1. Chapter 26 Magnetic Fields

2. Magnets In each magnet there are two poles present (the ends where objects are most strongly attracted): north and south Like (unlike) poles repel (attract) each other (similar to electric charges), and the force between two poles varies as the inverse square of the distance between them Magnetic poles cannot be isolated – if a permanent magnetic is cut in half, you will still have a north and a south pole (unlike electric charges) There is some theoretical basis for monopoles, but none have been detected

3. Magnets The poles received their names due to the way a magnet behaves in the Earth’s magnetic field If a bar magnet is suspended so that it can move freely, it will rotate The magnetic north pole points toward the Earth’s north geographic pole This means the Earth’s north geographic pole is a magnetic south pole Similarly, the Earth’s south geographic pole is a magnetic north pole

4. Magnets An unmagnetized piece of iron can be magnetized by stroking it with a magnet (like stroking an object to charge an object) Magnetism can be induced – if a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized Soft magnetic materials (such as iron) are easily magnetized and also tend to lose their magnetism easily Hard magnetic materials (such as cobalt and nickel) are difficult to magnetize and they tend to retain their magnetism

5. Magnetic Fields The region of space surrounding a moving charge includes a magnetic field (the charge will also be surrounded by an electric field) A magnetic field surrounds a properly magnetized magnetic material A magnetic field is a vector quantity symbolized by B Its direction is given by the direction a north pole of a compass needle pointing in that location Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look

6. Magnetic Field Lines A compass can be used to show the direction of the magnetic field lines

7. Magnetic Field Lines Iron filings can also be used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point Unlike poles (compare to the electric field produced by an electric dipole)

8. Magnetic Field Lines Iron filings can also be used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point Unlike poles (compare to the electric field produced by an electric dipole) Like poles (compare to the electric field produced by like charges)

9. Magnetic Fields When moving through a magnetic field, a charged particle experiences a magnetic force This force has a maximum (zero) value when the charge moves perpendicularly to (along) the magnetic field lines Magnetic field is defined in terms of the magnetic force exerted on a test charge moving in the field with velocity v The SI unit: Tesla (T) Nikola Tesla 1856 – 1943

10. Magnetic Fields Conventional laboratory magnets: ~ 2.5 T Superconducting magnets ~ 30 T Earth’s magnetic field ~ 5 x 10 -5 T

11. Direction of Magnetic Force Experiments show that the direction of the magnetic force is always perpendicular to both v and B F max occurs when v is perpendicular to B and F = 0 when v is parallel to B Right Hand Rule #1 (for a + charge): Place your fingers in the direction of v and curl the fingers in the direction of B – your thumb points in the direction of F If the charge is negative, the force points in the opposite direction

12. Direction of Magnetic Force The x’s indicate the magnetic field when it is directed into the page (the x represents the tail of the arrow) The dots would be used to represent the field directed out of the page (the represents the head of the arrow)

13. Differences Between Electric and Magnetic Fields The electric force acts along the direction of the electric field, whereas the magnetic force acts perpendicular to the magnetic field The electric force acts on a charged particle regardless of whether the particle is moving, while the magnetic force acts on a charged particle only when the particle is in motion The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced (because the force is perpendicular to the displacement)

14. Force on a Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field so that its velocity is perpendicular to the field The force is always directed toward the center of the circular path The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

15. This expression is known as the cyclotron equation r is proportional to the momentum of the particle and inversely proportional to the magnetic field If the particle’s velocity is not perpendicular to the field, the path followed by the particle is a spiral (helix) Force on a Charged Particle in a Magnetic Field

16. The motion is complex Particle in a Nonuniform Magnetic Field

17. In many applications, charged particles move in the presence of both magnetic and electric fields In that case, the total force is the sum of the forces due to the individual fields: Charged Particles Moving in Electric and Magnetic Fields

18. Chapter 26 Problem 23 Microwaves in a microwave oven are produced by electrons circling in a magnetic field at a frequency of 2.4 GHz. (a) What’s the magnetic field strength? (b) The electrons’ motion takes place inside a special tube called a magnetron. If the magnetron can accommodate electron orbits with maximum diameter 2.5 mm, what’s the maximum electron energy?

19. Magnetic Force on a Current Carrying Wire The current is a collection of many charged particles in motion The magnetic force is exerted on each moving charge in the wire The total force is the sum of all the magnetic forces on all the individual charges producing the current Therefore a force is exerted on a current- carrying wire placed in a magnetic field:

20. The direction of the force is given by right hand rule #1, placing your fingers in the direction of I instead of v Magnetic Force on a Current Carrying Wire

21. For a small segment of the wire, the force exerted on this segment is The total force is Magnetic Force on a Current Carrying Wire of an Arbitrary Shape

22. Chapter 26 Problem 28 A wire with mass per unit length 75 g/m runs horizontally at right angles to a horizontal magnetic field. A 6.2-A current in the wire results in its being suspended against gravity. What’s the magnetic field strength?

