Chapter 19
Magnetism
Magnets have two ends – poles – called
north and south.
Like poles repel; unlike poles attract.
If you cut a magnet in half, you don’t get a
north pole and a south pole – you get two
smaller magnets.
Types of Magnetic
Materials

Soft magnetic materials, such as
iron, are easily magnetized


They also tend to lose their
magnetism easily
Hard magnetic materials, such as
cobalt and nickel, are difficult to
magnetize

They tend to retain their magnetism
Sources of 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
Magnetic Fields




A vector quantity
Symbolized by B
Direction is given by the direction
a north pole of a compass needle
points in that location
Magnetic field lines can be used to
show how the field lines, as traced
out by a compass, would look
Magnetic Field Lines,
sketch


A compass can be used to show the
direction of the magnetic field lines (a)
A sketch of the magnetic field lines (b)
Magnetic Field Lines, Bar
Magnet


Iron filings are
used to show the
pattern of the
magnetic field
lines
The direction of
the field is the
direction a north
pole would point
Magnetic Field Lines,
Unlike Poles


Iron filings are used to
show the pattern of
the magnetic field lines
The direction of the
field is the direction a
north pole would point

Compare to the
magnetic field produced
by an electric dipole
Magnetic Field Lines, Like
Poles


Iron filings are used
to show the pattern
of the electric field
lines
The direction of the
field is the direction
a north pole would
point

Compare to the
electric field produced
by like charges
Earth’s Magnetic Field


The Earth’s geographic north pole
corresponds to a magnetic south
pole
The Earth’s geographic south pole
corresponds to a magnetic north
pole

Strictly speaking, a north pole should
be a “north-seeking” pole and a south
pole a “south-seeking” pole
Earth’s Magnetic Field

The Earth’s
magnetic field
resembles that
achieved by
burying a huge
bar magnet deep
in the Earth’s
interior
More About the Earth’s
Magnetic Poles

The magnetic and geographic
poles are not in the same exact
location

The difference between true north, at
the geographic north pole, and
magnetic north is called the magnetic
declination

The amount of declination varies by
location on the earth’s surface
Earth’s Magnetic
Declination
Source of the Earth’s
Magnetic Field


There cannot be large masses of
permanently magnetized materials
since the high temperatures of the core
prevent materials from retaining
permanent magnetization
The most likely source of the Earth’s
magnetic field is believed to be electric
currents in the liquid part of the core
Reversals of the Earth’s
Magnetic Field

The direction of the Earth’s
magnetic field reverses every few
million years


Evidence of these reversals are found
in basalts resulting from volcanic
activity
The origin of the reversals is not
understood
Magnetic Fields

When moving through a magnetic
field, a charged particle
experiences a magnetic force


This force has a maximum value
when the charge moves
perpendicularly to the magnetic field
lines
This force is zero when the charge
moves along the field lines
Magnetic Fields, cont

One can define a magnetic field in
terms of the magnetic force
exerted on a test charge moving in
the field with velocity v


Similar to the way electric fields are
defined
F
B
qv sin 
Units of Magnetic Field

The SI unit of magnetic field is the
Tesla (T)
Wb
N
N
T 2 

m
C  (m / s) A  m


Wb is a Weber
The cgs unit is a Gauss (G)

1 T = 104 G
A Few Typical B Values

Conventional laboratory magnets


Superconducting magnets


25000 G or 2.5 T
300000 G or 30 T
Earth’s magnetic field

0.5 G or 5 x 10-5 T
Finding the 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
F = 0 when v is
parallel to B
Right Hand Rule #1



Place your fingers in the
direction of v
Curl the fingers in the
direction of the
magnetic field, B
Your thumb points in
the direction of the
force, F , on a positive
charge

