Magnetic fields
The symbol we use for a magnetic field is B.
The unit is the tesla (T).
The Earth’s 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 Earth’s geographic
north pole?
What are some similarities between electric and magnetic
fields? What are some differences?
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.
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.
Observing a charge in a magnetic field
The force exerted on a charge in an electric field is given by
F  qE
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.
Observing a charge in a magnetic field
The force exerted on a charge in an electric field is given by
F  qE
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.
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.
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.
Observing a charge in a magnetic field
Simulation
Case 4: The same as case 3, except the magnetic field is
doubled.
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.
mv
F
r
2
Thus, doubling the magnetic field doubles the force the force is proportional to the magnetic field.
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.
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.
2
mv
F
r
Thus, doubling the charge doubles the force, and tripling the
charge triples the force the force is proportional to the charge.
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.
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:
2
mv
r
F
What does this tell us about how the force depends on
speed?
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:
2
mv
r
F
What does this tell us about how the force depends on
speed?
The force is proportional to the speed.
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).
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(θ), where θ 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.
Something to keep in mind
A force perpendicular to the velocity, such as the magnetic
force, can not change an object’s speed (or the kinetic
energy). All it can do is make the object change direction.
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
The right-hand rule
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
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.
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
Practice with the right-hand rule, II
Remember that with a negative charge, your right hand lies to
you – take the opposite direction.
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
Practice with the right-hand rule, III
Magnetic fields exert no force on stationary charges.
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
Practice with the right-hand rule, IV
We don’t need the right-hand rule for an electric field, we
need F  qE. The force is opposite to the field, for a negative
charge.
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
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 can’t tell us anything about a component parallel to the
field – that component is unaffected by the field.
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.
mv 2
F  qvB 
r
The radius of the circular path is:
mv
r
qB
Charges moving perpendicular to the field
The radius of the circular path is:
mv
r
qB
The time for the object to go once around the circle (the
period, T) is:
2 r 2 mv / qB 2 m
T


v
v
qB
Interestingly, the time is independent of the speed. The faster
the speed, the larger the radius, but the period is unchanged.
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 > 2 > 3
2. 2 > 1 > 3
3. 3 > 2 > 1
4. 3 > 1 > 2
5. None of the above
Circular paths
The radius of the path is proportional to the speed, so the
correct ranking by speed is choice 2, 2 > 1 > 3.
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 > 2 > 3
2. 2 > 1 > 3
3. 3 > 2 > 1
4. 3 > 1 > 2
5. None of the above
Circular paths, II
The force is proportional to the speed, so the correct ranking
by force is also choice 2, 2 > 1 > 3.
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
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
Spiraling charges
Charges spiral around magnetic field lines.
Charged particles near the Earth are trapped by the Earth’s
magnetic field, spiraling around the Earth’s magnetic field
down toward the Earth at the magnetic poles.
The energy deposited by such particles gives rise to ??
Spiraling charges
Charges spiral around magnetic field lines.
Charged particles near the Earth are trapped by the Earth’s
magnetic field, spiraling around the Earth’s 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
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.
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 |
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.
mv 2
F  qvB 
r
mv
 r
qB
Step 3: The Mass Separator
mv
F  qvB 
r
2
mv
 r
qB
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.
2(m2  m1 )v
s  2r2  2r1 
qB
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
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 FE  qE on the ions.
The magnetic field is perpendicular to both the ion velocity
and the electric field. The magnetic force, FM  qvB , exactly
balances the electric force when:
qE  qvB  E  vB
E
Ions with a speed of v 
B
pass straight through.
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
Magnetic field in the velocity selector
The right-hand rule tells us that the magnetic field is directed
into the screen.
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
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 don’t have to change a thing.
Faster ions in the velocity selector
Let’s 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
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.
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 Dshaped 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.
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.
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
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 ΔK
each time the charge
crosses the gap, and stays
constant in the dees.
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.
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:
0 I
B
2 r
0 , the permeability of free space, is:
0  4 10
7
Tm/A
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.
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
The net magnetic field
We add the individual fields to find the net field, which is
directed right.
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.
F  qvB sin 
Using q = I t, this becomes:
F  IvtB sin 
But a velocity multiplied by a time is a length L, so this can be
written:
F  ILB sin 
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.
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
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
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.
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?
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.
2.
3.
4.
5.
1
2
3
4
0 or more than 4
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?
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.