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Magnetism
The Magnetic Force


 
F  qE  qv  B
B
x x x x x x
x x x x x x
v
x x x x x x
F q
B
B

v

 q
F
v
q
F=0
Today...
• Introduction to Magnetic Phenomena
• Bar magnets & Magnetic Field Lines
• Source of Fields: Monopoles? Currents?
• Zip disks and refrigerators
• Magnetic forces: The Lorentz Force equation
• Motion of charged particle in a Constant Magnetic
Field.
Magnetism
• Magnetic effects from natural magnets have been known
for a long time. Recorded observations from the Greeks
more than 2500 years ago.
• The word magnetism comes from the Greek word for a
certain type of stone (lodestone) containing iron oxide
found in Magnesia, a district in northern Greece.
• Properties of lodestones: could exert forces on similar
stones and could impart this property (magnetize) to a
piece of iron it touched.
• Small sliver of lodestone suspended with a string will
always align itself in a north-south direction—it detects
the earth’s magnetic field.
Bar Magnet
• Bar magnet ... two poles: N and S
Like poles repel; Unlike poles attract.
• Magnetic Field lines: (defined in same way as electric
field lines, direction and density)
S
N
You will map this
field (and others)
in lab !!
• Does this remind you of a similar case in electrostatics?
Electric Field Lines
of an Electric Dipole
Magnetic Field Lines of
a bar magnet
S
N
Magnetic Monopoles
• Perhaps there exist magnetic charges, just like electric
charges. Such an entity would be called a magnetic
monopole (having + or - magnetic charge).
• How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
S
N
S
N
S
N
Even an individual
electron has a
magnetic “dipole”!
• Many searches for magnetic monopoles—the existence of
which would explain (within framework of QM) the
quantization of electric charge (argument of Dirac)
• No monopoles have ever been found: ˜ B  dS  0
Source of Magnetic Fields?
• What is the source of magnetic fields, if not magnetic
charge?
• Answer: electric charge in motion!
– e.g., current in wire surrounding cylinder (solenoid)
produces very similar field to that of bar magnet.
• Therefore, understanding source of field generated by bar
magnet lies in understanding currents at atomic level
within bulk matter.
Orbits of electrons about nuclei
Intrinsic “spin” of
electrons (more
important effect)
Magnetic Materials
(a simple look at an advanced topic)
• Materials can be classified by how they respond to an
applied magnetic field, Bapp.
• Paramagnetic (aluminum, tungsten, oxygen,…)
• Atomic magnetic dipoles (~atomic bar magnets) tend to line up
with the field, increasing it. But thermal motion randomizes
their directions, so only a small effect persists: Bind ~ Bapp •10-5
• Diamagnetic (gold, copper, water,…)
• The applied field induces an opposing field; again, this is
usually very weak; Bind ~ -Bapp •10-5 [Exception: Superconductors
exhibit perfect diamagnetism  they exclude all magnetic fields]
• Ferromagnetic (iron, cobalt, nickel,…)
• Somewhat like paramagnetic, the dipoles prefer to line up with
the applied field. But there is a complicated collective effect
due to strong interactions between neighboring dipoles 
they tend to all line up the same way.
• Very strong enhancement. Bind ~ Bapp •10+5
Ferromagnets, cont.
• Even in the absence of an applied B, the dipoles tend to
strongly align over small patches – “domains”.
Applying an external field, the domains align to produce
a large net magnetization.
Magnetic
Domains
• “Soft” ferromagnets
• The domains re-randomize when the field is removed
• “Hard” ferromagnets
• The domains persist even when the field is removed
• “Permanent” magnets
• Domains may be aligned in a different direction by applying
a new field
• Domains may be re-randomized by sudden physical shock
1
• If the temperature is raised above the “Curie point” (770˚
for iron), the domains will also randomize  paramagnet
Lecture 12, Act 1
1A
1B
•Which kind of material would you use in a video tape?
(a) diamagnetic
(c) “soft” ferromagnetic
(b) paramagnetic
(d) “hard” ferromagnetic
•How does a magnet attract screws, paper clips,
refrigerators, etc., when they are not “magnetic”?
Lecture 12, Act 1
1A
•Which kind of material would you use in a video tape?
(a) diamagnetic
(c) “soft” ferromagnetic
(b) paramagnetic
(d) “hard” ferromagnetic
Diamagnetism and paramagnetism are far too weak to be
used for a video tape. Since we want the information to
remain on the tape after recording it, we need a “hard”
ferromagnet. These are the key to the information age—
cassette tapes, hard drives, ZIP disks, credit card strips,…
Lecture 12, Act 1
•How does a magnet attract screws, paper clips,
refrigerators, etc., when they are not “magnetic”?
1B
The materials are all “soft” ferromagnets. The external field
temporarily aligns the domains so there is a net dipole,
which is then attracted to the bar magnet.
- The effect vanishes with no applied B field
- It does not matter which pole is used.
S
N
End of paper clip
A “bit” of history
IBM introduced the
first hard disk in 1957,
when data usually
was stored on tapes.
It consisted of 50
platters, 24 inch
diameter, and was
twice the size of a
refrigerator.
It cost $35,000 annually in leasing fees (IBM
would not sell it outright). It’s total storage
capacity was 5 MB, a huge number for its time!
Magnetic Fields
We know about the existence of magnetic fields by their
effect on moving charges. The magnetic field exerts a force
on the moving charge.
• What is the "magnetic force"? How is it distinguished from
the "electric" force?
Let’s start with some experimental observations about
the magnetic force:
a) magnitude:  to velocity of q
b) direction: ^ to direction of q’s velocity
q
v
c) direction: ^ to direction of B
Fmag
B is the magnetic field vector
Lorentz Force
• The force F on a charge q moving with velocity v
through a region of space with electric field E and
magnetic field B is given by:




