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Magnetic Materials
Ferromagnetic materials (ferrum – iron) have strong magnetic
properties. Loadstone contains magnetite which contains iron.
Magnetite is an oxide of iron and has the composition Fe3O4.
Magnetizing Materials
Magnetic Domains become aligned in External Magnetic Field.
B
Magnetic Materials
Moving Charges
Produce Magnetic Fields
Atom
Smallest Magnet:
Electron
Atomic Sources of Magnetic Fields
B: The Magnetic Field
Magnetic Field Lines point from North (+) to South (-)
Magnetic Compass
North seeking compass points to Earth’s Magnetic North which
is really a South Pole. A compass’ arrow head is the north pole.
Magnets
• a magnet is electrically neutral
• a magnet has two poles:
north & south
• like poles repel
• unlike poles attract
• No Magnetic Monopoles!
Use a compass to map out the field lines.
How does a compass work?
A compass is a small magnet.
Outside Field lines point from North to South.
A compass is a small magnet.
Inside the Magnet, the Field points from South to North.
Which way will the magnet rotate?
N
N
S
S
External B Field
Like Poles repel, Unlike Poles Attract.
Which way will the magnet rotate?
S
N
S
N
External B Field
Magnets align themselves so that their internal field is aligned with
the external B field.
Magnetic Force
A charged particle moving in an EXTERNAL magnetic field
experiences a force that is perpendicular to both the velocity and field.
Only the perpendicular components give rise to a force.
F  qv  B
F  qvB sin 
F  qv  B
F  qv  B
Review the Vector Product
• The magnitude of C is
AB sin  and is equal to the
area of the parallelogram
formed by A and B
• The direction of C is
perpendicular to the plane
formed by A and B
• The best way to determine
this direction is to use the
right-hand rule
=rxF
Lr p
A  B   Ay Bz  Az By  ˆi   Ax Bz  Az Bx  ˆj   Ax By  Ay Bx  kˆ
Direction: Right-Hand Rule #1
• The fingers point in the
direction of v
• B comes out of your palm
– Curl your fingers in the
direction of B
• The thumb points in the
direction of v x B which is
the direction of FB
Direction: Right-Hand Rule #2
• Alternative to Rule #1
• Thumb is in the direction
of v
• Fingers are in the
direction of B
• Palm is in the direction of
FB
– On a positive particle
– You can think of this as
your hand pushing the
particle
Direction: Right Hand Rule #2
For Protons!
Fingers – Field, Thumb – Beam, Palm – Force
Direction of the Force?
Into the Page.
Same
Greatest
Same
Rank the Force Magnitudes.
Direction: Left Hand Rule #1
For electrons!
Fingers – Field, Thumb – Beam, Palm – Force
Which way will a proton move?
An electron?
Which way will a proton move?
An electron?
All three charges have the same mass.
Determine if each charge is positive, negative or neutral.
F
+
0
-
F
Magnetic Field Strength
(Magnitude Only)
F  qvB sin 
F
B
qv sin 
N
N

 B  Tesla 
C  m / s A m
The cgs unit is a gauss (G)
1 T = 104 G
Some typical B fields:
surface of the Earth
small bar magnet
MRI
LHC
Magnet lab
surface of neutron star
10-4 T
10-2 T
2-3 T
10T
30 T
108 T
Magnetic Force
A charged particle moving in an EXTERNAL magnetic field
experiences a force that is perpendicular to both the velocity and field.
Only the perpendicular components give rise to a force.
F  qvB sin 
 0
No Force
  90
Max Force
F  qvB sin 
Example
A proton in a particle accelerator has a speed of
6x106m/s. The magnetic field is 0.40T and
makes an 30 degree angle with the velocity of
the proton as shown. What is the acceleration?
F  qvB sin 
Ns
 1.6 x10 C (6 x10 m / s)(.40
sin 30)
Cm
p: direction UP
 1.92 x1013 N
19
6
e: direction DOWN
a  F /m
13
 1.92 x10
N /1.67 x10
27
kg  1.2 x10 m / s
14
2
Circular Trajectory
The magnetic force always remains
perpendicular to the velocity and is
directed toward the center of
the circular path.