If the charge is negative,
the force is opposite that
determined by the right
hand rule
1) out of the page
2) into the page
3) downwards
A positive charge
enters a uniform
magnetic field as
shown. What is
the direction of
the magnetic
force?
4) to the right
5) to the left
x x x x x x
v
x x x x x x
x x x xq x x
A positive charge enters a
uniform magnetic field as shown.
What is the direction of the
magnetic force?
1) out of the page
2) into the page
3) downwards
4) to the right
5) to the left
Using the right-hand rule, you can
see that the magnetic force is
directed to the left. Remember
that the magnetic force must be
perpendicular to BOTH the B field
and the velocity.
x x x x x x
v
x x x x x x
x xFx xq x x
1) out of the page
2) into the page
3) downwards
4) upwards
5) to the left
A positive charge
enters a uniform
magnetic field as
shown. What is
the direction of
the magnetic
force?
x x x x x x
x x x x vx x
q
x x x x x x
A positive charge enters a
uniform magnetic field as
shown. What is the
direction of the magnetic
force?
1) out of the page
2) into the page
3) downwards
4) upwards
5) to the left
Using the right-hand rule, you can
see that the magnetic force is
directed upwards. Remember
that the magnetic force must be
perpendicular to BOTH the B field
and the velocity.
x x x
F
x x x
q
x x x
x x x
x vx x
x x x
1) out of the page
2) into the page
3) zero
A positive
charge enters a
uniform
magnetic field
as shown.
What is the
direction of the
magnetic force?
4) to the right
5) to the left


v

q

A positive charge enters a
uniform magnetic field as
shown. What is the
direction of the magnetic
force?
1) out of the page
2) into the page
3) zero
4) to the right
5) to the left
Using the right-hand rule, you can
see that the magnetic force is
directed into the page. Remember
that the magnetic force must be
perpendicular to BOTH the B field
and the velocity.


v

 q
F

1) out of the page
2) into the page
3) zero
A positive
charge enters a
uniform
magnetic field
as shown.
What is the
direction of the
magnetic force?
4) to the right
5) to the left
v
q
A positive charge enters a
uniform magnetic field as
shown. What is the
direction of the magnetic
force?
1) out of the page
2) into the page
3) zero
4) to the right
5) to the left
The charge is moving parallel to
the magnetic field, so it does not
v
experience any magnetic force.
Remember that the magnetic force
is given by: F = v B sin() .
q
F=0
Quick Quiz
A charged particle moves in a straight line through a
region of space. Which of the following answers must be
true? (Assume any other fields are negligible.) The
magnetic field (a) has a magnitude of zero (b) has a zero
component perpendicular to the particle’s velocity (c) has a
zero component parallel to the particle’s velocity in that
region.
Answer
(b). The force that a magnetic field exerts on a charged
particle moving through it is given by F  qvB sin  qvB ,
where B is the component of the field perpendicular to
the particle’s velocity. Since the particle moves in a
straight line, the magnetic force (and hence B ,
since qv  0 ) must be zero.
Quick Quiz
The north-pole end of a bar magnet is held near a
stationary positively charged piece of plastic. Is the
plastic (a) attracted, (b) repelled, or (c) unaffected by
the magnet?
Answer
(c). The magnetic force exerted by a magnetic
field on a charge is proportional to the charge’s
velocity relative to the field. If the charge is
stationary, as in this situation, there is no
magnetic force.
x x x x x x x x x 1x 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
2
x x x x x x x x x x x x
A beam of
x x x x x x x x x x x x
atoms enters
x x x x x x x x x x x x
a magnetic
field region.
What path will
the atoms
follow?
4
3
A beam of atoms
enters a magnetic
x x x x x x x x x 1x x x
field region. What
x x x x x x x x x x x x
path will the atoms
x x x x x x x x x x x x
follow?
x x x x x x x x x x x x
x x x x x x x x x x x x
4
x x x x x x x x x x x x
Atoms are neutral objects whose net charge is zero.
Thus they do not experience a magnetic force.
2
3
Example 1
An electron gun fires electrons into a magnetic field
directed straight downward. Find the direction of the force
exerted by the field on an electron for each of the
following directions of the electron’s velocity: (a) horizontal
and due north; (b) horizontal and 30° west of north; (c)
due north, but at 30° below the horizontal; (d) straight
upward. (Remember that an electron has a negative
charge.)
Example 2
(a) Find the direction of the force
on a proton (a positively charged
particle) moving through the
magnetic fields in Figure P19.2,
as shown. (b) Repeat part (a),
assuming the moving particle is
an electron.
Example 3
An electron is accelerated through 2 400 V from rest and
then enters a region where there is a uniform 1.70-T
magnetic field. What are (a) the maximum and (b) the
minimum magnitudes of the magnetic force acting on this
electron?
Practice 1
A proton moves perpendicularly to a uniform magnetic
field at 1.0 × 107 m/s and exhibits an acceleration of 2.0
× 1013 m/s2 in the +x-direction when its velocity is in the
+z-direction. Determine the magnitude and direction of
the field.
Magnetic Force on a Current
Carrying Conductor