23. Biot-Savart Law Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space due to a current The magnetic field is dB at some point P; the length element is ds; the wire is carrying a steady current of I Jean-Baptiste Biot 1774 – 1862 Félix Savart 1791 – 1841

24. Biot-Savart Law Vector dB is perpendicular to both ds and to the unit vector directed from ds toward P The magnitude of dB is inversely proportional to r 2, where r is the distance from ds to P The magnitude of dB is proportional to the current and to the magnitude ds of the length element

25. Biot-Savart Law The magnitude of dB is proportional to sin  where  is the angle between the vectors ds and The observations are summarized in the mathematical equation called the Biot- Savart law (magnetic field due to the current-carrying conductor): µ o = 4  x 10 -7 T. m / A: permeability of free space

26. Biot-Savart Law To find the total field, sum up the contributions from all the current elements

27. Biot-Savart Law The magnitude of the magnetic field varies as the inverse square of the distance from the ds element The electric field due to a point charge also varies as the inverse square of the distance from the charge The electric field created by a point charge is radial in direction The magnetic field created by a current element is perpendicular to both the length element and the unit vector The current element producing a magnetic field is part of an extended current distribution

28. A Long, Straight Conductor The thin, straight wire is carrying a constant current

29. A Long, Straight Conductor The thin, straight wire is carrying a constant current If the conductor is an infinitely long, straight wire, θ 1 = π/2 and θ 2 = – π/2, and the field becomes

30. A Long, Straight Conductor The magnetic field lines are circles concentric with the wire The field lines lie in planes perpendicular to the wire The magnitude of the field is constant on any circle of radius a Right Hand Rule #2: Grasp the wire in your right hand and point your thumb in the direction of the current and your fingers will curl in the direction of the field

31. A Curved Wire Segment Find the field at point O due to the wire segment (I, a are constants) The field at the center of the full circle loop

32. Magnetic Field of a Current Loop

33. The field contribution from a current element I dl = I dx For large distances (x >> a), this reduces to

34. Chapter 26 Problem 30 A single-turn wire loop is 2.0 cm in diameter and carries a 650-mA current. Find the magnetic field strength (a) at the loop center and (b) on the loop axis, 20 cm from the center.

35. Torque on a Current Loop

36. Applies to any shape loop Torque has a maximum value when  = 90° Torque is zero when the field is perpendicular to the plane of the loop

37. Magnetic Moment The vector is called the magnetic dipole moment of the coil Its magnitude is given by μ = IAN The vector always points perpendicular to the plane of the loop(s) The equation for the magnetic torque can be written as τ = BIAN sinθ = μB sinθ The angle is between the moment and the field

38. Potential Energy The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field U min = – μB and occurs when the dipole moment is in the same direction as the field U max = + μB and occurs when the dipole moment is in the direction opposite the field

39. Chapter 26 Problem 35 A single-turn square wire loop 5.0 cm on a side carries a 450-mA current. (a) What’s the loop’s magnetic dipole moment? (b) If the loop is in a uniform 1.4-T magnetic field with its dipole moment vector at 40° to the field, what’s the magnitude of the torque it experiences?

40. Electric Motor An electric motor converts electrical energy to mechanical energy (rotational kinetic energy) An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field The torque acting on the loop will tend to rotate the loop to smaller values of θ until the torque becomes 0 at θ = 0°

41. Electric Motor If the loop turns past this point and the current remains in the same direction, the torque reverses and turns the loop in the opposite direction To provide continuous rotation in one direction, the current in the loop must periodically reverse In ac motors, this reversal naturally occurs In dc motors, a split-ring commutator and brushes are used

42. Electric Motor Just as the loop becomes perpendicular to the magnetic field and the torque becomes 0, inertia carries the loop forward and the brushes cross the gaps in the ring, causing the current loop to reverse its direction This provides more torque to continue the rotation The process repeats itself Actual motors would contain many current loops and commutators

43. Magnetic Force Between Two Parallel Conductors

44. The force (per unit length) on wire 1 due to the current in wire 1 and the magnetic field produced by wire 2: Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in the opposite directions repel each other

45. Chapter 26 Problem 63 A long, straight wire carries 20 A. A 5.0-cm by 10-cm rectangular wire loop carrying 500 mA is 2.0 cm from the wire, as shown in the figure. Find the net magnetic force on the loop.