2
mv
FC 
r
F  qvB sin 
Radius of Circular Path:
mv
r
qB
Cyclotron Frequency
The angular speed of a particle
is referred to as the cyclotron
frequency:
v qB
 
r m
2
2 m
T


qB
mv
r
qB
The angular speed and
period do not depend on
the linear speed or on the
radius of the orbit!
Example
A proton with a speed of 2.2 x 106 m/s is shot into a region of
constant B separated by a distance of 0.18m.
What is B so that the proton misses the opposite plate?
Example
A proton with a speed of 2.2 x 106 m/s is shot into a region of
constant B separated by a distance of 0.18m.
What is B so that the proton misses the opposite plate?
mv
r
qB
mv
B
qr
1.67 x1027 kg  2.2 x106 m / s
B
1.6 x1019 C  .18m
 0.13kg
m/ s s
x
C m s
 0.13
B  0.13T
N
A m
Example
The electric potential difference
between the plates is 2100V.
What is the speed of the proton
when it exits the capacitor?
KE  PE
1 2
mv  qV
2
2qV
v
m
Mass Spectrometer
An ion source in a spectrometer produces doubly ionized gold ions
(Au 2+) each with a mass of 3.27 x10–25 kg. The ions are accelerated
from rest through a potential difference of 1kV. Then, a 0.500T
magnetic field causes the ions to follow a circular path.
Determine the radius of curvature.
2qV
v
m
mv
r
qB
Eliminate v to get:
2mV
r
qB 2
2  3.27 x1025 kg 103 J / C
r
2 1.6 x1019 C (.5 Ns / Cm) 2
r = 0.09m
Three ions with the same charge
move through a constant Magnetic Field.
Rank their masses.
 qr
m
 2V
2
1>3>2
 2
B

Simple Motor
http://www.youtube.com/watch?v=oRSU4FnUSrA&
feature=fvwp&NR=1
MHD Propulsion
Magnetohydrodynamics (MHD) (magnetofluiddynamics or
hydromagnetics) is the academic discipline which studies the
dynamics of electrically conducting fluids
Yamato 1
Last Time: Magnetic Force
A charged particle moving in an EXTERNAL magnetic field
experiences a force that is perpendicular to both the velocity and field.
Only the perpendicular components give rise to a force.
F  qv  B
F  qvB sin 
Charged Particle in a Magnetic Field
with velocity component parallel to the field
moves in a Helical path. The pitch is the
distance between loops.
mv
r
qB
v  v v
2
y
2
z
Radius of Gyration
mv
r
qB
Particle in a Nonuniform
Magnetic Field
• The motion is
complex
• For example, the
particles can oscillate
back and forth
between two positions
• This configuration is
known as a magnetic
bottle
Magnetic Bottles: Tokamaks
“Such a plasma-confinement scheme could fulfill a crucial role in the
control of nuclear fusion, a process that could supply us with an almost
endless source of energy” Serway, page 838
Problems with Fusion?
Tritium is RARE!
Tritium is RARE!
Breeding it:
Breathing it:
Cost: ~ $200 million/kg
Need ~ 56kg/year for the ITER
Cost of Fusion?
Average US Budget for Fusion Energy per year
over past 50 years:
$~250 Million x 50 years ~ $13 Billion so far
2008 Budget: Fusion Energy Science: $428 Million
2012 Budget: Fusion Energy Science: $398 Million
Confined Fusion: The Energy of
the Future…and it ALWAYS will be!
World Cost for ITER: $25 Billion
FREE: Solar Fusion:
The Energy of the
Present and it ALWAYS
will be!
Solar Wind
The Earth’s magnetic field shield us
from the stream of charged particles
from the Sun called the solar wind
Van Allen Radiation Belts
•
•
•
The particles are trapped by the
Earth’s magnetic field and spiral
from pole to pole, creating auroras
The inner Van Allen Belt extends
from an altitude of 700–10,000 km
(0.1 to 1.5 Earth radii) above the
Earths surface, and contains high
concentrations of energetic protons
with energies exceeding 100 MeV
and electrons in the range of 100's
of keV, trapped by the strong
(relative to the outer belts) magnetic
fields in the region.
