copyright © 2009 pearson education, inc. chapter 28 sources of magnetic field

Copyright © 2009 Pearson Education, Inc.

Chapter 28Sources of Magnetic Field


28-4 Ampère’s Law

Example 28-8: A nice use for Ampère’s law.

Use Ampère’s law to show that in any region of space where there are no currents the magnetic field cannot be both unidirectional and nonuniform as shown in the figure.


28-4 Ampère’s Law

Solving problems using Ampère’s law:

• Ampère’s law is only useful for solving problems when there is a great deal of symmetry. Identify the symmetry.

• Choose an integration path that reflects the symmetry (typically, the path is along lines where the field is constant and perpendicular to the field where it is changing).

• Use the symmetry to determine the direction of the field.

• Determine the enclosed current.


28-5 Magnetic Field of a Solenoid and a Toroid

A solenoid is a coil of wire containing many loops. To find the field inside, we use Ampère’s law along the path indicated in the figure.



The field is zero outside the solenoid, and the path integral is zero along the vertical lines, so the field is (n is the number of loops per unit length)



Example 28-9: Field inside a solenoid.

A thin 10-cm-long solenoid used for fast electromechanical switching has a total of 400 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center.



Example 28-10: Toroid.

Use Ampère’s law to determine the magnetic field (a) inside and (b) outside a toroid, which is like a solenoid bent into the shape of a circle as shown.


28-6 Biot-Savart Law

The Biot-Savart law gives the magnetic field due to an infinitesimal length of current; the total field can then be found by integrating over the total length of all currents:



Example 28-11: B due to current I in straight wire.

For the field near a long straight wire carrying a current I, show that the Biot-Savart law gives B = μ0I/2πR.

B��



Example 28-12: Current loop.

Determine B for points on the axis of a circular loop of wire of radius R carrying a current I.

B��


28-6 Biot-Savart LawExample 28-13: B due to a wire segment.

One quarter of a circular loop of wire carries a current I. The current I enters and leaves on straight segments of wire, as shown; the straight wires are along the radial direction from the center C of the circular portion. Find the magnetic field at point C.

B��


Ferromagnetic materials are those that can become strongly magnetized, such as iron and nickel.

These materials are made up of tiny regions called domains; the magnetic field in each domain is in a single direction.

28-7 Magnetic Materials – Ferromagnetism


When the material is unmagnetized, the domains are randomly oriented. They can be partially or fully aligned by placing the material in an external magnetic field.



A magnet, if undisturbed, will tend to retain its magnetism. It can be demagnetized by shock or heat.

The relationship between the external magnetic field and the internal field in a ferromagnet is not simple, as the magnetization can vary.


ConcepTest 28.3 ConcepTest 28.3 Current LoopCurrent Loop

P

I

What is the direction of the What is the direction of the

magnetic field at the center magnetic field at the center

(point P) of the square loop (point P) of the square loop

of current?of current?

1) left1) left

2) right2) right

3) zero3) zero

4) into the page4) into the page

5) out of the page5) out of the page

Use the right-hand rule for each

wire segment to find that each

segment has its B field pointing

out of the pageout of the page at point P.

ConcepTest 28.3 ConcepTest 28.3 Current LoopCurrent Loop

P

I


magnetic field at the center magnetic field at the center

(point P) of the square loop (point P) of the square loop

of current?of current?

1) left1) left

2) right2) right

3) zero3) zero

4) into the page4) into the page

5) out of the page5) out of the page


Remember that a solenoid is a long coil of wire. If it is tightly wrapped, the magnetic field in its interior is almost uniform.

28-8 Electromagnets and Solenoids – Applications


If a piece of iron is inserted in the solenoid, the magnetic field greatly increases. Such electromagnets have many practical applications.

28-8 Electromagnets and Solenoids – Applications


28-9 Magnetic Fields in Magnetic Materials; Hysteresis

If a ferromagnetic material is placed in the core of a solenoid or toroid, the magnetic field is enhanced by the field created by the ferromagnet itself. This is usually much greater than the field created by the current alone.

If we write

B = μI

where μ is the magnetic permeability, ferromagnets have μ >> μ0, while all other materials have μ ≈ μ0.



Not only is the permeability very large for ferromagnets, its value depends on the external field.


