Magnetic fields

1
Magnetic fields

• The symbol we use for a magnetic field is B.
• The unit is the tesla (T).
• The Earths magnetic field is about 5 x 10-5 T.
• Which pole of a magnet attracts the north pole of
  a compass? Which way does a compass point on the
  Earth?
• What kind of magnetic pole is near the Earths
  geographic north pole?
• What are some similarities between electric and
  magnetic fields? What are some differences?

2
Similarities between electric and magnetic fields

• Electric fields are produced by two kinds of
  charges, positive and negative. Magnetic fields
  are associated with two magnetic poles, north and
  south, although they are also produced by charges
  (but moving charges).
• Like poles repel unlike poles attract.
• Electric field points in the direction of the
  force experienced by a positive charge. Magnetic
  field points in the direction of the force
  experienced by a north pole.

3
Differences between electric and magnetic fields

• Positive and negative charges can exist
  separately. North and south poles always come
  together. Single magnetic poles, known as
  magnetic monopoles, have been proposed
  theoretically, but a magnetic monopole has never
  been observed.
• Electric field lines have definite starting and
  ending points. Magnetic field lines are
  continuous loops. Outside a magnet the field is
  directed from the north pole to the south pole.
  Inside a magnet the field runs from south to
  north.

4
Observing a charge in a magnetic field

• The force exerted on a charge in an electric
  field is given by
• Is there an equivalent equation for the force
  exerted on a charge in a magnetic field?
  Simulation
• Case 1 The charge is initially stationary in the
  field.
• Case 2 The velocity of the charge is parallel to
  the field.

5
Observing a charge in a magnetic field

• The force exerted on a charge in an electric
  field is given by
• Is there an equivalent equation for the force
  exerted on a charge in a magnetic field?
  Simulation
• Case 1 The charge is initially stationary in the
  field.
• The charge feels no force.
• Case 2 The velocity of the charge is parallel to
  the field.
• The charge feels no force.

6
Observing a charge in a magnetic field

• Simulation
• Case 3 Three objects, one , one -, and one
  neutral, have an initial velocity perpendicular
  to the field. The field is directed out of the
  screen.

7
Observing a charge in a magnetic field

• Simulation
• Case 3 Three objects, one , one -, and one
  neutral, have an initial velocity perpendicular
  to the field. The field is directed out of the
  screen.
• Magnetic fields exert no force on neutral
  particles.
• The force exerted on a charge is opposite to
  that exerted on a charge.
• The force on a charged particle is perpendicular
  to the velocity and the field. In this special
  case where the velocity and field are
  perpendicular to one another, we get uniform
  circular motion.

8
Observing a charge in a magnetic field

• Simulation
• Case 4 The same as case 3, except the magnetic
  field is doubled.

9
Observing a charge in a magnetic field

• Simulation
• Case 4 The same as case 3, except the magnetic
  field is doubled.
• We observe the radius of the path to be half as
  large.
• Thus, doubling the magnetic field doubles the
  force -
• the force is proportional to the magnetic field.

10
Observing a charge in a magnetic field

• Simulation
• Case 5 Three positive charges q, 2q, and 3q
  are initially moving perpendicular to the field
  with the same velocity.

11
Observing a charge in a magnetic field

• Simulation
• Case 5 Three positive charges q, 2q, and 3q
  are initially moving perpendicular to the field
  with the same velocity.
• We observe the radius of the path to vary
  inversely with the charge.
• Thus, doubling the charge doubles the force, and
  tripling the charge triples the force -
• the force is proportional to the charge.

12
Observing a charge in a magnetic field

• Simulation
• Case 6 Three identical charges, are initially
  moving perpendicular to the field with initial
  velocities of v, 2v, and 3v, respectively.

13
Observing a charge in a magnetic field

• Simulation
• Case 6 Three identical charges, are initially
  moving perpendicular to the field with initial
  velocities of v, 2v, and 3v, respectively.
• We observe the radius of the path to be
  proportional to the speed. However
• What does this tell us about how the force
  depends on speed?

