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Magnetism
The Magnetic Force
r
r r
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
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
• 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
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
F mag
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:
r
r
r
r
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
B
v
q
F=0
Question
Three points are arranged in a uniform
magnetic field. The B field points into
the screen.
r
r
Magnetic Force: F  qv  B
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
e) zero
If v = 0  F = 0.
The positive charge moves from point A toward B. The direction
r
of the magnetic force on the particle is:
If v ^ B
a) right
b) left
d) out of the screen
then F = qvB
c) into the screen
If v is up, and B is into the
e) zero page, then F is to the left.
Question:
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
r r
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.
Question
• Two independent protons each move at
speed v (as 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.
– 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
z
(c) F1 > F2
• The magnetic force is given by:
r
r
r
F  q v  B  F  qvB sinq
• In both cases the angle between v and B is 90!!
Therefore F1 = F2.
v
x
Question
• Two independent protons each move at
speed v (as 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.
– What is F2x, the x-component of the
force on the second proton?
(a) F2x < 0
(b) F2x = 0
F1
F2
y
1
v
B
2
v
z
x
(c) F2x > 0
• To determine the direction of the force, we use the
right-hand rule.
r r
F  qv  B
• As shown in the diagram, F2x < 0.
Question
• Two protons each move at speed v (as
shown in the diagram) in a region of space
which contains a constant B field in the -zdirection. 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
r experience a force (which deflects
it), this is always
to v. Therefore, there is no possibility to
r
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 !
Question:
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.
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
Question
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
Question
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
Question
L
• 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
r r
Definition of work W  F gdx

The magnetic force is:
r
r
r
F  qv  B
• Therefore, the work done is ZERO in each case:
r r
r r
 F gdx   F gv dt  0
Charges in a conductor - current
Now you know how a single charged particle moves in a
magnetic field, what about a group?
For a piece of a conductor we know there are
n
number of charges per unit volume
A
area of the conductor
dl
r
vd
length of the element
dl
vd
n
drift velocity of a charge
then
r r
r
r
r
FB  nAdl qvd  B  I dl  B
r r
r r
 I  dl  B  I L  B


A
I
Implications
To get the sum of a number of vectors - put them all head to
tail and connect the initial (a) and final point (b).
So
b
r r
 dl  Lba
b
a
a
If the initial and final points are the same, the integral is zero!
That is, the net magnetic force on a closed current
loop in a uniform magnetic field is zero!
BUT……..
Torque
Although the net magnetic force acting on a closed current
loop in a uniform magnetic field is zero, the forces are not
In the same place, so there can be a net torque.
F
Loop has length dimension a into the page
(and normal to B)
Loop has width b.
A
q
B
X
F
b
  2F sin q  I abBsin q
2
r r
  IA B
r
Define the magnetic moment   I A
r r r
 B
r
Question
Each of the two turns of a circular loop of wire conductor
(diameter 20 cm) carries a current of I = 2 amps. If it is placed in
a magnetic field of 0.1 T at 450 to the plane of the loop, the torque
is;
a) 22.3 x10-4 N-m
b) 44.6 x10-4 N-m
c) 88. x10-4 N-m
   Bsin q   B  0.1T sin q  0.71
  I A  2 turns 2 amps  0.01 m 2 
  0.126 amp gm 2
  89.  10 4 Ngm
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
VH
q
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.