and N turns
ε = N B A ω sin ω t
ε = εmax = NBAω when loop is parallel to the field
ε = 0 when when the loop is perpendicular to the field

27. Problem A coil of area 0.10 m2 is rotating at 60 rev/s with its axis of rotation perpendicular to a 0.20-T magnetic field. (a) If there are turns on the loop, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the loop with respect to the magnetic field?
Using, emax = NBA w
=1000 (0.2T) (0.10m2) (120 p rad/s) = V
Plane of the loop is parallel to the magnetic field

28. DC Generators
Components are essentially the same as that of an ac generator
The major difference is the contacts to the rotating loop are made by a split ring, or commutator

29. DC Generators The output voltage always has the same polarity
The current is a pulsing current

30. Self-inductance
Self-inductance occurs when the changing flux through a circuit arises from the circuit itself
As the current increases, the magnetic flux through a loop due to this current also increases
The increasing flux induces an emf that opposes the current
As the magnitude of the current increases, the rate of increase lessens and the induced emf decreases
This opposing emf results in a gradual increase of the current

31. Self-inductance cont
The self-induced emf must be proportional to the time rate of change of the current
L is inductance of the device, unit Henry
1 H = 1 (V · s) / A

32. Self inductance of a solenoid
= BA cos q, when q =90o
F = BA
[A is the cross-sectional area of the solenoid]
For a solenoid B = mo n I = mo (N/l)I,
F = mo A NI/l
L = N F /I = mo A N2 /l
L depends only on geometric factors A and l, number of turns squared, and on mo

33. Problem 20.39
A solenoid of radius 2.5 cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75 mV.

34. QUICK QUIZ 20.5
The switch in the circuit shown in the figure below is closed and the lightbulb glows steadily. The inductor is a simple air-core solenoid. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As the rod is inserted into the solenoid, the brightness of the lightbulb (a) increases, (b) decreases, or (c) remains the same.

35. Energy Stored in a Magnetic Field
PEL = ½ L I2

36. Conceptual question
13. If the current in an inductor is doubled, by what factor does the stored energy change?

37. Problem 20-56
A novel method of storing electrical energy has been proposed. A huge underground superconducting coil, 1.00 km in diameter, would be fabricated. It would carry a maximum current of 50.0 kA through each winding of a 150-turn Nb3Sn solenoid.
If the inductance of this huge coil is 50.0 H, what is the total energy stored?
(b) What is the compressive force per meter length acting between two adjacent windings m apart? (Hint: Because the radius of the coil is so large, the magnetic field created by one winding and acting on an adjacent turn can be considered to be that of a long, straight wire.)

38. Review questions
1. A heavy permanent magnet is moving toward a current carrying circular loop of wire. Which is correct?
The coil will push or pull the magnet just as hard as the magnet pulls or pushes the coil.
The magnet pushes harder on the coil than the coil pushes on the magnet because the magnet is more massive than the coil.
The magnet will push or pull on the coil, but the coil will not push or pull on the magnet at all because the coil is not a magnet.

39. A conducting bar is sliding at a constant velocity along two conducting horizontal rods. The rods are separated by a distance l and connected across by a resistor R. The entire apparatus is placed inside a magnetic field B directed into the page.
How will the current in the apparatus be generated?
a. sinusoidally b. clockwise
c. counterclockwise d. not enough information v

40. 3. A conducting coil is rotated at a constant speed in an external magnetic field. Which of the following most likely represents the current generated within the coil as a function of time? t i t i c d a t i t i b