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Chapter 10 Review

Key Terms

back emf
emf generated by a running motor, because it consists of a coil turning in a magnetic field; it opposes the voltage powering the motor



eddy current
current loop in a conductor caused by motional emf



electric generator
device for converting mechanical work into electric energy; it induces an emf by rotating a coil in a magnetic field



Faraday’s law
induced emf is created in a closed loop due to a change in magnetic flux through the loop



induced electric field
created based on the changing magnetic flux with time



induced emf
short-lived voltage generated by a conductor or coil moving in a magnetic field



Lenz’s law
direction of an induced emf opposes the change in magnetic flux that produced it; this is the negative sign in Far



day’s law
magnetic damping
drag produced by eddy currents



magnetic flux
measurement of the amount of magnetic field lines through a given area



motionally induced emf
voltage produced by the movement of a conducting wire in a magnetic field



peak emf
maximum emf produced by a generator



Key Equations


Magnetic flux
Faraday’s law
Motionally induced emf
Motional emf around a circuit
Emf produced by an electric generator


Summary

10.1 Faraday’s Law

  • The magnetic flux through an enclosed area is defined as the amount of field lines cutting through a surface area defined by the unit area vector.
  • The units for magnetic flux are webers, where
  • The induced emf in a closed loop due to a change in magnetic flux through the loop is known as Faraday’s law. If there is no change in magnetic flux, no induced emf is created.

10.2 Lenz’s Law

  • We can use Lenz’s law to determine the directions of induced magnetic fields, currents, and emfs.
  • The direction of an induced emf always opposes the change in magnetic flux that causes the emf, a result known as Lenz’s law.

10.3 Motional Emf

  • The relationship between an induced emf in a wire moving at a constant speed through a magnetic field is given by
  • An induced emf from Faraday’s law is created from a motional emf that opposes the change in flux.

10.4 Induced Electric Fields

  • A changing magnetic flux induces an electric field.
  • Both the changing magnetic flux and the induced electric field are related to the induced emf from Faraday’s law.

10.5 Eddy Currents

  • Current loops induced in moving conductors are called eddy currents. They can create significant drag, called magnetic damping.
  • Manipulation of eddy currents has resulted in applications such as metal detectors, braking in trains or roller coasters, and induction cooktops.

10.6 Electric Generators and Back Emf

  • An electric generator rotates a coil in a magnetic field, inducing an emf given as a function of time by where is the area of an -turn coil rotated at a constant angular velocity in a uniform magnetic field
  • The peak emf of a generator is
  • Any rotating coil produces an induced emf. In motors, this is called back emf because it opposes the emf input to the motor.

10.7 Applications of Electromagnetic Induction

  • Hard drives utilize magnetic induction to read/write information.
  • Other applications of magnetic induction can be found in graphics tablets, electric and hybrid vehicles, and in transcranial magnetic stimulation.


Answers to Check Your Understanding

10.1


10.2 To the observer shown, the current flows clockwise as the magnet approaches, decreases to zero when the magnet is centered in the plane of the coil, and then flows counterclockwise as the magnet leaves the coil.



10.4

, with

at a higher potential than


10.5


10.6 a. yes; b. Yes; however there is a lack of symmetry between the electric field and coil, making

a more complicated relationship that can’t be simplified as shown in the example.


10.7


10.8


10.9 a.



Conceptual Questions

10.1 Faraday’s Law

1. A stationary coil is in a magnetic field that is changing with time. Does the emf induced in the coil depend on the actual values of the magnetic field?


2. In Faraday’s experiments, what would be the advantage of using coils with many turns?


3. A copper ring and a wooden ring of the same dimensions are placed in magnetic fields so that there is the same change in magnetic flux through them. Compare the induced electric fields and currents in the rings.


4. Discuss the factors determining the induced emf in a closed loop of wire.


5. (a) Does the induced emf in a circuit depend on the resistance of the circuit? (b) Does the induced current depend on the resistance of the circuit?