A force is exerted on a currentcarrying wire placed in a magnetic
field


The current is a collection of many
charged particles in motion
The direction of the force is given
by right hand rule #1
Force on a Wire

The blue x’s indicate the
magnetic field is
directed into the page


Blue dots would be used
to represent the field
directed out of the page


The x represents the tail
of the arrow
The • represents the head
of the arrow
In this case, there is no
current, so there is no
force
Force on a Wire,
cont



B is into the page
The current is up
the page
The force is to the
left
Force on a Wire,
final



B is into the page
The current is
down the page
The force is to the
right
Force on a Wire, equation



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
F = B I ℓ sin θ
 θ is the angle between B and the direction

of I
The direction is found by the right hand
rule, placing your fingers in the direction of
I instead of v
1) left
2) right
3) zero
A horizontal wire
4) into the page
5) out of the page
carries a current
and is in a vertical
magnetic field.
I
What is the
direction of the
force on the wire?
B
A horizontal wire carries a
current and is in a vertical
1) left
magnetic field. What is the
2) right
direction of the force on the
3) zero
wire?
4) into the page
5) out of the page
Using the right-hand rule, we
see that the magnetic force
must point out of the page.
Since F must be perpendicular
to both I and B, you should
realize that F cannot be in the
plane of the page at all.
I
B
Torque on a Current Loop




Applies to any shape
loop
N is the number of
turns in the coil
Torque has a
maximum value of
NBIA


When  = 90°
Torque is zero when
the field is parallel to
the plane of the loop
Magnetic Moment





The vector m is called the magnetic
moment of the coil
Its magnitude is given by m = IAN
The vector always points perpendicular
to the plane of the loop(s)
The angle is between the moment and
the field
The equation for the magnetic torque
can be written as t = mB sin
Electric Motor

An electric motor
converts electrical
energy to
mechanical energy


The mechanical
energy is in the form
of rotational kinetic
energy
An electric motor
consists of a rigid
current-carrying loop
that rotates when
placed in a magnetic
field
A galvanometer
takes advantage of
the torque on a
current loop to
measure current.
Quick Quiz
A square and a circular loop with the same area lie in the
xy-plane, where there is a uniform magnetic field pointing
at some angle θ with respect to the positive z-direction.
Each loop carries the same current, in the same direction.
Which magnetic torque is larger? (a) the torque on the
square loop (b) the torque on the circular loop (c) the
torques are the same (d) more information is needed
Answer
(c). The torque that a planar current loop will experience
when it is in a magnetic field is given by t  BIA sin  .
Note that this torque depends on the strength of the field,
the current in the coil, the area enclosed by the coil, and
the orientation of the plane of the coil relative to the
direction of the field. However, it does not depend on the
shape of the loop.
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
Force on a Charged
Particle

Equating the magnetic and centripetal
forces:
2
mv
F  qvB 
r

mv
Solving for r: r 
qB


r is proportional to the momentum of the
particle and inversely proportional to the
magnetic field
Sometimes called the cyclotron equation
Particle Moving in an
External Magnetic Field

