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Chapter 22
Magnetic Forces
and
Magnetic Fields
A Brief History of Magnetism

13th century BC

Chinese used a compass



Uses a magnetic needle
Probably an invention of Arab or Indian origin
800 BC

Greeks

Discovered magnetite attracts pieces of iron
A Brief History
of Magnetism, 2

1269



Pierre de Maricourt found that the direction
of a needle near a spherical natural
magnet formed lines that encircled the
sphere
The lines also passed through two points
diametrically opposed to each other
He called the points poles
A Brief History
of Magnetism, 3

1600

William Gilbert



Expanded experiments with magnetism to a variety of
materials
Suggested the earth itself was a large permanent magnet
1750

John Michell


Magnetic poles exert attractive or repulsive forces on
each other
These forces vary as the inverse square of the
separation
A Brief History
of Magnetism, 4

1819

Hans Christian Oersted




Pictured, 1777 – 1851
Discovered the relationship between
electricity and magnetism
An electric current in a wire
deflected a nearby compass needle
André-Marie Ampère


Deduced quantitative laws of
magnetic forces between currentcarrying conductors
Suggested electric current loops of
molecular size are responsible for
all magnetic phenomena
A Brief History
of Magnetism, final

1820’s

Faraday and Henry



Further connections between electricity and
magnetism
A changing magnetic field creates an electric
field
Maxwell

A changing electric field produces a magnetic
field
Electric and Magnetic Fields


An electric field surrounds any
stationary electric charge
The region of space surrounding a
moving charge includes a magnetic field



In addition to the electric field
A magnetic field also surrounds any
material with permanent magnetism
Both fields are vector fields
Magnetic Poles

Every magnet, regardless of its shape,
has two poles


Called north and south poles
Poles exert forces on one another


Similar to the way electric charges exert forces
on each other
Like poles repel each other


N-N or S-S
Unlike poles attract each other

N-S
Magnetic Poles, cont


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
Magnetic Poles, final


The force between two poles varies as
the inverse square of the distance
between them
A single magnetic pole has never been
isolated


In other words, magnetic poles are always
found in pairs
There is some theoretical basis for the
existence of monopoles – single poles
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,
Bar Magnet Example


The compass can
be used to trace the
field lines
The lines outside
the magnet point
from the North pole
to the South pole
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
electric field produced
by an electric dipole
Magnetic Field Lines,
Like 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
electric field produced
by like charges
Definition of Magnetic Field


The magnetic field at some point in
space can be defined in terms of the
magnetic force, FB
The magnetic force will be exerted on a
charged particle moving with a velocity,
v
Characteristics of the
Magnetic Force


The magnitude of the force exerted on
the particle is proportional to the charge,
q, and to the speed, v, of the particle
When a charged particle moves parallel
to the magnetic field vector, the
magnetic force acting on the particle is
zero
Characteristics of the
Magnetic Force, cont

When the particle’s velocity vector
makes any angle q 0 with the field, the
magnetic force acts in a direction
perpendicular to both the speed and the
field

The magnetic force is perpendicular to the
plane formed by v and B
Characteristics of the
Magnetic Force, final


The force exerted on a negative charge
is directed opposite to the force on a
positive charge moving in the same
direction
If the velocity vector makes an angle q
with the magnetic field, the magnitude
of the force is proportional to sin q
More About Direction


The force is perpendicular to both the field
and the velocity
Oppositely directed forces exerted on
oppositely charged particles will cause the
particles to move in opposite directions
Force on a Charge Moving
in a Magnetic Field, Formula
The characteristics can be summarized
in a vector equation
 FB  qv  B
 FB is the magnetic force




q is the charge
v is the velocity of the moving charge
B is the magnetic field
Units of Magnetic Field

The SI unit of magnetic field is the
Tesla (T)
N s
T
C m

The cgs unit is a Gauss (G)

1 T = 104 G
Directions –
Right Hand Rule #1



Depends on the right-hand rule
for cross products
The fingers point in the
direction of the velocity
The palm faces the field


Curl your fingers in the direction of
field
The thumb points in the
direction of the cross product,
which is the direction of force

For a positive charge, opposite
the direction for a negative charge
Direction –
Right Hand Rule #2




Alternative to Rule #1
Thumb is the direction
of the velocity
Fingers are in the
direction of the field
Palm is in the direction
of force



On a positive particle
Force on a negative
charge is opposite
You can think of this as
your hand pushing the
particle
More About Magnitude
of the Force