6. How would changing the radius of loop D shown below affect its emf, assuming C and D are much closer together compared to their radii?



7. Can there be an induced emf in a circuit at an instant when the magnetic flux through the circuit is zero?


8. Does the induced emf always act to decrease the magnetic flux through a circuit?


9. How would you position a flat loop of wire in a changing magnetic field so that there is no induced emf in the loop?


10. The normal to the plane of a single-turn conducting loop is directed at an angle

to a spatially uniform magnetic field

It has a fixed area and orientation relative to the magnetic field. Show that the emf induced in the loop is given by

where

is the area of the loop.


10.2 Lenz’s Law

11. The circular conducting loops shown in the accompanying figure are parallel, perpendicular to the plane of the page, and coaxial. (a) When the switch S is closed, what is the direction of the current induced in D? (b) When the switch is opened, what is the direction of the current induced in loop D?



12. The north pole of a magnet is moved toward a copper loop, as shown below. If you are looking at the loop from above the magnet, will you say the induced current is circulating clockwise or counterclockwise?



13. The accompanying figure shows a conducting ring at various positions as it moves through a magnetic field. What is the sense of the induced emf for each of those positions?



14. Show that

and

have the same units.


15. State the direction of the induced current for each case shown below, observing from the side of the magnet.



10.3 Motional Emf

16. A bar magnet falls under the influence of gravity along the axis of a long copper tube. If air resistance is negligible, will there be a force to oppose the descent of the magnet? If so, will the magnet reach a terminal velocity?


17. Around the geographic North Pole (or magnetic South Pole), Earth’s magnetic field is almost vertical. If an airplane is flying northward in this region, which side of the wing is positively charged and which is negatively charged?


18. A wire loop moves translationally (no rotation) in a uniform magnetic field. Is there an emf induced in the loop?



10.4 Induced Electric Fields

19. Is the work required to accelerate a rod from rest to a speed

in a magnetic field greater than the final kinetic energy of the rod? Why?


20. The copper sheet shown below is partially in a magnetic field. When it is pulled to the right, a resisting force pulls it to the left. Explain. What happen if the sheet is pushed to the left?



10.5 Eddy Currents

21. A conducting sheet lies in a plane perpendicular to a magnetic field

that is below the sheet. If

oscillates at a high frequency and the conductor is made of a material of low resistivity, the region above the sheet is effectively shielded from

Explain why. Will the conductor shield this region from static magnetic fields?


22. Electromagnetic braking can be achieved by applying a strong magnetic field to a spinning metal disk attached to a shaft. (a) How can a magnetic field slow the spinning of a disk? (b) Would the brakes work if the disk was made of plastic instead of metal?


23. A coil is moved through a magnetic field as shown below. The field is uniform inside the rectangle and zero outside. What is the direction of the induced current and what is the direction of the magnetic force on the coil at each position shown?



Problems

10.1 Faraday’s Law

24. A

-turn coil has a diameter of

The coil is placed in a spatially uniform magnetic field of magnitude

so that the face of the coil and the magnetic field are perpendicular. Find the magnitude of the emf induced in the coil if the magnetic field is reduced to zero uniformly in (a)

(b)

and (c)


25. Repeat your calculations of the preceding problem’s time of

with the plane of the coil making an angle of (a)

(b)

and (c)

with the magnetic field.


26. A square loop whose sides are

long is made with copper wire of radius

If a magnetic field perpendicular to the loop is changing at a rate of

what is the current in the loop?


27. The magnetic field through a circular loop of radius

varies with time as shown below. The field is perpendicular to the loop. Plot the magnitude of the induced emf in the loop as a function of time.



28. The accompanying figure shows a single-turn rectangular coil that has a resistance of

The magnetic field at all points inside the coil varies according to

where

and

What is the current induced in the coil at (a)

(b)

(c)

?