14
Observing a charge in a magnetic field

• Simulation
• Case 6 Three identical charges, are initially
  moving perpendicular to the field with initial
  velocities of v, 2v, and 3v, respectively.
• We observe the radius of the path to be
  proportional to the speed. However
• What does this tell us about how the force
  depends on speed?
• The force is proportional to the speed.

15
Summarizing the observations

• There is no force applied on a stationary charge
  by a magnetic field, or on a charge moving
  parallel to the field.
• Reversing the sign of the charge reverses the
  direction of the force.
• The force is proportional to q (charge), to B
  (field), and to v (speed).

16
Summarizing the observations

• There is no force applied on a stationary charge
  by a magnetic field, or on a charge moving
  parallel to the field.
• Reversing the sign of the charge reverses the
  direction of the force.
• The force is proportional to q (charge), to B
  (field), and to v (speed).
• The magnitude of the force is F q v B sin(?),
  where ? is the angle between the velocity vector
  v and the magnetic field B.
• The direction of the force, which is
  perpendicular to both v and B, is given by the
  right-hand rule.

17
Something to keep in mind

• A force perpendicular to the velocity, such as
  the magnetic force, can not change an objects
  speed (or the kinetic energy). All it can do is
  make the object change direction.

18
The right-hand rule

• Point the fingers of your right hand in the
  direction of the velocity.
• Curl your fingers into the direction of the
  magnetic field (if v and B are perpendicular,
  pointing your palm in the direction of the field
  will orient your hand properly).
• Stick out your thumb, and your thumb points in
  the direction of the force experienced by a
  positive charge.
• If the charge is negative your right-hand lies to
  you. In that case, the force is opposite to what
  your thumb says.
• Simulation

19
The right-hand rule
20
Practice with the right-hand rule
In what direction is the force on a positive
charge with a velocity to the left in a uniform
magnetic field directed down and to the left? 1.
up 2. down 3. left 4. right 5. into the
screen 6. out of the screen 7. a combination of
two of the above 8. the force is zero 9. this
case is ambiguous - we can't say for certain
21
Practice with the right-hand rule

• v and B define a plane, and the force is
  perpendicular to that plane. The right-hand rule
  tells us the force is out of the screen. We use a
  dot symbol to represent out of the screen (or
  page), and an x symbol to represent into the
  screen.

22
Practice with the right-hand rule, II
In what direction is the force on a negative
charge, with a velocity down, in a uniform
magnetic field directed out of the screen? 1. up
2. down 3. left 4. right 5. into the screen
6. out of the screen 7. a combination of two of
the above 8. the force is zero 9. this case is
ambiguous - we can't say for certain
23
Practice with the right-hand rule, II

• Remember that with a negative charge, your right
  hand lies to you take the opposite direction.

24
Practice with the right-hand rule, III
In what direction is the force on a positive
charge that is initially stationary in a uniform
magnetic field directed into the screen? 1. up
2. down 3. left 4. right 5. into the screen
6. out of the screen 7. a combination of two of
the above 8. the force is zero 9. this case is
ambiguous - we can't say for certain
25
Practice with the right-hand rule, III

• Magnetic fields exert no force on stationary
  charges.

26
Practice with the right-hand rule, IV
In what direction is the force on a negative
charge with a velocity to the left in a uniform
electric field directed out of the screen? 1. up
2. down 3. left 4. right 5. into the screen
6. out of the screen 7. a combination of two of
the above 8. the force is zero 9. this case is
ambiguous - we can't say for certain
27
Practice with the right-hand rule, IV

• We dont need the right-hand rule for an electric
  field, we need . The force is
  opposite to the field, for a negative charge.

28
Practice with the right-hand rule, V
In what direction is the velocity of a positive
charge if it feels a force directed into the
screen from a magnetic field directed right? 1.
up 2. down 3. left 4. right 5. into the
screen 6. out of the screen 7. a combination of
two of the above 8. the force is zero 9. this
case is ambiguous - we can't say for certain
29
Practice with the right-hand rule, V

• This is ambiguous. The right-hand rule tells us
  about the component of the velocity that is
  perpendicular to the field, but it cant tell us
  anything about a component parallel to the field
  that component is unaffected by the field.