The magnitude of the magnetic force on a
charged particle is FB = |q| v B sin q


q is the angle between the velocity and the field
The force is zero when the velocity and the field
are parallel or antiparallel


q = 0 or 180o
The force is a maximum when the velocity and the
field are perpendicular

q = 90o
Differences Between
Electric and Magnetic Fields

Direction of force



The electric force acts parallel or antiparallel to the
electric field
The magnetic force acts perpendicular to the
magnetic field
Motion


The electric force acts on a charged particle
regardless of its velocity
The magnetic force acts on a charged particle only
when the particle is in motion and the force is
proportional to the velocity
More Differences Between
Electric and Magnetic Fields

Work


The electric force does work in displacing a
charged particle
The magnetic force associated with a
steady magnetic field does no work when a
particle is displaced

This is because the force is perpendicular to
the displacement
Work in Fields, cont


The kinetic energy of a charged particle
moving through a constant magnetic field
cannot be altered by the magnetic field alone
When a charged particle moves with a
velocity v through a magnetic field, the field
can alter the direction of the velocity, but not
the speed or the kinetic energy
Notation Note

The dots indicate the
direction is out of the
page


The dots represent the
tips of the arrows coming
toward you
The crosses indicate
the direction is into the
page

The crosses represent
the feathered tails of the
arrows
Charged Particle
in a Magnetic Field



Consider a particle
moving in an external
magnetic field with its
velocity 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


Using Newton’s Second Law, you can equate
the magnetic and centripetal forces:
2
mv
 F  ma  qvB  r
mv
Solving for r: r 
qB

r is proportional to the linear momentum of the
particle and inversely proportional to the magnetic
field and the charge
More About Motion
of Charged Particle

The angular speed of the particle is
v qB
w 
r
m


The angular speed, w, is also referred to as
the cyclotron frequency
The period of the motion is
2 r 2 2 m
T


v
w
qB
Motion of a Particle, General


If a charged particle
moves in a magnetic
field at some
arbitrary angle with
respect to the field,
its path is a helix
Same equations
apply, with
v  v y2  v z2
Bending of an Electron Beam



Electrons are
accelerated from
rest through a
potential difference
Conservation of
Energy will give v
Other parameters
can be found
Charged Particle Moving in
Electric and Magnetic Fields



In many applications, the charged particle will
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
In general: F  qE  qv  B


This force is called the Lorenz force
It is the vector sum of the electric force and the
magnetic force
Velocity Selector


Used when all the
particles need to
move with the same
velocity
A uniform electric
field is perpendicular
to a uniform
magnetic field
Velocity Selector, cont


When the force due
to the electric field is
equal but opposite
to the force due to
the magnetic field,
the particle moves in
a straight line
This occurs for
velocities of value
v=E/B
Velocity Selector, final



Only those particles with the given
speed will pass through the two fields
undeflected
The magnetic force exerted on particles
moving at speed greater than this is
stronger than the electric field and the
particles will be deflected upward
Those moving more slowly will be
deflected downward
Mass Spectrometer


A mass
spectrometer
separates ions
according to their
mass-to-charge ratio
A beam of ions
passes through a
velocity selector and
enters a second
magnetic field
Mass Spectrometer, cont




After entering the second magnetic field, the
ions move in a semicircle of radius r before
striking a detector at P
If the ions are positively charged, they deflect
upward
If the ions are negatively charged, they
deflect downward
This version is known as the Bainbridge Mass
Spectrometer
Mass Spectrometer, final

Analyzing the forces on the particles in
the mass spectrometer gives
m rBo B

q
E

Typically, ions with the same charge are
used and the mass is measured
Thomson’s e/m Experiment



Electrons are
accelerated from the
cathode
They are deflected
by electric and
magnetic fields
The beam of
electrons strikes a
fluorescent screen
Thomson’s e/m
Experiment, cont


Thomson’s variation found e/me by
measuring the deflection of the beam
and the fields
This experiment was crucial in the
discovery of the electron
Cyclotron



A cyclotron is a device that can
accelerate charged particles to very
high speeds
The energetic particles produced are
used to bombard atomic nuclei and
thereby produce reactions
These reactions can be analyzed by
researchers
Cyclotron, 2



D1 and D2 are called
dees because of
their shape
A high frequency
alternating potential
is applied to the
dees
A uniform magnetic
field is perpendicular
to them
Cyclotron, 3



A positive ion is released near the
center and moves in a semicircular path
The potential difference is adjusted so
that the polarity of the dees is reversed
in the same time interval as the particle
travels around one dee
This ensures the kinetic energy of the
particle increases each trip
Cyclotron, final

