Chapter 28 Sources of Magnetic Field

1. Chapter 28Sources of Magnetic Field

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

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

10. 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.

11. 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.

12. 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.

13. Chapter 29Electromagnetic Induction and Faraday’s Law

14. 29-1 Induced EMF Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

15. 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.

16. 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.

17. 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.

18. 29-2 Faraday’s Law of Induction; Lenz’s Law This drawing shows the variables in the flux equation:

19. 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.

20. 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?

21. 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

22. 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?

23. 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?

24. 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.

25. ConcepTest 29.1 Magnetic Flux II 1) tilt the loop 2) change the loop area 3) use thicker wires 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?

26. ConcepTest 29.1 Magnetic Flux II 1) tilt the loop 2) change the loop area 3) use thicker wires 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? Since F = BA cos q , changing the area or tilting the loop (which varies the projected area) would change the magnetic flux through the loop.

27. 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. Nature hates change!

28. 29-2 Faraday’s Law of Induction; Lenz’s Law How can the flux change? Change any one and the flux changes.

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

30. 29-2 Faraday’s Law of Induction; Lenz’s Law Magnetic flux will change if the angle between the loop and the field changes.

31. 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?

32. 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.

33. 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?

34. ConcepTest 29.2 Moving Bar Magnet 1) clockwise 2) counterclockwise 3) no induced current If a north pole moves toward the loop from above the page, in what direction is the induced current?

35. ConcepTest 29.2 Moving Bar Magnet 1) clockwise 2) counterclockwise 3) no induced current If a north pole moves toward the loop from above the page, in what direction is the induced current? The magnetic field of the moving bar magnet is pointing into the page and getting larger as the magnet moves closer to the loop. Thus the induced magnetic field has to point out of the page. A counterclockwise induced current will give just such an induced magnetic field. Follow-up: What happens if the magnet is stationary but the loop moves?

36. 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)?

37. 29-3 EMF Induced in a Moving Conductor This image shows another way the magnetic flux can change:

38. 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.

39. 29-3 EMF Induced in a Moving Conductor 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).

40. 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 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?

41. 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 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?

42. 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? Remind you of the Hall effect?

43. 29-3 EMF Induced in a Moving Conductor Example 29-8: Force on the rod. To make the rod 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?

44. 29-4 Electric Generators A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator: The axle is rotated by an external force such as falling water or steam. The brushes are in constant electrical contact with the slip rings.

45. ConcepTest 29.5 Rotating Wire Loop 1) clockwise 2) counterclockwise 3) no induced current If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current?

46. ConcepTest 29.5 Rotating Wire Loop 1) clockwise 2) counterclockwise 3) no induced current If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current? As the coil is rotated into the B field, the magnetic flux through it increases. According to Lenz’s law, the induced B field has to oppose this increase, thus the new B field points to the right. An induced counterclockwise current produces just such a B field.

47. 29-4 Electric Generators If the loop is rotating with constant angular velocity ω, the induced emf is sinusoidal: For a coil of N loops,

48. 29-4 Electric Generators Example 29-9: An ac generator. The armature of a 60-Hz ac generator rotates in a 0.15-T magnetic field. If the area of the coil is 2.0 x 10-2 m2, how many loops must the coil contain if the peak output is to be V0 = 170 V?

49. 29-5 Back EMF and Counter Torque; Eddy Currents

50. 29-5 Back EMF and Counter Torque; Eddy Currents Induced currents can flow in bulk material as well as through wires. These are called eddy currents, and can dramatically slow a conductor moving into or out of a magnetic field.