29. How would the answers to the preceding problem change if the coil consisted of

closely spaced turns?


30. A long solenoid with

turns per centimetre has a cross-sectional area of

and carries a current of

A coil with five turns encircles the solenoid. When the current through the solenoid is turned off, it decreases to zero in

What is the average emf induced in the coil?


31. A rectangular wire loop with length

and width

lies in the

-plane, as shown below. Within the loop there is a time-dependent magnetic field given by

with

in tesla. Determine the emf induced in the loop as a function of time.



32. The magnetic field perpendicular to a single wire loop of diameter

decreases from

to zero. The wire is made of copper and has a diameter of

and length

How much charge moves through the wire while the field is changing?



10.2 Lenz’s Law

33. A single-turn circular loop of wire of radius

lies in a plane perpendicular to a spatially uniform magnetic field. During a

time interval, the magnitude of the field increases uniformly from

to

(a) Determine the emf induced in the loop. (b) If the magnetic field is directed out of the page, what is the direction of the current induced in the loop?


34. When a magnetic field is first turned on, the flux through a

-turn loop varies with time according to

where

is in milliwebers,

is in seconds, and the loop is in the plane of the page with the unit normal pointing outward. (a) What is the emf induced in the loop as a function of time? What is the direction of the induced current at (b)

(c)

(d)

and (e)

?


35. The magnetic flux through the loop shown in the accompanying figure varies with time according to

where

is in milliwebers. What are the direction and magnitude of the current through the

resistor at (a)

(b)

and (c)

?



36. Use Lenz’s law to determine the direction of induced current in each case.



10.3 Motional Emf

37. An automobile with a radio antenna

long travels at

in a location where the Earth’s horizontal magnetic field is

What is the maximum possible emf induced in the antenna due to this motion?


38. The rectangular loop of

turns shown below moves to the right with a constant velocity

while leaving the poles of a large electromagnet. (a) Assuming that the magnetic field is uniform between the pole faces and negligible elsewhere, determine the induced emf in the loop. (b) What is the source of work that produces this emf?



39. Suppose the magnetic field of the preceding problem oscillates with time according to

What then is the emf induced in the loop when its trailing side is a distance d from the right edge of the magnetic field region?


40. A coil of

turns encloses an area of

It is rotated in

from a position where its plane is perpendicular to Earth’s magnetic field to one where its plane is parallel to the field. If the strength of the field is

what is the average emf induced in the coil?


41. In the circuit shown in the accompanying figure, the rod slides along the conducting rails at a constant velocity

The velocity is in the same plane as the rails and directed at an angle

to them. A uniform magnetic field

is directed out of the page. What is the emf induced in the rod?



42. The rod shown in the accompanying figure is moving through a uniform magnetic field of strength

with a constant velocity of magnitude

What is the potential difference between the ends of the rod? Which end of the rod is at a higher potential?



43. A

rod moves at

in a plane perpendicular to a magnetic field of strength

The rod, velocity vector, and magnetic field vector are mutually perpendicular, as indicated in the accompanying figure. Calculate (a) the magnetic force on an electron in the rod, (b) the electric field in the rod, and (c) the potential difference between the ends of the rod. (d) What is the speed of the rod if the potential difference is

?



44. In the accompanying figure, the rails, connecting end piece, and rod all have a resistance per unit length of

The rod moves to the left at

If

everywhere in the region, what is the current in the circuit (a) when

? (b) when

? Specify also the sense of the current flow.



45. The rod shown below moves to the right on essentially zero-resistance rails at a speed of

If

everywhere in the region, what is the current through the

resistor? Does the current circulate clockwise or counterclockwise?



46. Shown below is a conducting rod that slides along metal rails. The apparatus is in a uniform magnetic field of strength

which is directly into the page. The rod is pulled to the right at a constant speed of

by a force

The only significant resistance in the circuit comes from the