30
Charges moving perpendicular to the field

• The force exerted on a charge moving in a
  magnetic field is always perpendicular to both
  the velocity and the field.
• If v is perpendicular to B, the charge follows a
  circular path.
• The radius of the circular path is

31
Charges moving perpendicular to the field

• The radius of the circular path is
• The time for the object to go once around the
  circle (the period, T) is
• Interestingly, the time is independent of the
  speed. The faster the speed, the larger the
  radius, but the period is unchanged.

32
Circular paths

• Three charged objects with the same mass and the
  same magnitude charge have initial velocities
  directed right. Here are the trails they follow
  through a region of uniform magnetic field. Rank
  the objects based on their speeds.
• 1. 1 gt 2 gt 3
• 2. 2 gt 1 gt 3
• 3. 3 gt 2 gt 1
• 4. 3 gt 1 gt 2
• 5. None of the above

33
Circular paths

• The radius of the path is proportional to the
  speed, so the correct ranking by speed is choice
  2, 2 gt 1 gt 3.

34
Circular paths, II

• Three charged objects with the same mass and the
  same magnitude charge have initial velocities
  directed right. Rank the objects based on the
  magnitude of the force they experience as they
  travel through the magnetic field.
• 1. 1 gt 2 gt 3
• 2. 2 gt 1 gt 3
• 3. 3 gt 2 gt 1
• 4. 3 gt 1 gt 2
• 5. None of the above

35
Circular paths, II

• The force is proportional to the speed, so the
  correct ranking by force is also choice 2, 2 gt 1
  gt 3.

36
Possible paths of a charge in a magnetic field

• If the velocity of a charge is parallel to the
  magnetic field, the charge moves with constant
  velocity because there's no net force.
• If the velocity is perpendicular to the magnetic
  field, the path is circular because the force is
  always perpendicular to the velocity.
• What happens when the velocity is not one of
  these special cases, but has a component parallel
  to the field and a component perpendicular to the
  field?
• The parallel component produces straight-line
  motion. The perpendicular component produces
  circular motion. The net motion is a combination
  of these, a spiral. Simulation

37
Which way is the field?
The charge always spirals around the magnetic
field. Assuming the charge in this case is
positive, which way does the field point in the
simulation? 1. Left 2. Right
38
Spiraling charges

• Charges spiral around magnetic field lines.
• Charged particles near the Earth are trapped by
  the Earths magnetic field, spiraling around the
  Earths magnetic field down toward the Earth at
  the magnetic poles.
• The energy deposited by such particles gives rise
  to ??

39
Spiraling charges

• Charges spiral around magnetic field lines.
• Charged particles near the Earth are trapped by
  the Earths magnetic field, spiraling around the
  Earths magnetic field down toward the Earth at
  the magnetic poles.
• The energy deposited by such particles gives rise
  to the aurora borealis (northern lights) and the
  aurora australis (southern lights). The colors
  are usually dominated by emissions from oxygen
  atoms.

Photos from Wikipedia
40
A mass spectrometer

• A mass spectrometer is a device for separating
  particles based on their mass. There are
  different types we will investigate one that
  exploits electric and magnetic fields.
• Step 1 Accelerate charged particles via an
  electric field.
• Step 2 Use an electric field and a magnetic
  field to select particles of a particular
  velocity.
• Step 3 Use a magnetic field to separate
  particles based on mass.

41
Step 1 The Accelerator

• Simulation
• The simplest way to accelerate ions is to place
  them between a set of charged parallel plates.
  The ions are repelled by one plate and attracted
  to the other. If we cut a hole in the second
  plate, the ions emerge with a kinetic energy
  determined by the potential difference between
  the plates.
• K q DV

42
Step 3 The Mass Separator

• Simulation
• In the last stage, the ions enter a region of
  uniform magnetic field B/. The field is
  perpendicular to the velocity. Everything is the
  same for the ions except for mass, so the radius
  of each circular path depends only on mass.

43
Step 3 The Mass Separator

• The ions are collected after traveling through
  half-circles, with the separation s between two
  ions is equal to the difference in the diameters
  of their respective circles.