Furthermore, the induced field depends on the history of the material. Starting with unmagnetized material and no magnetic field, the magnetic field can be increased, decreased, reversed, and the cycle repeated. The resulting plot of the total magnetic field within the ferromagnet is called a hysteresis loop.



28-10 Paramagnetism and Diamagnetism

All materials exhibit some level of magnetic behavior; most are either paramagnetic (μ slightly greater than μ0) or diamagnetic (μ slightly less than μ0). The following is a table of magnetic susceptibility χm, where

χm = μ/μ0 – 1.


28-10 Paramagnetism and Diamagnetism

Molecules of paramagnetic materials have a small intrinsic magnetic dipole moment, and they tend to align somewhat with an external magnetic field, increasing it slightly.

Molecules of diamagnetic materials have no intrinsic magnetic dipole moment; an external field induces a small dipole moment, but in such a way that the total field is slightly decreased.


• Magnitude of the field of a long, straight current-carrying wire:

• The force of one current-carrying wire on another defines the ampere.

• Ampère’s law:

Summary of Chapter 28


• Magnetic field inside a solenoid:

• Biot-Savart law:

Summary of Chapter 28

• Ferromagnetic materials can be made into strong permanent magnets.


Chapter 29Electromagnetic Induction

and Faraday’s Law


• Induced EMF

• Faraday’s Law of Induction; Lenz’s Law

• EMF Induced in a Moving Conductor

• Electric Generators

• Back EMF and Counter Torque; Eddy Currents

Units of Chapter 29


• Transformers and Transmission of Power

• A Changing Magnetic Flux Produces an Electric Field

• Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI

Units of Chapter 29


Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

29-1 Induced EMF


He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.

29-1 Induced EMF

ConcepTest 29.1 ConcepTest 29.1 Magnetic Flux IMagnetic Flux I

In order to change the In order to change the

magnetic flux through magnetic flux through

the loop, what would the loop, what would

you have to do?you have to do?

1) drop the magnet1) drop the magnet

2) move the magnet upward2) move the magnet upward

3) move the magnet sideways3) move the magnet sideways

4) only (1) and (2)4) only (1) and (2)

5) all of the above5) all of the above

Moving the magnet in any directionany direction would

change the magnetic field through the

loop and thus the magnetic flux.

ConcepTest 29.1 ConcepTest 29.1 Magnetic Flux IMagnetic Flux I

In order to change the In order to change the

magnetic flux through magnetic flux through

the loop, what would the loop, what would

you have to do?you have to do?

1) drop the magnet1) drop the magnet

2) move the magnet upward2) move the magnet upward

3) move the magnet sideways3) move the magnet sideways

4) only (1) and (2)4) only (1) and (2)

5) all of the above5) all of the above


Therefore, a changing magnetic field induces an emf.

Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.

29-1 Induced EMF


The induced emf in a wire loop is proportional to the rate of change of magnetic flux through the loop.

Magnetic flux:

Unit of magnetic flux: weber, Wb:

1 Wb = 1 T·m2.

29-2 Faraday’s Law of Induction; Lenz’s Law


This drawing shows the variables in the flux equation:



The magnetic flux is analogous to the electric flux – it is proportional to the total number of magnetic field lines passing through the loop.




Conceptual Example 29-1: 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?

B��


Faraday’s law of induction: the emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit:


or



Example 29-2: 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. What is the magnetic flux in the loop (a) when B is perpendicular to the face of the loop and (b) when B is at an angle of 30° to the area A of the loop? (c) What is the magnitude of the average current in the loop if it has a resistance of 0.012 Ω and it is rotated from position (b) to position (a) in 0.14 s?

B��

B��

A��


The minus sign gives the direction of the induced emf:

A current produced by an induced emf moves in a direction so that the magnetic field it produces tends to

restore the changed field.

or:

An induced emf is always in a direction that opposes the original change in flux that caused it.



Magnetic flux will change if the area of the loop changes.



Magnetic flux will change if the angle between the loop and the field changes.




Conceptual Example 29-3: 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?


Problem Solving: Lenz’s Law

1. Determine whether the magnetic flux is increasing, decreasing, or unchanged.

2. 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.

3. Use the right-hand rule to determine the direction of the current.

4. Remember that the external field and the field due to the induced current are different.




Conceptual Example 29-4: Practice with Lenz’s law.