F  qE  qv  B
B
x x x x x x
B

x x x x x x
v
x x x x x x
q
F
v

 q
F
2
What’s that??
B
v
q
F=0
Preflight 12:
Three points are arranged in a uniform
magnetic field. The B field points into
the screen.


Magnetic Force: F  qv  B
1) A positively charged particle is located at point A and is stationary.
The direction of the magnetic force on the particle is:
a) right
b) left
d) out of the screen
c) into the screen
If v = 0  F = 0.
e) zero
2) The positive charge moves from point A toward B. The direction of
the magnetic force on the particle is:
 
If
a) right
b) left
d) out of the screen
v^B
then F = qvB
c) into the screen
e) zero
If v is up, and B is into the
page, then F is to the left.
Preflight 12:
3) The positive charge moves from point A toward C. The direction
of the magnetic force on the particle is:
a) up and right
b) up and left
c) down and right
d) down and left

 
Magnetic Force: F  qv  B
If v is up and to the right, it is still perpendicular to B,
hence F = qvB then and F is up and to the left.
Lecture 12, Act 2
• Two protons each move at speed v (as
shown in the diagram) in a region of
space which contains a constant B field
2A
in the -z-direction. Ignore the interaction
between the two protons.
– What is the relation between the
magnitudes of the forces on the two
protons?
(a) F1 < F2
2B
1
v
B
2
z
v
x
(c) F1 > F2
– What is F2x, the x-component of the force on the second
proton?
(a) F2x < 0
2C
(b) F1 = F2
y
(b) F2x = 0
(c) F2x > 0
– Inside the B field, the speed of each proton:
(a) decreases
(b) increases
(c) stays the same
Lecture 12, Act 2
• Two independent protons each move at
speed v (as shown in the diagram) in a
region of space which contains a
2A
constant B field in the -z-direction.
Ignore the interaction between the two
protons.
– What is the relation between the
magnitudes of the forces on the two
protons?
(a) F1 < F2
(b) F1 = F2
y
1
v
B
2
v
z
(c) F1 > F2
• The magnetic force is given by:



F  q v  B  F  qvB sin θ
• In both cases the angle between v and B is 90!!
Therefore F1 = F2.
x
Lecture 12, Act 2
• Two independent protons each move at
speed v (as shown in the diagram) in a
region of space which contains a
2B
constant B field in the -z-direction.
Ignore the interaction between the two
protons.
– What is F2x, the x-component of the
force on the second proton?
(a) F2x < 0
(b) F2x = 0
F2
F1
y
1
v
B
2
v
z
x
(c) F2x > 0
• To determine the direction of the force, we use the
right-hand rule.

 
F  qv  B
• As shown in the diagram, F2x < 0.
Lecture 12, Act 2
• Two protons each move at speed v (as
2C
shown in the diagram) in a region of
space which contains a constant B field
in the -z-direction. Ignore the interaction
between the two protons.
– Inside the B field, the speed of each
proton:
(a) decreases
(b) increases
y
1
v
B
2
z
v
x
(c) stays the same
Although the proton does experience a force
(which deflects it), this is always ^ to v .
Therefore, there is no possibility to do work, so
kinetic energy is constant and v is constant.
Trajectory in
Constant B Field
• Suppose charge q enters B-field with velocity v as
shown below. What will be the path q follows?
x x x x x x x x x x x x
x x x x x x x x x x x vx B
x x x x x x x x x x x x
q
v
F
F
R
• Force is always ^ to velocity and B. What is path?
– Path will be circle. The magnetic force provides the centripetal
force needed to keep the charge in its circular orbit. Calculate
R:
Radius of Circular Orbit
• Lorentz force:
F  qvB
x x x x x x x x x x x x
• centripetal acc:
v2
a 
R
• Newton's 2nd Law:
F  ma 

