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Chapter 28 Sources of Magnetic Field

Chapter 28 Sources 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.

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Chapter 28 Sources of Magnetic Field

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  1. Chapter 28Sources of Magnetic Field

  2. 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.

  3. 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.

  4. 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.

  5. 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)

  6. 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.

  7. 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.

  8. 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:

  9. 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 = μ0I/2πR.

  10. 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.

  11. 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.

  12. 28-7 Magnetic Materials – Ferromagnetism 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.

  13. 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.

  14. 28-7 Magnetic Materials – Ferromagnetism 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.

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

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

  17. 28-8 Electromagnets and Solenoids – Applications Remember that a solenoid is a long coil of wire. If it is tightly wrapped, the magnetic field in its interior is almost uniform.

  18. 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.

  19. 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.

  20. 28-9 Magnetic Fields in Magnetic Materials; Hysteresis Not only is the permeability very large for ferromagnets, its value depends on the external field.

  21. 28-9 Magnetic Fields in Magnetic Materials; Hysteresis 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.

  22. 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.

  23. 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.

  24. Summary of Chapter 28 • 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:

  25. Summary of Chapter 28 • Magnetic field inside a solenoid: • Biot-Savart law: • Ferromagnetic materials can be made into strong permanent magnets.

  26. Chapter 29Electromagnetic Induction and Faraday’s Law

  27. Units of Chapter 29 • 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

  28. 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

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

  30. 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.

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

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

  33. 29-1 Induced EMF 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.

  34. 29-2 Faraday’s Law of Induction; Lenz’s Law 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.

  35. 29-2 Faraday’s Law of Induction; Lenz’s Law This drawing shows the variables in the flux equation:

  36. 29-2 Faraday’s Law of Induction; Lenz’s Law The magnetic flux is analogous to the electric flux – it is proportional to the total number of magnetic field lines passing through the loop.

  37. 29-2 Faraday’s Law of Induction; Lenz’s Law 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?

  38. 29-2 Faraday’s Law of Induction; Lenz’s Law 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

  39. 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?

  40. 29-2 Faraday’s Law of Induction; Lenz’s Law 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.

  41. 29-2 Faraday’s Law of Induction; Lenz’s Law Magnetic flux will change if the area of the loop changes.

  42. 29-2 Faraday’s Law of Induction; Lenz’s Law Magnetic flux will change if the angle between the loop and the field changes.

  43. 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?

  44. 29-2 Faraday’s Law of Induction; Lenz’s Law • Problem Solving: Lenz’s Law • Determine whether the magnetic flux is increasing, decreasing, or unchanged. • 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. • Use the right-hand rule to determine the direction of the current. • Remember that the external field and the field due to the induced current are different.

  45. 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?

  46. 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 (Fext)?

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

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

  49. 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 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?

  50. 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 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? 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 clockwisedirection in the wire loop. Follow-up: What happens when the loop is completely out of the field?

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