44
Magnetic field in the mass separator
In what direction is the magnetic field in the
mass separator? The paths shown are for positive
charges. 1. up 2. down 3. left 4. right 5.
into the screen 6. out of the screen
45
Step 2 The Velocity Selector

• Simulation
• To ensure that the ions arriving at step 3 have
  the same velocity, the ions pass through a
  velocity selector, a region with uniform electric
  and magnetic fields.
• The electric field comes from a set of parallel
  plates, and exerts a force of
  on the ions.
• The magnetic field is perpendicular to both the
  ion velocity and the electric field. The magnetic
  force, , exactly balances the
  electric force when
• Ions with a speed of
• pass straight through.

46
Magnetic field in the velocity selector
In what direction is the magnetic field in the
velocity selector, if the positive charges pass
through undeflected? The electric field is
directed down. 1. up 2. down 3. left 4.
right 5. into the screen 6. out of the screen
47
Magnetic field in the velocity selector

• The right-hand rule tells us that the magnetic
  field is directed into the screen.

48
Negative ions in the velocity selector
If the charges passing through the velocity
selector were negative, what (if anything) would
have to be changed for the velocity selector to
allow particles of just the right speed to pass
through undeflected? 1. reverse the direction of
the electric field 2. reverse the direction of
the magnetic field 3. reverse the direction of
one field or the other 4. reverse the directions
of both fields 5. none of the above, it would
work fine just the way it is
49
Negative ions in the velocity selector

• If the charges are negative, both the electric
  force and the magnetic force reverse direction.
  The forces still balance, so we dont have to
  change a thing.

50
Faster ions in the velocity selector
Lets go back to positive ions. If the ions are
traveling faster than the ions that pass
undeflected through the velocity selector, what
happens to them? They get deflected 1. up 2.
down 3. into the screen 4. out of the screen

51
Faster ions in the velocity selector

• For ions with a larger speed, the magnetic force
  exceeds the electric force and those ions are
  deflected up of the beam. The opposite happens
  for slower ions, so they are deflected down out
  of the beam.

52
A cyclotron

• Simulation
• A cyclotron is a particle accelerator that is so
  compact a small one can fit in your pocket. It
  consists of two D-shaped regions known as dees.
  In each dee there is a magnetic field
  perpendicular to the plane of the page. In the
  gap separating the dees, there is a uniform
  electric field pointing from one dee to the
  other. When a charge is released from rest, it is
  accelerated by the electric field and carried
  into a dee. The magnetic field in the dee causes
  the charge to follow a half-circle that carries
  it back to the gap.

53
A cyclotron

• Ernest Lawrence won the 1939 Nobel Prize in
  Physics for inventing the cyclotron. The
  cyclotron has many applications, including
  accelerating ions to high energies for medical
  treatments. A good example is the Proton Therapy
  Center at Mass General Hospital (see below).
  After leaving the cyclotron, the beam is steered
  using magnetic fields.

54
Magnetic fields in the dees

• In what direction is the magnetic
• field in each of the dees? The
• path shown is for a positive
• charge.
• 1. out of the screen in both dees
• 2. into the screen in both dees
• 3. out of the screen in the left dee into the
  screen in the right dee
• 4. into the screen in the left dee out of the
  screen in the right dee

55
Increasing the energy
You want to increase the speed of the particles
when they emerge from the cyclotron. Which is
more effective, increasing the potential
difference across the gap or increasing the
magnetic field in the dees? 1. increasing the
potential difference in the gap 2. increasing
the magnetic field in the dees 3. either one,
they're equally effective
The energy increases by ?K each time the charge
crosses the gap, and stays constant in the dees.
56
Producing a magnetic field

• Electric fields are produced by charges.
• Magnetic fields are produced by moving charges.
• In practice, we generally produce magnetic fields
  from currents.

57
The magnetic field from a long straight wire

• The long straight current-carrying wire, for
  magnetism, is analogous to the point charge for
  electric fields.
• The magnetic field a distance r
• from a wire with current I is
• , the permeability of free space, is

58
The magnetic field from a long straight wire

• Magnetic field lines from a long straight
  current-carrying wire are circular loops centered
  on the wire.
• The direction is given by another
• right-hand rule.
• Point your right thumb in the
• direction of the current
• (out of the screen in the
• diagram, and the fingers on
• your right hand, when you curl
• them, show the field direction.