The outer belt consists mainly of
high energy (0.1–10 MeV) electrons
trapped by the Earth's
magnetosphere.
Starfish Prime: Man Made Belts
Starfish Prime was a high-altitude
(400 km) nuclear bomb test conducted
by the United States of America on
July 9, 1962. It produced a yield of
1.4 megatons of TNT. While some of
the energetic beta particles followed
the earth's magnetic field and
illuminated the sky, other high-energy
electrons became trapped in manmade radiation belts around the earth.
These man-made radiation belts
eventually crippled one-third of all
satellites in low orbit. Seven satellites
were destroyed as radiation knocked
out their solar arrays or electronics,
including the first commercial
communication satellite ever, Telstar.
The flash created by the explosion
as seen through heavy cloud cover
from Honolulu 1,300 km away
Auroras
The colors are caused by energetic electrons colliding with
oxygen and nitrogen molecules in the atmosphere. This excites
the molecules, and when they decay from the excited states they
emit the light that we see in the aurora.
Aurora
surrounding
the north
geomagnetic
pole as seen
from space
Aurora as seen from space
Radius of Gyration
mv
r
qB
A charged particle from the solar wind traveling with a speed of
9.0x106 m/s encounters the Earth’s Magnetic Field at an altitude
where the field has a magnitude of 1.2x10-7 T. Assuming that the
particle’s velocity is perpendicular to the magnetic field, find the
radius of curvature of the path if it is a) a proton and b) an electron.
Great Demo: Wire JUMP
Magnetic Force on a Current Carrying Wire
F  qv  B
F  qvd B sin 
x
q
B sin 
t
q
 xB sin 
t
F  ILB sin 
OR:
Ftotal  qv d  B(#charges)
Ftotal  qv d  B(nAL)
I  nqvd A
ˆ
L  LL
F  IL  B
Magnetic Force on a
Current Carrying Wire
F  IL  B
ˆ
L  LL
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
Use Vector Notation!
F  IL  B
Use Vector Notation!
F  IL  B
HW 37. A rod of mass m and radius R rests on
two parallel rails that are a distance d apart and
have a length L. The rod carries a current I (in
the direction shown) and rolls along the rails
without slipping. A uniform magnetic field B is
directed perpendicular to the rod and the rails.
If it starts from rest, what is the speed of the
rod as it leaves the rails?
F  IL  B
Review! Review! Review!
Great Demo: Jumping Ring
The magnetic force on each bit of ring is radially inward and upward, at
angle  above the radial line. The radially inward components tend to
squeeze the ring but all cancel out as forces. The upward components add:
F  IL  B  dF  IdsB sin   F  IB sin   ds
F  I 2 rB sin  up
The Motor Effect: Torque
An electric current in a
magnetic field experiences
a force that is
perpendicular to both the
direction of current and the
direction of the field. If
that current is bent into the
shape of a loop, the
magnetic force produces a
torque that causes the loop
to rotate. This is the basis
of The Motor Effect.
F  IL  B
Torque on a Current Loop
• The rectangular loop
carries a current I in a
uniform magnetic field
• No magnetic force acts on
sides 1 & 3
– The wires are parallel to
the field and L x B = 0
F  IL  B
Torque on a Current Loop
• There is a force on sides 2
& 4 perpendicular to the
field
• The magnitude of the
magnetic force on these
sides will be:
F 2 = F4 = IaB
• The direction of F2 is out
of the page
• The direction of F4 is into
the page
F  IL  B
Torque on a Current Loop
• The forces are equal
and in opposite
directions, but not
along the same line of
action
• The forces produce a
torque around point O
F  IL  B
Torque on a Current Loop
• The maximum torque is
found by:
b
b
b
b
 F4  (I aB )  (I aB )
2
2
2
2
 I abB
τ max  F2
• The area enclosed by the
loop is ab, so
max = IAB
– This maximum value occurs only
when the field is parallel to the
plane of the loop
Torque on a Current Loop
• 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
= IAB sin 
τ  IAB sin 
τ  IA  B
Problem
τ  IAB sin 
τ  IA  B
A coil of wire has an area of 2.0x10-4 m2 consists of
100 loops or turns, and contains a current of 0.045 A.