In which direction is the current induced in the circular loop for each situation?



Example 29-5: Pulling a coil from a magnetic field.

A 100-loop square coil of wire, with side l = 5.00 cm and total resistance 100 Ω, is positioned perpendicular to a uniform 0.600-T magnetic field. It is quickly pulled from the field at constant speed (moving perpendicular to B) to a region where B drops abruptly to zero. At t = 0, the right edge of the coil is at the edge of the field. It takes 0.100 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) How much energy is dissipated in the coil? (d) What was the average force required (Fext)?

B��

x x x x x x x x x x x x







A wire loop is being pulled A wire loop is being pulled

through a uniform magnetic through a uniform magnetic

field. What is the direction field. What is the direction

of the induced current? of the induced current?

1) clockwise1) clockwise

2) counterclockwise2) counterclockwise

3) no induced current3) no induced current

ConcepTest 29.3 ConcepTest 29.3 Moving Wire Loop IMoving Wire Loop I

Since the magnetic field is uniform, the

magnetic flux through the loop is not magnetic flux through the loop is not

changingchanging. Thus no current is inducedno current is induced.









through a uniform magnetic through a uniform magnetic

field. What is the direction field. What is the direction

of the induced current? of the induced current?




ConcepTest 29.3 ConcepTest 29.3 Moving Wire Loop IMoving Wire Loop I

Follow-up:Follow-up: What happens if the loop moves out of the page? What happens if the loop moves out of the page?





through a through a uniform magnetic uniform magnetic

field that suddenly endsfield that suddenly ends. .


induced current? induced current?

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

ConcepTest 29.3 ConcepTest 29.3 Moving Wire Loop IIMoving Wire Loop II





through a through a uniform magnetic uniform magnetic

field that suddenly endsfield that suddenly ends. .


induced current? induced current?

The BB field into the page field into the page is disappearing in

the loop, so it must be compensated by an

induced flux also into the pageinduced flux also into the page. This can

be accomplished by an induced current in induced current in

the clockwisethe clockwise directiondirection in the wire loop.

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

x x x x x

ConcepTest 29.3 ConcepTest 29.3 Moving Wire Loop IIMoving Wire Loop II

Follow-up:Follow-up: What happens when the loop is completely out of the field? What happens when the loop is completely out of the field?


This image shows another way the magnetic flux can change:

29-3 EMF Induced in a Moving Conductor


The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving.


A conducting rod slides on a

conducting track in a constant

B field directed into the page.

What is the direction of the

induced current?

x x x x x x x x x x x




v




ConcepTest 29.9 ConcepTest 29.9 Motional EMFMotional EMF

A conducting rod slides on a

conducting track in a constant

B field directed into the page.

What is the direction of the

induced current?





v

The B field points into the pageinto the page. The flux is increasingincreasing since the area is increasing. The induced B field opposes this change and therefore points out of the pageout of the page. Thus, the induced current runs counterclockwise,counterclockwise, according to the right-hand rule.




ConcepTest 29.9 ConcepTest 29.9 Motional EMFMotional EMF

Follow-up:Follow-up: What direction is the magnetic force on the rod as it moves? What direction is the magnetic force on the rod as it moves?


The induced emf has magnitude


This equation is valid as long as B, l, and v are mutually perpendicular (if not, it is true for their perpendicular components).



Example 29-6: Does a moving airplane develop a large emf?

An airplane travels 1000 km/h in a region where the Earth’s magnetic field is about 5 x 10-5 T and is nearly vertical. What is the potential difference induced between the wing tips that are 70 m apart?



Example 29-7: Electromagnetic blood-flow measurement.

The rate of blood flow in our body’s vessels can be measured using the apparatus shown, since blood contains charged ions. Suppose that the blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the measured emf is 0.10 mV. What is the flow velocity of the blood?



Example 29-8: Force on the rod.

To make the rod (having resistance R) move to the right at speed v, you need to apply an external force on the rod to the right. (a) Explain and determine the magnitude of the required force. (b) What external power is needed to move the rod?


Homework # 9

Chapter 28 – 28, 31, 37Chapter 29 – 6, 18, 30

copyright © 2009 pearson education, inc. chapter 28 sources of magnetic field

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