46. Ampère’s Law Ampère’s Circuital Law: a procedure for deriving the relationship between the current in an arbitrarily shaped wire and the magnetic field produced by the wire Choose an arbitrary closed path around the current and sum all the products of B || Δℓ around the closed path (put the thumb of your right hand in the direction of the current through the loop and your fingers curl in the direction you should integrate around the loop)

47. Ampère’s Law for a Long Straight Wire Use a closed circular path The circumference of the circle is 2  r

48. Ampère’s Law for a Long Straight Wire

49. Magnetic Field of a Solenoid

50. If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid It is also known as an electromagnet since it acts like a magnet only when it carries a current The field inside the solenoid is nearly uniform and strong – the field lines are nearly parallel, uniformly spaced, and close together The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

51. Magnetic Field of a Solenoid The field lines of the solenoid resemble those of a bar magnet The magnitude of the field inside a solenoid is approximately constant at all points far from its ends B = µ o n I n = N / ℓ : the number of turns per unit length This result can be obtained by applying Ampère’s Law to the solenoid

52. Magnetic Field of a Solenoid A cross-sectional view of a tightly wound solenoid If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero Apply Ampère’s Law to the blue dashed rectangle

53. Magnetic Effects of Electrons – Orbits An individual atom should act like a magnet because of the motion of the electrons about the nucleus Each electron circles the atom once in about every 10 -16 seconds; this would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

54. Magnetic Effects of Electrons – Spins Electrons also have spin (it is a quantum effect) The classical model is to consider the electrons to spin like tops The field due to the spinning is generally stronger than the field due to the orbital motion Electrons usually pair up with their spins opposite each other, so their fields cancel each other, hence most materials are not naturally magnetic

55. Magnetic Effects of Electrons – Domains In some materials – ferromagnetic – the spins do not naturally cancel Large groups of atoms in which the spins are aligned are called domains When an external field is applied, it causes the material to become magnetized: the domains that are aligned with the field tend to grow at the expense of the others

56. Domains and Permanent Magnets In hard magnetic materials, the domains remain aligned after the external field is removed The result is a permanent magnet In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

57. Ferromagnetism Some substances exhibit strong magnetic effects called ferromagnetism (e.g., iron, cobalt, nickel, gadolinium, dysprosium) They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

58. Paramagnetism Paramagnetic substances have small but positive magnetism, which results from the presence of atoms that have permanent magnetic moments These moments interact weakly with each other When placed in an external magnetic field, atomic moments tend to line up with the field and the alignment process competes with thermal motion which randomizes the moment orientations

59. Diamagnetism When an external magnetic field is applied to a diamagnetic substance, a weak magnetic moment is induced in the direction opposite the applied field Diamagnetic substances are weakly repelled by a magnet

60. Earth’s Magnetic Field The Earth’s geographic north (south) pole corresponds to a magnetic south (north) pole – a north (south) pole should be a “north- (south-) seeking” pole The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior The most likely source of the Earth’s magnetic field – electric currents in the liquid part of the core

61. Earth’s Magnetic Field The magnetic and geographic poles are not in the same exact location – magnetic declination is the difference between true north (geographic north pole) and magnetic north pole The amount of declination varies by location on the earth’s surface The direction of the Earth’s magnetic field reverses every few million years (the origin of these reversals is not understood)

62. Earth’s Magnetic Field If a compass is free to rotate vertically as well as horizontally, it points to the earth’s surface The angle between the horizontal and the direction of the magnetic field is called the dip angle The farther north the device is moved, the farther from horizontal the compass needle would be The compass needle would be horizontal at the equator and the dip angle would be 0° The compass needle would point straight down at the south magnetic pole and the dip angle would be 90°

63. Magnetic Flux Magnetic flux associated with a magnetic field is defined in a way similar to electric flux SI unit of flux: Weber Wb = T. m² Wilhelm Eduard Weber 1804 – 1891

64. For a flat surface with an area A in a uniform magnetic field, the flux is (θ is the angle between B and the normal to the plane): Φ B = B  A = B A cos θ When the field is perpendicular to the plane, θ = 0 and Φ B = Φ B, max = BA When the field is parallel to the plane, θ = 90° and Φ B = 0 The flux can be negative, for example if θ = 180° Magnetic Flux

65. The value of the magnetic flux is proportional to the total number of magnetic field lines passing through area When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is parallel to the lines, no lines pass through the area and the flux is 0 Magnetic Flux

66. Magnetic fields do not begin or end at any point The number of lines entering a surface equals the number of lines leaving the surface Gauss’ law in magnetism says the magnetic flux through any closed surface is always zero: Gauss’ Law in Magnetism

67. Answers to Even Numbered Problems Chapter 26: Problem 16 (a) 3.4 × 10 5 m/s (b) does not change

68. Answers to Even Numbered Problems Chapter 26: Problem 20 3.9 mm

69. Answers to Even Numbered Problems Chapter 26: Problem 32 4.0 A

70. Answers to Even Numbered Problems Chapter 26: Problem 36 480 mT

71. Answers to Even Numbered Problems Chapter 26: Problem 38 24 A

72. Answers to Even Numbered Problems Chapter 26: Problem 42 (a) (−1.1iˆ + 1.5 ˆj + 1.7kˆ) × 10 −3 N (b) 0

73. Answers to Even Numbered Problems Chapter 26: Problem 58 10 m