The cyclotron’s operation is based on the fact
that T is independent of the speed of the
particles and of the radius of their path
2 2 2
1
q
BR
2
K  mv 
2
2m

When the energy of the ions in a cyclotron
exceeds about 20 MeV, relativistic effects
come into play
First Cyclotron


Invented by E. O.
Lawrence and M. S.
Livingston
Invented in 1934
Force on a
Current-Carrying Conductor



A current carrying conductor
experiences a force when placed in an
external magnetic field
The current represents a collection of
many charged particles in motion
The resultant magnetic force on the wire
is due to the sum of the magnetic forces
on the charged particles
Force on a Wire


In this case, there is
no current, so there
is no force
Therefore, the wire
remains vertical
Force on a Wire,cont



The magnetic field
is into the page
The current is
upward, along
the page
The force is to
the left
Force on a Wire, final



The field is into the
page
The current is
downward along 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
 F  qv  B
B
d
The total force is the
product of the force
on one charge and
the number of
charges
 F  qv  B nA
B
d


Force on a Wire, cont

In terms of the current, this becomes
FB  I  B

l is a vector that points in the direction
of the current


Its magnitude is the length of the segment
This applies only to a straight segment of wire
in a uniform external magnetic field
Force on a Wire,
Arbitrary Shape

Consider a small
segment of the wire,
ds

The force exerted
on this segment is
dFB  I ds  B

The total force is
b
FB  I  ds  B
a
Torque on a Current Loop


The rectangular loop
carries a current I in
a uniform magnetic
field
No magnetic force
acts on sides  & 

The wires are
parallel to the field
and cross product is
zero
Torque on a Current Loop, 2

There is a force on sides  & 


The magnitude of the magnetic force on
these sides will be:



These sides are perpendicular to the field
F2 = F4 = I a B
The direction of F2 is out of the page
The direction of F4 is into the page
Torque on a Current Loop, 3


The forces are equal
and in opposite
directions, but not
along the same line
of action
The forces produce
a torque around
point O
Torque on a
Current Loop, Equation

The maximum torque is found by:
b
b
b
b
t max  F2  F4  (IaB )  (IaB )
2
2
2
2
 IabB

The area enclosed by the loop is ab, so
tmax = I A B

This maximum value occurs only when the
field is parallel to the plane of the loop
Torque on a
Current Loop, General


Assume the
magnetic field
makes an angle of
q<90o with a line
perpendicular to the
plane of the loop
The net torque
about point O will be
t = I A B sin q
Torque on a
Current Loop, Summary



The torque has a maximum value when the
field is perpendicular to the normal to the
plane of the loop
The torque is zero when the field is parallel to
the normal to the plane of the loop
t  I A  B where A is perpendicular to the
plane of the loop and has a magnitude equal
to the area of the loop
Direction of A



The right-hand rule
can be used to
determine the
direction of A
Curl your fingers in
the direction of the
current in the loop
Your thumb points in
the direction of A
Magnetic Dipole Moment

The product I A is defined as the
magnetic dipole moment,  of
the loop



Often called the magnetic moment
SI units: A m2
Torque in terms of magnetic moment:
t   B
Biot-Savart Law – Introduction


Biot and Savart conducted experiments
on the force exerted by an electric
current on a nearby magnet
They arrived at a mathematical
expression that gives an expression for
the magnetic field at some point in
space due to a current
Biot-Savart Law – Set-Up



The magnetic field is
dB at some point P
The length element
is ds
The wire is carrying
a steady current of I
Biot-Savart Law –
Observations


The vector dB is perpendicular to both
ds and to the unit vector r̂ directed from
ds toward P
The magnitude of dB is inversely
proportional to r2, where r is the
distance from ds to P
Biot-Savart Law –
Observations, cont


The magnitude of dB is proportional to
the current and to the magnitude ds of
the length element ds
The magnitude of dB is proportional to
sin q, where q is the angle between the
vectors ds and r̂
Biot-Savart Law, Equation

The observations are summarized in the
mathematical equation called BiotSavart Law:
I ds  rˆ
dB  km
2
r

The Biot-Savart law gives the magnetic
field only for a small length of the
conductor
Permeability of Free Space




o
7 T  m
km 
 10
4
A
The constant o is called the
permeability of free space
o = 4  x 10-7 T. m / A
The Biot-Savart Law can be written as
o I ds  rˆ
dB 
2
4 r
Total Magnetic Field

To find the total field, you need to sum
up the contributions from all the current
elements


You need to evaluate the field by
integrating over the entire current
distribution
The magnitude of the field will be
oI
B
2 r
B Compared to E