59
The net magnetic field

• In which direction is the net magnetic field at
  the origin in the situation shown below? All the
  wires are the same distance from the origin.
• 1. Left
• 2. Right
• 3. Up
• 4. Down
• 5. Into the page
• 6. Out of the page
• 7. The net field is zero

60
The net magnetic field

• We add the individual fields to find the net
  field, which is directed right.

61
Four wires

• Think about the net magnetic field at the center
  of the square because of four wires, one at each
  corner, that carry currents of the same
  magnitude. Can you choose current directions for
  the four wires so that the net magnetic field at
  the center is directed toward the top right
  corner of the square?

62
How many ways?

• You can choose the direction of the currents at
  each corner. How many configurations give a net
  magnetic field at the center that is directed
  toward the top-right corner?
• 1. 1
• 2. 2
• 3. 3
• 4. 4
• 5. 0 or more than 4

63
How many ways?

• First, think about the four fields we need to add
  to get a net field toward the top right. How many
  ways can we create this set of four fields?

64
How many ways?

• How many ways can we create this set of four
  fields?
• Two. Wires 2 and 4 have to
• have the currents shown.
• Wires 1 and 3 have to
• match, so their fields cancel.
• The right-hand rule
• Point your thumb in the
• direction of the current,
• and your curled fingers
• show the direction of the field.

65
The force on a current-carrying wire

• A magnetic field exerts a force on a single
  moving charge, so it's not surprising that it
  exerts a force on a current-carrying wire, seeing
  as a current is a set of moving charges.
• Using q I t, this becomes
• But a velocity multiplied by a time is a length
  L, so this can be written
• The direction of the force is given by the
  right-hand rule, where your fingers point in the
  direction of the current. Current is defined to
  be the direction of flow of positive charges, so
  your right hand always gives the correct
  direction.

66
The right-hand rule

• A wire carries current into the page in a
  magnetic field directed down the page. In which
  direction is the force?
• 1. Left
• 2. Right
• 3. Up
• 4. Down
• 5. Into the page
• 6. Out of the page
• 7. The net force is zero

67
Three wires
Consider three wires carrying identical currents
between two points, a and b. The wires are
exposed to a uniform magnetic field. Wire 1 goes
directly from a to b. Wire 2 consists of two
straight sections, one parallel to the magnetic
field and one perpendicular to the field. Wire 3
takes a meandering path from a to b. Which wire
experiences more force? 1. Wire 1 2. Wire 2
3. Wire 3 4. equal for all three
68
Three wires

• The force is equal for all three. What matters is
  the displacement perpendicular to the field, and
  that's equal for all wires carrying equal
  currents between the same two points in a uniform
  magnetic field.

69
The force on a current-carrying loop

• A wire loop carries a clockwise current in a
  uniform magnetic field directed into the page. In
  what direction is the net force on the loop?
• 1. Left
• 2. Right
• 3. Up
• 4. Down
• 5. Into the page
• 6. Out of the page
• 7. The net force is zero

70
The force on a current-carrying loop

• The net force is always zero on a
  current-carrying loop in a UNIFORM magnetic
  field.

71
Is there a net anything on the loop?

• Lets change the direction of the uniform
  magnetic field. Is the net force on the loop
  still zero? Is there a net anything on the loop?

72
Is there a net anything on the loop?

• Lets change the direction of the uniform
  magnetic field. Is the net force on the loop
  still zero? Is there a net anything on the loop?
• The net force is still zero, but there is a net
  torque that tends to make the loop spin.

