# Copyright © 2009 Pearson Education, Inc. Chapter 28 Sources of Magnetic Field.

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Copyright © 2009 Pearson Education, Inc. Chapter 28 Sources of Magnetic Field

Copyright © 2009 Pearson Education, Inc. 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.

Copyright © 2009 Pearson Education, Inc. 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.

Copyright © 2009 Pearson Education, Inc. 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.

Copyright © 2009 Pearson Education, Inc. 28-5 Magnetic Field of a Solenoid and a Toroid 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)

Copyright © 2009 Pearson Education, Inc. 28-5 Magnetic Field of a Solenoid and a Toroid 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.

Copyright © 2009 Pearson Education, Inc. 28-5 Magnetic Field of a Solenoid and a Toroid 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.

Copyright © 2009 Pearson Education, Inc. 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:

Copyright © 2009 Pearson Education, Inc. 28-6 Biot-Savart Law 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 = μ 0 I/2πR.

Copyright © 2009 Pearson Education, Inc. 28-6 Biot-Savart Law 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.

Copyright © 2009 Pearson Education, Inc. 28-6 Biot-Savart Law Example 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.

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. 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. 28-7 Magnetic Materials – Ferromagnetism

Copyright © 2009 Pearson Education, Inc. 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. 28-7 Magnetic Materials – Ferromagnetism

ConcepTest 28.3 Current Loop P I What is the direction of the magnetic field at the center (point P) of the square loop of current? 1) left 2) right 3) zero 4) into the page 5) out of the page

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 page at point P. ConcepTest 28.3 Current Loop P I What is the direction of the magnetic field at the center (point P) of the square loop of current? 1) left 2) right 3) zero 4) into the page 5) out of the page

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. 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 >> μ 0, while all other materials have μ ≈ μ 0.

Copyright © 2009 Pearson Education, Inc. 28-9 Magnetic Fields in Magnetic Materials; Hysteresis Not only is the permeability very large for ferromagnets, its value depends on the external field.

Copyright © 2009 Pearson Education, Inc. 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. 28-9 Magnetic Fields in Magnetic Materials; Hysteresis

Copyright © 2009 Pearson Education, Inc. 28-10 Paramagnetism and Diamagnetism All materials exhibit some level of magnetic behavior; most are either paramagnetic ( μ slightly greater than μ 0 ) or diamagnetic ( μ slightly less than μ 0 ). The following is a table of magnetic susceptibility χ m, where χ m = μ/μ 0 – 1.

Copyright © 2009 Pearson Education, Inc. 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.

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. Magnetic field inside a solenoid: Biot-Savart law: Summary of Chapter 28 Ferromagnetic materials can be made into strong permanent magnets.

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus: 29-1 Induced EMF

Copyright © 2009 Pearson Education, Inc. 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 Magnetic Flux I In order to change the magnetic flux through the loop, what would you have to do? 1) drop the magnet 2) move the magnet upward 3) move the magnet sideways 4) only (1) and (2) 5) all of the above

any direction Moving the magnet in any direction would change the magnetic field through the loop and thus the magnetic flux. ConcepTest 29.1 Magnetic Flux I In order to change the magnetic flux through the loop, what would you have to do? 1) drop the magnet 2) move the magnet upward 3) move the magnet sideways 4) only (1) and (2) 5) all of the above

Copyright © 2009 Pearson Education, Inc. 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

Copyright © 2009 Pearson Education, Inc. 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·m 2. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. This drawing shows the variables in the flux equation: 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. The magnetic flux is analogous to the electric flux – it is proportional to the total number of magnetic field lines passing through the loop. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. 29-2 Faraday’s Law of Induction; Lenz’s Law Conceptual Example 29-1: Determining flux. A square loop of wire encloses area A 1. A uniform magnetic field B perpendicular to the loop extends over the area A 2. What is the magnetic flux through the loop A 1 ?

Copyright © 2009 Pearson Education, Inc. Faraday’s law of induction: the emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit: 29-2 Faraday’s Law of Induction; Lenz’s Law or

Copyright © 2009 Pearson Education, Inc. 29-2 Faraday’s Law of Induction; Lenz’s Law 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?

Copyright © 2009 Pearson Education, Inc. 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. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. Magnetic flux will change if the area of the loop changes. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. Magnetic flux will change if the angle between the loop and the field changes. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. 29-2 Faraday’s Law of Induction; Lenz’s Law 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?

Copyright © 2009 Pearson Education, Inc. 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. 29-2 Faraday’s Law of Induction; Lenz’s Law

Copyright © 2009 Pearson Education, Inc. 29-2 Faraday’s Law of Induction; Lenz’s Law Conceptual Example 29-4: Practice with Lenz’s law. In which direction is the current induced in the circular loop for each situation?

Copyright © 2009 Pearson Education, Inc. 29-2 Faraday’s Law of Induction; Lenz’s Law 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 ( F ext )?

x x x x x x A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1) clockwise 2) counterclockwise 3) no induced current ConcepTest 29.3 Moving Wire Loop I

magnetic flux through the loop is not changingno current is induced Since the magnetic field is uniform, the magnetic flux through the loop is not changing. Thus no current is induced. x x x x x x A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1) clockwise 2) counterclockwise 3) no induced current ConcepTest 29.3 Moving Wire Loop I Follow-up: What happens if the loop moves out of the page?

1) clockwise 2) counterclockwise 3) no induced current A wire loop is being pulled through a uniform magnetic field that suddenly ends. What is the direction of the induced current? x x x x x ConcepTest 29.3 Moving Wire Loop II

1) clockwise 2) counterclockwise 3) no induced current A wire loop is being pulled through a uniform magnetic field that suddenly ends. What is the direction of the induced current? B field into the page induced flux also into the page induced current in the clockwisedirection The B field into the page is disappearing in the loop, so it must be compensated by an induced flux also into the page. This can be accomplished by an induced current in the clockwise direction in the wire loop. x x x x x ConcepTest 29.3 Moving Wire Loop II Follow-up: What happens when the loop is completely out of the field?

Copyright © 2009 Pearson Education, Inc. This image shows another way the magnetic flux can change: 29-3 EMF Induced in a Moving Conductor

Copyright © 2009 Pearson Education, Inc. The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving. 29-3 EMF Induced in a Moving Conductor

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 1) clockwise 2) counterclockwise 3) no induced current ConcepTest 29.9 Motional 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? x x x x x x x x x x x v into the page increasing out of the page counterclockwise, The B field points into the page. The flux is increasing since the area is increasing. The induced B field opposes this change and therefore points out of the page. Thus, the induced current runs counterclockwise, according to the right-hand rule. 1) clockwise 2) counterclockwise 3) no induced current ConcepTest 29.9 Motional EMF Follow-up: What direction is the magnetic force on the rod as it moves?

Copyright © 2009 Pearson Education, Inc. The induced emf has magnitude 29-3 EMF Induced in a Moving Conductor This equation is valid as long as B, l, and v are mutually perpendicular (if not, it is true for their perpendicular components).

Copyright © 2009 Pearson Education, Inc. 29-3 EMF Induced in a Moving Conductor 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?

Copyright © 2009 Pearson Education, Inc. 29-3 EMF Induced in a Moving Conductor 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?

Copyright © 2009 Pearson Education, Inc. 29-3 EMF Induced in a Moving Conductor 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?

Copyright © 2009 Pearson Education, Inc. Homework # 9 Chapter 28 – 28, 31, 37 Chapter 29 – 6, 18, 30