If the particle’s
velocity is not
perpendicular to
the field, the path
followed by the
particle is a spiral

The spiral path is
called a helix
Quick Quiz
As a charged particle moves freely in a circular path in the
presence of a constant magnetic field applied perpendicular
to the particle’s velocity, its kinetic energy (a) remains
constant, (b) increases, or (c) decreases.
Answer
(a). The magnetic force acting on the particle is always
perpendicular to the velocity of the particle, and hence to
the displacement the particle is undergoing. Under these
conditions, the force does no work on the particle and the
particle’s kinetic energy remains constant.
Hans Christian Oersted


1777 – 1851
Best known for
observing that a
compass needle
deflects when placed
near a wire carrying
a current

First evidence of a
connection between
electric and magnetic
phenomena
Magnetic Fields –
Long Straight Wire


A current-carrying
wire produces a
magnetic field
The compass needle
deflects in directions
tangent to the circle

The compass needle
points in the direction
of the magnetic field
produced by the
current
Direction of the Field of a
Long Straight Wire

Right Hand Rule
#2



Grasp the wire in
your right hand
Point your thumb
in the direction of
the current
Your fingers will
curl in the
direction of the
field
Magnitude of the Field of a
Long Straight Wire


The magnitude of the field at a
distance r from a wire carrying a
current of I is
mo I
B
2 r
µo = 4  x 10-7 T.m / A

µo is called the permeability of free
space
Ampère’s Law


André-Marie Ampère found a procedure
for deriving the relationship between
the current in an arbitrarily shaped wire
and the magnetic field produced by the
wire
Ampère’s Circuital Law


B|| Δℓ = µo I
Sum over the closed path
Ampère’s Law, cont


Choose an
arbitrary closed
path around the
current
Sum all the
products of B|| Δℓ
around the closed
path
Ampère’s Law to Find B for
a Long Straight Wire


Use a closed circular
path
The circumference of
the circle is 2  r
mo I
B
2 r


This is identical to the
result previously
obtained
Example 4
In 1962, measurements of the magnetic field of a large
tornado were made at the Geophysical Observatory in
Tulsa, Oklahoma. If the magnitude of the tornado’s field
was B = 1.50 × 10−8 T pointing north when the
tornado was 9.00 km east of the observatory, what
current was carried up or down the funnel of the
tornado? Model the vortex as a long, straight wire
carrying a current.
Example 5
The two wires in Figure P19.40 carry currents of 3.00 A
and 5.00 A in the direction indicated. (a) Find the direction
and magnitude of the magnetic field at a point midway
between the wires. (b) Find the magnitude and direction of
the magnetic field at point P, located 20.0 cm above the
wire carrying the 5.00-A current.
Practice 2
A long, straight wire lies on a horizontal table and carries a
current of 1.20 μA. In a vacuum, a proton moves parallel to
the wire (opposite the direction of the current) with a
constant velocity of 2.30 × 104 m/s at a constant distance d
above the wire. Determine the value of d. (You may ignore
the magnetic field due to Earth.)
André-Marie Ampère


1775 – 1836
Credited with the
discovery of
electromagnetism


Relationship
between electric
currents and
magnetic fields
Mathematical
genius evident by
age 12
Magnetic Force Between
Two Parallel Conductors


The force on wire 1
is due to the
current in wire 1
and the magnetic
field produced by
wire 2
The force per unit
length is:
F
mo I1 I2