73
The torque on a current loop

• The magnetic field is in the plane of the loop
  and parallel to two sides. If the loop has a
  width a, a height b, and a current I, then the
  force on each of the left and right sides is F
  IbB. The other sides experience no force because
  the field is parallel to the current in those
  sides. Simulation
• The torque ( ) about an
  axis running through the center of the loop is

74
The torque on a current loop

• ab is the area of the loop, so the torque here is
  .
• This is the maximum possible torque, when the
  field is in the plane of the loop. When the field
  is perpendicular to the loop the torque is zero.
  In general, the torque is given by

where q is the angle between the area vector, A,
(which is perpendicular to the plane of the loop)
and the magnetic field, B.
75
A DC motor

• A direct current (DC) motor is one application of
  the torque exerted on a current loop by a
  magnetic field. The motor converts electrical
  energy into mechanical energy.
• If the current always went the same way around
  the loop, the torque would be clockwise for half
  a revolution and counter-clockwise during the
  other half. To keep the torque (and the rotation)
  going the same way, a DC motor usually has a
  "split-ring commutator" that reverses the current
  every half rotation. Simulation

76
The force between two wires

• A long-straight wire carries current out of the
  page. A second wire, to the right of the first,
  carries current into the page. In which direction
  is the force that the second wire feels because
  of the first wire?
• 1. Left
• 2. Right
• 3. Up
• 4. Down
• 5. Into the page
• 6. Out of the page
• 7. The net force is zero

77
The force between two wires

• In this situation, opposites repel and likes
  attract!
• Parallel currents going the same direction
  attract.
• If they are in opposite directions they repel.

78
A loop and a wire

• A loop with a clockwise current is placed below a
  long straight wire carrying a current to the
  right. In which direction is the net force
  exerted by the wire on the loop?
• 1. Left
• 2. Right
• 3. Up
• 4. Down
• 5. Into the page
• 6. Out of the page
• 7. The net force is zero

79
A loop and a wire

• The long straight wire creates a non-uniform
  magnetic field, pictured below.

80
A loop and a wire

• The forces on the left and right sides cancel,
  but the forces on the top and bottom only partly
  cancel the net force is directed up, toward the
  long straight wire.

81
Five wires

• Four long parallel wires carrying equal currents
  perpendicular to your page pass through the
  corners of a square drawn on the page, with one
  wire passing through each corner. You get to
  decide whether the current in each wire is
  directed into the page or out of the page.
• We also have a fifth parallel wire, carrying
  current into the page, that passes through the
  center of the square. Can you choose current
  directions for the other four wires so that the
  fifth wire experiences a net force directed
  toward the top right corner of the square?

82
How many ways?

• You can choose the direction of the currents at
  each corner. How many configurations give a net
  force on the center wire that is directed toward
  the top-right corner?
• 1. 1
• 2. 2
• 3. 3
• 4. 4
• 5. 0 or more than 4

83
How many ways?

• First, think about the four forces we need to add
  to get a net force toward the top right. How many
  ways can we create this set of four forces?

84
How many ways?

• How many ways can we create this set of four
  forces?
• Two. Wires 1 and 3 have to
• have the currents shown.
• Wires 2 and 4 have to
• match, so they either both
• attract or both repel.
• Currents going the same
• way attract opposite
• currents repel.

85
The field from a solenoid

• A solenoid is simply a coil of wire with a
  current going through it. It's basically a bunch
  of loops stacked up. Inside the coil, the field
  is very uniform (not to mention essentially
  identical to the field from a bar magnet).
• For a solenoid of length L, current I, and total
  number of turns N, the magnetic field inside the
  solenoid is given by

86
The field from a solenoid

• We can make this simpler by using n N/L as the
  number of turns per unit length, to get
  .
• The magnetic field is almost uniform - the
  solenoid is the magnetic equivalent of the
  parallel-plate capacitor. If we put a piece of
  ferromagnetic material (like iron or steel)
  inside the solenoid, we can magnify the magnetic
  field by a large factor (like 1000 or so).

87
Magnetism on the atomic level

• Currents in wires produce magnetic fields. What
  produces the magnetic field from a bar magnet,
  where there are no wires? Why does that field
  look like the field of a solenoid?
• Consider the Bohr model of the atom, where
  electrons travel in circular orbits around the
  nucleus. An electron in a circular orbit looks
  like a current loop, so it produces a magnetic
  field. In some materials (ferromagnetic
  materials) the magnetic moments associated with
  the atoms align, leading to a large net magnetic
  field.

88
Whiteboard