Magnetic Fields

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 Earths magnetic field
• If a bar magnet is suspended so that it can move
  freely, it will rotate
• The magnetic north pole points toward the Earths
  north geographic pole
• This means the Earths north geographic pole is a
  magnetic south pole
• Similarly, the Earths 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)

10
Magnetic Fields

• Conventional laboratory magnets 2.5 T
• Superconducting magnets 30 T
• Earths 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
• Fmax 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 xs 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
Force on a Charged Particle in a Magnetic Field

• 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 particles velocity is not perpendicular
  to the field, the path followed by the particle
  is a spiral (helix)

16
Particle in a Nonuniform Magnetic Field

• The motion is complex

17
Charged Particles Moving in Electric and Magnetic
Fields

• 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

18
Chapter 26Problem 23

• Microwaves in a microwave oven are produced by
  electrons circling in a magnetic field at a
  frequency of 2.4 GHz. (a) Whats 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, whats 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
Magnetic Force on a Current Carrying Wire

• The direction of the force is given by right hand
  rule 1, placing your fingers in the direction of
  I instead of v

21
Magnetic Force on a Current CarryingWire of an
Arbitrary Shape

• For a small segment of the wire, the force
  exerted on this segment is
• The total force is

22
Chapter 26Problem 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.
  Whats 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

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
  r2, 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 sinq,
  where q 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)
• µo 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, ?1 p/2 and
• ?2 p/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
Magnetic Field of a Current Loop

• The field contribution from a current element I
  dl I dx
• For large distances (x gtgt a), this reduces to

34
Chapter 26Problem 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
Torque on a Current Loop

• Applies to any shape loop
• Torque has a maximum value when q 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
• t BIAN sin? µB sin?
• 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
• Umin µB and occurs when the dipole moment is
  in the same direction as the field
• Umax µB and occurs when the dipole moment is
  in the direction opposite the field

39
Chapter 26Problem 35

• A single-turn square wire loop 5.0 cm on a side
  carries a 450-mA current. (a) Whats the loops
  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, whats 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 ? until the torque
  becomes 0 at ? 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
Magnetic Force Between Two Parallel Conductors

• 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 26Problem 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
Ampères Law

• Ampères 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 ?l
  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
Ampères Law for a Long Straight Wire

• Use a closed circular path
• The circumference of the circle is 2? r

48
Ampères Law for a Long Straight Wire
49
Magnetic Field of a Solenoid
50
Magnetic Field of a Solenoid

• 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 µo n I
• n N / l the number of turns per unit length
• This result can be obtained by applying Ampères
  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 Ampères 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
Earths Magnetic Field

• The Earths geographic north (south) pole
  corresponds to a magnetic south (north) pole a
  north (south) pole should be a north- (south-)
  seeking pole

• The Earths magnetic field resembles that
  achieved by burying a huge bar magnet deep in the
  Earths interior
• The most likely source of the Earths magnetic
  field electric currents in the liquid part of
  the core

61
Earths 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 earths surface
• The direction of the Earths magnetic field
  reverses every few million years (the origin of
  these reversals is not understood)

62
Earths Magnetic Field

• If a compass is free to rotate vertically as well
  as horizontally, it points to the earths 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

64
Magnetic Flux

• For a flat surface with an area A in a uniform
  magnetic field, the flux is (? is the angle
  between B and the normal to the plane)
• FB B?A B A cos ?
• When the field is perpendicular to the plane, ?
  0 and FB FB, max BA
• When the field is parallel to the plane, ? 90
  and FB 0
• The flux can be negative, for example if ? 180

65
Magnetic Flux

• 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

66
Gauss Law in Magnetism

• 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

67

• Answers to Even Numbered Problems
• Chapter 26
• Problem 16
• 3.4 105 m/s
• 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
• (-1.1iˆ 1.5 ˆj 1.7kˆ) 10-3 N
• 0

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