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 current-carrying 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 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
B
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,
The magnetic force will be exerted on a charged particle moving with a velocity,
BF
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 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 andv 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 with the magnetic field, the magnitude of the force is proportional to sin
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
is the magnetic force q is the charge is the velocity of the moving charge is the magnetic field
B q F v B
BF
v
B
Units of Magnetic Field The SI unit of magnetic field is the
Tesla (T)
The cgs unit is a Gauss (G) 1 T = 104 G
N sT
C m
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 is the angle between the velocity and the field The force is zero when the velocity and the field
are parallel or antiparallel = 0 or 180o
The force is a maximum when the velocity and the field are perpendicular
= 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 through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy
v
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:
Solving for r:
r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field and the charge
2mvF ma qvB
r
mvr
qB
More About Motion of Charged Particle The angular speed of the particle is
The angular speed, , is also referred to as the cyclotron frequency
The period of the motion is
v qB
r m
2 2 2r mT
v 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
2 2y zv v v
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: This force is called the Lorenz force It is the vector sum of the electric force and the
magnetic force
q q F E v B
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
Typically, ions with the same charge are used and the mass is measured
orB Bm
q E
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
When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play
2 2 221
2 2
q B RK mv
m
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
The total force is the
product of the force on one charge and the number of charges
B dq F v B
B dq nA F v B
Force on a Wire, cont In terms of the current, this becomes
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
B I F B
Force on a Wire, Arbitrary Shape Consider a small
segment of the wire,
The force exerted on
this segment is
The total force is
b
B I d aF s B
ds
Bd I d F s B
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 &
These sides are perpendicular to the field The magnitude of the magnetic force on
these sides will be: 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:
The area enclosed by the loop is ab, so max = I A B This maximum value occurs only when the
field is parallel to the plane of the loop
max 2 4 ( ) ( )2 2 2 2
b b b bF F IaB IaB
IabB
Torque on a Current Loop, General Assume the
magnetic field makes an angle of <90o with a line perpendicular to the plane of the loop
The net torque about point O will be = I A B sin
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
where A is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop
I A B
Direction of A The right-hand rule
can be used to determine the direction of
Curl your fingers in the direction of the current in the loop
Your thumb points in the direction of
A
A
Magnetic Dipole Moment The product I 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:
A
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
at some point P The length element
is The wire is carrying
a steady current of I
dB
ds
Biot-Savart Law – Observations The vector is perpendicular to both
ds and to the unit vector directed from
toward P The magnitude of is inversely
proportional to r2, where r is the distance from to P
r̂dB
dBds
ds
Biot-Savart Law – Observations, cont The magnitude of is proportional to
the current and to the magnitude ds of the length element ds
The magnitude of is proportional to sin where is the angle between the vectors and r̂
dB
dB
ds
The observations are summarized in the mathematical equation called Biot-Savart Law:
The Biot-Savart law gives the magnetic field only for a small length of the conductor
Biot-Savart Law, Equation
2
ˆm
I dd k
r
s rB
Permeability of Free Space
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
7104
om
T mk
A
2
ˆ
4o I d
dr
s rB
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
2oIBr
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 and the unit vector r̂ds
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 Compared to E, 3
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 at point P
2
32 2 22
ox
IRB
x R
B
Field at the Center of a Loop Consider the field at the center of the
current loop At this special point, x = 0 Then,
2
32 2 2 22
o ox
IR IB
Rx R
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 due to the current in wire 2 exerts a force on wire 1 of F1 = I1l B2
2B
Magnetic Force Between Two Parallel Conductors, cont
Substituting the equation for B2 gives
Parallel conductors carrying currents in the same direction attract each other
Parallel conductors carrying current in opposite directions repel each other
1 21 2
oI IF
a
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
a2IIF 21oB
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 can be evaluated for
small length elements on the circular path defined by the compass needles for the long straight wire
Ampere’s Law states that the line integral of
around any closed path equals oI where I is the total steady current passing through any surface bounded by the closed path
od I B s
dB s
ds
dB s
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
Inside the wire, we need I’, the current inside the amperian circle
(2 )
2
o
o
d B r I
IB
r
B s
2
2
2
(2 ) ' '
2
o
o
rd B r I I I
RI
B rR
B s
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
(2 )
2
o
o
d B r NI
NIB
r
B s
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
The total current through the rectangular path equals the current through each turn multiplied by the number of turns
1 1path path
d d B ds B B s B s
od B NI B s
Magnetic Field of a Solenoid, final Solving Ampere’s Law for the magnetic
field is
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
o o
NB I nI
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
Compensated for by the proximity detector Relatively small separation between the magnets
and the tracks 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