Distance


The magnitude of the magnetic field varies
as the inverse square of the distance from
the source
The electric field due to a point charge also
varies as the inverse square of the
distance from the charge
B Compared to E, 2

Direction


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 ds and the unit vector r̂
B Compared to E, 3

Source


An electric field is established by an
isolated electric charge
The current element that produces a
magnetic field must be part of an extended
current distribution

Therefore you must integrate over the entire
current distribution
B for a Long, Straight
Conductor, Direction




The magnetic field lines
are circles concentric
with the wire
The field lines lie in
planes perpendicular to
to wire
The magnitude of the
field is constant on any
circle of radius a
The right hand rule for
determining the
direction of the field is
shown
B for a Circular Current Loop


The loop has a
radius of R and
carries a steady
current of I
Find B at point P
oIR 2
Bx 
3
2
2
2 x  R  2
Field at the Center of a Loop



Consider the field at the center of the
current loop
At this special point, x = 0
Then,
Bx 
oIR 2

2 x2  R2

3

2
o I
2R
Magnetic Field
Lines for a Loop



Figure a shows the magnetic field lines surrounding a
current loop
Figure b shows the field lines in the iron filings
Figure c compares the field lines to that of a bar
magnet
Magnetic Force Between
Two Parallel Conductors


Two parallel wires
each carry a steady
current
The field B2 due to
the current in wire 2
exerts a force on
wire 1 of F1 = I1l B2
Magnetic Force Between
Two Parallel Conductors, cont

Substituting the equation for B2 gives
oI1I2
F1 
2 a


Parallel conductors carrying currents in the
same direction attract each other
Parallel conductors carrying current in
opposite directions repel each other
Magnetic Force Between
Two Parallel Conductors, final


The result is often expressed as the
magnetic force between the two wires,
FB
This can also be given as the force per
unit length, FB/l
FB  oI1I2

 2a
Definition of the Ampere


The force between two parallel wires
can be used to define the ampere
When the magnitude of the force per
unit length between two long parallel
wires that carry identical currents and
are separated by 1 m is 2 x 10-7 N/m,
the current in each wire is defined to
be 1 A
Definition of the Coulomb


The SI unit of charge, the coulomb, is
defined in terms of the ampere
When a conductor carries a steady
current of 1 A, the quantity of charge
that flows through a cross section of the
conductor in 1 s is 1 C
Magnetic Field of a Wire



A compass can be used
to detect the magnetic
field
When there is no
current in the wire,
there is no field due to
the current
The compass needles
all point toward the
earth’s north pole

Due to the earth’s
magnetic field
Magnetic Field of a Wire, 2



The wire carries a
strong current
The compass needles
deflect in a direction
tangent to the circle
This shows the
direction of the
magnetic field
produced by the wire
Magnetic Field of a Wire, 3

The circular
magnetic field
around the wire is
shown by the iron
filings
André-Marie Ampère


1775 –1836
Credited with the
discovery of
electromagnetism


The relationship
between electric
currents and
magnetic fields
Died of pneumonia
Ampere’s Law


The product of B  ds can be evaluated for
small length elements ds on the circular path
defined by the compass needles for the long
straight wire
Ampere’s Law states that the line integral of
B  ds around any closed path equals oI
where I is the total steady current passing
through any surface bounded by the closed
path
 B  ds   o I
Ampere’s Law, cont

Ampere’s Law describes the creation of
magnetic fields by all continuous current
configurations


Most useful for this course if the current
configuration has a high degree of symmetry
Put the thumb of your right hand in the
direction of the current through the amperian
loop and your figures curl in the direction you
should integrate around the loop
Amperian Loops

Each portion of the path satisfies one or
more of the following conditions:


The value of the magnetic field can be
argued by symmetry to be constant over
the portion of the path
The dot product can be expressed as a
simple algebraic product B ds

The vectors are parallel
Amperian Loops, cont

Conditions:

The dot product is zero


The vectors are perpendicular
The magnetic field can be argued to be
zero at all points on the portion of the path
Field Due to a Long Straight
Wire – From Ampere’s Law


Want to calculate
the magnetic field at
a distance r from the
center of a wire
carrying a steady
current I
The current is
uniformly distributed
through the cross
section of the wire
Field Due to a Long Straight Wire
– Results From Ampere’s Law

Outside of the wire, r > R
 B  ds  B(2 r )   I
o
o I
B
2 r

Inside the wire, we need I’, the current
inside the amperian circle
 B  ds  B(2 r )   I '
o
 I 
B   o 2 r
 2 R 
r2
I'  2 I
R
Field Due to a Long Straight Wire
– Results Summary