The coil is in a uniform magnetic field of 0.15 T.
Find the maximum torque acting on the coil.
This maximum value occurs only when the
field is parallel to the plane of the loop
  NIAB sin 
  100(0.045 A)2 x104 m2 .015T sin 90
  1.35x10 Nm
4
F  IL  B
Magnetic Dipole Moment
• The product IA is defined as the
magnetic dipole moment, m, of the
loop
– Often called the magnetic moment
• SI units: A · m2
• Torque in terms of magnetic moment:
τ  μB
• Potential Energy of a magnetic
moment in an external mag field:
U  μ  B
ˆ
μ  IAA
ˆ
A  AA
Compare Dipoles
p  2aq
m  IA
τ  pE
τ  μ×E
U  -p  E
U  -μ  B
ˆ
A  AA
Review! Review! Review!
Torque is a Vector obeying the Right Hand Rule!!
23.
A rectangular coil consists of N = 100 closely wrapped turns and has
dimensions a = 0.400 m and b = 0.300 m. The coil is hinged along the y axis, and
its plane makes an angle θ = 30.0° with the x axis (Fig. P29.23). What is the
magnitude of the torque exerted on the coil by a uniform magnetic field B = 0.800
T directed along the x axis when the current is I = 1.20 A in the direction shown?
What is the expected direction of rotation of the coil?
B
τ  μB
ˆ
μ  IAA
The loop will rotate so as to align the magnetic moment with the B field.
Looking down along the y-axis, the loop will rotate in a clockwise direction.
23.
A rectangular coil consists of N = 100 closely wrapped turns and has
dimensions a = 0.400 m and b = 0.300 m. The coil is hinged along the y axis, and
its plane makes an angle θ = 30.0° with the x axis (Fig. P29.23). What is the
magnitude of the torque exerted on the coil by a uniform magnetic field B = 0.800
T directed along the x axis when the current is I = 1.20 A in the direction shown?
What is the expected direction of rotation of the coil?
  NBAI sin 


  100  0.800 T  0.400  0.300 m 2 1.20 A  sin 60
  9.98 N  m
Clockwise
The Motor Effect: Meters
The Motor Effect: Motors
The loop will rotate so as to align the magnetic moment
with the external magnetic field.
A
  IAB sin 
Fsin
Loop Width: w, Length: l
The Motor Effect: Motors
The loop will rotate so as to align the magnetic moment
with the external magnetic field.
A
  IAB sin 
Loop Width: w, Length: l
The Motor Effect: Motors
The loop will rotate so as to align the magnetic moment
with the external magnetic field.
  IAB sin 
A
Loop Width: w, Length: l
The Motor Effect: Motors
The loop will rotate so as to align the magnetic moment
with the external magnetic field.
The loop with oscillate
back and forth unless there
are brushes….
  IAB sin 
A
Torque Reverses!
KILL CURRENT!
LET MOMENTUM
CARRY LOOP!
Loop Width: w, Length: l
Motors: Brushes
The split-ring commutator keeps the current in the
proper direction to yield a torque that produces a
continuous rotation of the coil.
Split-ring commutator
Motors: Brushes
The split-ring commutator keeps the current in the
proper direction to yield a torque that produces a
continuous rotation of the coil.
Brushless DC Motors
In a BLDC motor, the electromagnets do not move; instead, the
permanent magnets rotate and the armature remains static. The
brush-system/commutator assembly is replaced by an intelligent
electronic solid-state circuit controller rather than a
commutator/brush system.
Spindle motor from a 3.5"
floppy disk drive
A BLDC motor powering a micro
remote-controlled airplane. The
motor is connected to a
microprocessor-controlled BLDC
controller.
The poles on the stator of a twophase BLDC motor. This is part
of a computer cooling fan; the
rotor has been removed.
Fun Problems!