2 d
Force Between Two
Conductors, cont


Parallel conductors carrying
currents in the same direction
attract each other
Parallel conductors carrying
currents in the opposite directions
repel each other
Parallel currents attract; antiparallel currents
repel.
Quick Quiz
If, in Figure 19.28, I1 = 2 A and I2 = 6 A, which of the
following is true? (a) F1 = 3F2 (b) F1 = F2 or (c) F1 =
F2/3
Answer
(b). The two forces are an action-reaction pair. They act
on different wires and have equal magnitudes but
opposite directions.
1) toward each other
2) away from each other
3) there is no force
Two straight wires run
parallel to each other, each
carrying a current in the
direction shown below. The
two wires experience a
force in which direction?
Two straight wires run parallel to each
other, each carrying a current in the
direction shown below. The two wires
experience a force in which direction?
1) toward each other
2) away from each other
3) there is no force
The current in each wire produces a magnetic
field that is felt by the current of the other
wire. Using the right-hand rule, we find that
each wire experiences a force toward the
other wire (i.e., an attractive force) when the
currents are parallel (as shown).
Example 6
A wire with a weight per unit length of 0.080 N/m is
suspended directly above a second wire. The top wire
carries a current of 30.0 A and the bottom wire carries a
current of 60.0 A. Find the distance of separation
between the wires so that the top wire will be held in
place by magnetic repulsion.
Magnetic Field of a Current
Loop


The strength of a
magnetic field
produced by a wire
can be enhanced by
forming the wire into
a loop
All the segments,
Δx, contribute to the
field, increasing its
strength
Magnetic Field of a Current
Loop – Total Field
Magnetic Field of a Current
Loop – Equation

The magnitude of the magnetic field
at the center of a circular loop with a
radius R and carrying current I is
B

mo I
2R
With N loops in the coil, this becomes
BN
mo I
2R
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
Magnetic Field of a
Solenoid, 2

The field lines inside the solenoid
are nearly parallel, uniformly
spaced, and close together


This indicates that the field inside the
solenoid is nearly uniform and strong
The exterior field is nonuniform,
much weaker, and in the opposite
direction to the field inside the
solenoid
Magnetic Field in a
Solenoid, 3

The field lines of the solenoid resemble
those of a bar magnet
Magnetic Field in a
Solenoid, Magnitude


The magnitude of the field inside a
solenoid is constant at all points far
from its ends
B = µo n I



n is the number of turns per unit length
n=N/ℓ
The same result can be obtained by
applying Ampère’s Law to the solenoid
If a piece of iron is inserted in the solenoid, the
magnetic field greatly increases. Such electromagnets
have many practical applications.
Example 7
What current is required in the windings of a long solenoid
that has 1 000 turns uniformly distributed over a length of
0.400 m in order to produce a magnetic field of
magnitude 1.00 × 10−4 T at the center of the solenoid?
Practice 3
It is desired to construct a solenoid that will have a
resistance of 5.00 Ω (at 20°C) and produce a magnetic field
of 4.00 × 10−2 T at its center when it carries a current of
4.00 A. The solenoid is to be constructed from copper wire
having a diameter of 0.500 mm. If the radius of the
solenoid is to be 1.00 cm, determine (a) the number of
turns of wire needed and (b) the length the solenoid should
have.
Mass Spectrometer
A mass spectrometer measures the masses of
atoms. If a charged particle is moving through
perpendicular electric and magnetic fields,
there is a particular speed at which it will not
be deflected:
All the atoms
reaching the second
magnetic field will
have the same
speed; their radius of
curvature will depend
on their mass.
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
Magnetic Effects of
Electrons – Orbits, cont

The net result is that the magnetic
effect produced by electrons
orbiting the nucleus is either zero
or very small for most materials
Magnetic Effects of
Electrons – Spins

Electrons also
have spin


The classical
model is to
consider the
electrons to spin
like tops
It is actually a
quantum effect
Magnetic Effects of
Electrons – Spins, cont


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

That is why most materials are not
naturally magnetic
Magnetic Effects of
Electrons – Domains

In some materials, the spins do not
naturally cancel



Such materials are called ferromagnetic
Large groups of atoms in which the
spins are aligned are called domains
When an external field is applied, the
domains that are aligned with the field
tend to grow at the expense of the
others

This causes the material to become
magnetized
Domains, cont


Random alignment, a, shows an
unmagnetized material
When an external field is applied, the
domains aligned with B grow, b
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