The field is
proportional to r
inside the wire
The field varies as
1/r outside the wire
Both equations are
equal at r = R
Magnetic Field of a Toroid


Find the field at a
point at distance r
from the center of
the toroid
The toroid has N
turns of wire
 B  ds  B(2 r )   NI
o
o NI
B
2 r
Magnetic Field of a Solenoid



A solenoid is a long wire wound in the form
of a helix
A reasonably uniform magnetic field can be
produced in the space surrounded by the
turns of the wire
Each of the turns can be modeled as a
circular loop

The net magnetic field is the vector sum of all the
fields due to all the turns
Magnetic Field of a
Solenoid, Description

The field lines in the interior are




Approximately parallel to each other
Uniformly distributed
Close together
This indicates the field is strong and
almost uniform
Magnetic Field of a
Tightly Wound Solenoid


The field distribution
is similar to that of a
bar magnet
As the length of the
solenoid increases


The interior field
becomes more
uniform
The exterior field
becomes weaker
Ideal Solenoid –
Characteristics

An ideal solenoid is
approached when




The turns are closely
spaced
The length is much
greater than the radius of
the turns
For an ideal solenoid,
the field outside of
solenoid is negligible
The field inside is
uniform
Ampere’s Law Applied
to a Solenoid



Ampere’s Law can be used to find the
interior magnetic field of the solenoid
Consider a rectangle with side l parallel
to the interior field and side w
perpendicular to the field
The side of length l inside the solenoid
contributes to the field

This is path 1 in the diagram
Ampere’s Law Applied
to a Solenoid, cont

Applying Ampere’s Law gives
 B  ds  
B  ds  B
path1


ds  B
path1
The total current through the
rectangular path equals the current
through each turn multiplied by the
number of turns
 B  ds  B
 o NI
Magnetic Field of
a Solenoid, final

Solving Ampere’s Law for the magnetic
field is
N
B  o I  o nI


n = N / l is the number of turns per unit
length
This is valid only at points near the
center of a very long solenoid
Magnetic Moment –
Bohr Atom




The electrons move in
circular orbits
The orbiting electron
constitutes a tiny current
loop
The magnetic moment of
the electron is associated
with this orbital motion
The angular momentum
and magnetic moment are
in opposite directions due
to the electron’s negative
charge
Magnetic Moments
of Multiple Electrons


In most substances, the magnetic
moment of one electron is canceled by
that of another electron orbiting in the
opposite direction
The net result is that the magnetic effect
produced by the orbital motion of the
electrons is either zero or very small
Electron Spin

Electrons (and other particles) have an
intrinsic property called spin that also
contributes to its magnetic moment



The electron is not physically spinning
It has an intrinsic angular momentum as if
it were spinning
Spin angular momentum is actually a
relativistic effect
Electron Magnetic
Moment, final

In atoms with multiple
electrons, many
electrons are paired up
with their spins in
opposite directions



The spin magnetic
moments cancel
Those with an “odd”
electron will have a net
moment
Some moments are
given in the table
Ferromagnetic Materials

Some examples of ferromagnetic materials
are






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
Domains

All ferromagnetic materials are made up
of microscopic regions called domains


The domain is an area within which all
magnetic moments are aligned
The boundaries between various
domains having different orientations
are called domain walls
Domains,
Unmagnetized Material


The magnetic
moments in the
domains are
randomly aligned
The net magnetic
moment is zero
Domains,
External Field Applied



A sample is placed
in an external
magnetic field
The size of the
domains with
magnetic moments
aligned with the field
grows
The sample is
magnetized
Domains,
External Field Applied, cont



The material is
placed in a stronger
field
The domains not
aligned with the field
become very small
When the external
field is removed, the
material may retain
most of its
magnetism
Magnetic Levitation


The Electromagnetic System (EMS) is
one design model for magnetic
levitation
The magnets supporting the vehicle are
located below the track because the
attractive force between these magnets
and those in the track lift the vehicle
EMS, cont


The proximity detector uses magnetic induction
to measure the magnet-rail separation
The power supply is adjusted to maintain proper
separation
EMS, final

Disadvantages

Instability caused by the variation of magnetic
force with distance


Relatively small separation between the magnets
and the tracks



Compensated for by the proximity detector
Usually about 10 mm
Track needs high maintenance
Advantage

Independent of speed, so wheels are not needed

Wheels are in place for “emergency landing” system
German Transrapid –
EMS Example