Draw Vector Diagrams! Use Vector Notation!
The TOTAL FORCE
Electric & Magnetic Fields
The Electric Force acts parallel to the
Electric Field.
F  qE  qv  B
The Magnetic Force acts perpendicular
to the Magnetic Field and the velocity.
Cathode Ray Tube
Electric Force makes the Electron gun.
Magnetic Force directs the beam.
Cathode TV has 3
electron guns, one
for each color
RGB which scan
525 times in 1/30
of a second.
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
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
•
•
•
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
Bainbridge 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
In general, m/q can be determined by
measuring the radius of curvature and
knowing the magnitudes of the fields
JJ Thomson’s variation found e/me by
measuring the deflection of the beam
and compared it to mass-to-charge
ratio of protons, proving the existence
of the electron as a subatomic particle!
m rBo B

q
E
Hall Voltage
• This shows an
arrangement for observing
the Hall effect
• The Hall voltage is
measured between points
a and c
Hall Voltage
• When the charge carriers are negative, the upper edge of the
conductor becomes negatively charged
– c is at a lower potential than a
• When the charge carriers are positive, the upper edge becomes
positively charged
– c is at a higher potential than a
Hall Voltage
 VH = EHd = vd B d
– d is the width of the conductor
– vd is the drift velocity
– If B and d are known, vd can be found
I B RH I B
VH 

nqt
t
•
– RH = 1 / nq is called the Hall coefficient
– A properly calibrated conductor can be used to measure the
magnitude of an unknown magnetic field
The Hall Effect
When a current carrying conductor is placed in a magnetic field, the
magnetic force causes a separation of charge in the conductor which
produces a charge separation and voltage, VH, in a direction that is
perpendicular to both the current and the magnetic field.
VH  qE  qvd B
This Hall voltage produces an E field that opposes the Hall
Effect and tries to push the electrons back up. Eventually,
balance is reached. By measuring the voltage you can use a
hall probe to measure the magnetic field.
Earth’s interior consists of a rocky
mantle and an iron rich core
The Earth’s magnetic field is caused by
dynamo movements in Earth’s core
Earth’s magnetic field flips
poles. Consecutive reversals
are spaced 5 thousand years
to 50 million years apart. The
last reversal happened
740,000 years ago. Some
researchers think our planet is
overdue for another one, but
nobody knows exactly when
the next reversal might occur.
The Sun's north magnetic pole is
pointing through the Sun's
southern hemisphere, until the
year 2012 when they will reverse.
This transition happens, as far as
we know, at the peak of every 11year sunspot cycle -- like
clockwork.
Babcock’s magnetic dynamo is one possible explanation of the
sunspot cycle where magnetic field lines become complexly
entangled after many solar rotations
The Sun’s magnetic fields create sunspots
Zeeman effect - spectral
lines split in regions of high
magnetic fields
Magnetic field lines connect sunspots on the Sun’s photosphere
Solar magnetic fields also create
other atmospheric phenomena
• plages
• filaments
Solar magnetic fields also create
other atmospheric phenomena
• plages
• filaments
• prominences
Solar magnetic fields also create
other atmospheric phenomena
•
•
•
•
plages
filaments
prominences
solar flares
Solar magnetic fields also create
other atmospheric phenomena
•
•
•
•
•
plages
filaments
prominences
solar flares
coronal holes
Solar magnetic fields also create
other atmospheric phenomena
•
•
•
•
plages
filaments
prominences
solar flares
• coronal holes
• coronal mass
ejections
(CMEs)
NEXT: Currents
Produce Magnetic Fields Too!
Smallest Magnet:
Electron
Magnetic Fields due to Currents
Magnitude of the Field:
m0 I
B
2 r
m0  4 x10 T  m / A
7
“mew not” : The permeability of free space.
Permeability is a measure of a materials ability to be
permeated by magnetic fields – ferromagnetic materials
have a higher permeability.
This Week’s Lab
Using a Tangent Galvanometer to
Measure the Earth’s Magnetic Field
Magnetic Field at the center of the loop:
m0 I
B
2 r
Horizontal
Earth Field
Total Field