x x x x x x x x x x x vx B
x x x x x x x x x x x x
v
F
F q
R
v2
qvB  m
R
mv
R
qB
This is an important result,
with useful experimental
consequences !
Ratio of charge to mass
for an electron
e-
1) Turn on electron ‘gun’
1
mv
2
2
R
 qV
DV
2) Turn on magnetic field B
‘gun’
mv
R 
qB
3) Calculate B … next week; for now consider it a measurement
4) Rearrange in terms of measured values, V, R and B
q
v 2  2V
m

and
q
2V
 2 2
m R B
 q

v 2   RB 
m

2
Let’s Try It...
1) Do the expt. Adjust I and V to get a good circle
2) Measure R = .05 m
3) What was V? V = 230 V
B
8 m 0 NI
 7.8  10 - 4  I for our coils
5 5r
4) How about B? B ~10-3 T
q
2V
= 2 2 ≈ 1.8 1011 C/kg
m R B
(
e
me
= 1.761011 C/kg )
3
Preflight 12:
The drawing below shows the top
view of two interconnected chambers.
Each chamber has a unique magnetic
field. A positively charged particle is
fired into chamber 1, and observed to
follow the dashed path shown in the
figure.
5) What is the direction of the magnetic field in chamber 1?
a) Up
b) Down
c) Left
d) Right
e) Into page
f) Out of page
Preflight 12:
6) What is the direction of the magnetic field in chamber 2?
a) Up
b) Down
c) Left
d) Right
e) Into page
f) Out of page
In chamber 1, the velocity is initially up. Since the
particle’s path curves to the right, the force is to the
right as the particle enters the chamber.
Three ways to figure out the direction of B from
this:
1) Put your thumb in the direction of the F (right) and your fingers in the
direction of v (up) The way that your fingers curl is the direction of B.
2) Put your palm in the direction of F (right), and your thumb in the
direction of v (up), your fingers (keep them straight) point in the direction
of B.
3) Keep your thumb, index and middle fingers at right angles from each
other. Your thumb points in the direction of v (up), middle finger points
towards F (right), then the index finger gives the the direction of B (out of
page)
Preflight 12:
8) Compare the magnitude of the magnetic field in chamber 1 to the
magnitude of the magnetic field in chamber 2.
a) B1 > B2
b) B1 = B2
c) B1 < B2
The magnetic force is always perpendicular to v.
The force doesn’t change the magnitude of v, it
only changes the particle’s direction of motion.
The force gives rise to a centripetal acceleration.
The radius of curvature is given by:
R
mv
qB
L
Lecture 12, Act 3
• A proton, moving at speed v, enters a
region of space which contains a
constant B field in the -z-direction and is
deflected as shown.
B
v
• Another proton, moving at speed v1 = 2v,
enters the same region of space and is
deflected as shown.
B
B
v1
B
– Compare the work done by the magnetic field
(W for v, W1 for v1) to deflect the protons.
(a) W1 < W
(b) W1 = W
v
(c) W1 > W
v1
L
Lecture 12, Act 3
• A proton, moving at speed v, enters a
region of space which contains a
constant B field in the -z-direction and is
deflected as shown.
v
B
v
• Another proton, moving at speed v1 = 2v,
enters the same region of space and is
deflected as shown.
B
B
v1
v1
B
– Compare the work done by the magnetic field
(W for v, W1 for v1) to deflect the protons.
(a) W1 < W
(b) W1 = W
(c) W1 > W


• Remember that the work done W is defined as: W  F  dx

• Also remember that the magnetic force is always perpendicular to the

 
velocity:
F  qv  B
 
 
• Therefore, the work done is ZERO in each case:
 F  dx   F  vdt  0
Summary
• Lorentz force equation:
   
F  qE  qv  B
– Static B-field does no work
– Velocity-dependent force given by right hand rule
formula
• Next time: magnetic forces and dipoles
The Hall Effect
• Which charges carry current?
• Positive charges moving
• Negative charges moving clockwise
counterclockwise experience
experience upward force
upward force
• Upper plate at higher potential
• Upper plate at lower potential
Equilibrium between electrostatic & magnetic forces:
Fup  qvdrift B
Fdown  qEinduced  q
VH
w
VH  vdrift Bw  "Hall Voltage"
• This type of experiment led to the discovery (E. Hall, 1879) that current in
conductors is carried by negative charges (not always so in semiconductors).
• Can be used as a B-sensor; used in some ABS to detect shaft rotation speed –
ferromagnetic rotating blades interupt the magnetic field  oscillating voltage