Prelude to an Exam
Allegro con brio



Next Friday – EXAMINATION #2
New WebAssign on Board on Magnetism
TWO Old WebAssigns due this week
• No Extensions

Please take a look at the short memo on
vector multiplication that was put on
the website earlier … we will no start to
use this material.
Magnetism
1
Magnetism
A Whole New Topic
October 18, 2005
Magnetism
2
DEMO
Magnetism
3
Lodestone (Mineral)
• Lodestones attracted
iron filings.
• Lodestones seemed to
attract each other.
• Used as a compass.
– One end always
pointed north.
• Lodestone is a natural
magnet.
Magnetism
4
Magnetism
• Refrigerators are attracted to magnets!
Magnetism
5
Applications
• Motors
• Navigation – Compass
• Magnetic Tapes
– Music, Data
• Television
– Beam deflection Coil
• Magnetic Resonance Imaging
• High Energy Physics Research
Magnetism
6
Magnets
S N
Shaded End is NORTH Pole
Shaded End of a compass points
to the NORTH.
Magnetism
• Like Poles Repel
• Opposite Poles
Attract
• Magnetic Poles are
only found in pairs.
– No magnetic
monopoles have
ever been
observed.
7
Observations
• Bring a magnet to an electrically charged
object and nothing happens. No forces.
• Bring a magnet near some metals (Co, Fe, Ni
…) and it will be attracted to the magnet.
– The metal will be attracted to both the N and S
poles independently.
– Some metals are not attracted at all.
– Wood is NOT attracted to a magnet.
– Neither is water.
• A magnet will force a compass needle to align
with it. (No big Surprise.)
Magnetism
8
Magnets
Cutting a bar magnet in half produces TWO bar
magnets, each with N and S poles.
Magnetism
9
Consider a Permanent Magnet

B
N
Magnetism
S
10
Introduce Another Permanent Magnet

B
N
N
S
pivot
S
The bar magnet (a magnetic dipole) wants to align with the B-field.
Magnetism
11
Field of a Permanent Magnet

B
N
N
S
S
The south pole of the small bar magnet is attracted towards
the north pole of the big magnet.
Also, the small bar magnet (a magnetic dipole) wants to align
with the B-field.
The
field attracts and exerts a torque on the small magnet.
Magnetism
12
Field of a Permanent Magnet

B
N
N
S
S
The bar magnet (a magnetic dipole) wants to align with the B-field.
Magnetism
13
The Magnetic Field
• Similar to Electric Field … exists in
space.
– Has Magnitude AND Direction.
• The “stronger” this field, the greater is
the ability of the field to interact with
a magnet.
Magnetism
14
Convention For Magnetic Fields
X
Field INTO Paper
Magnetism
B

Field OUT of Paper
15
Experiments with Magnets Show
• Current carrying wire produces a
circular magnetic field around it.
• Force on Compass Needle (or magnet)
increases with current.
Magnetism
16
Current Carrying Wire
Current into
the page.
B
Right hand RuleThumb in direction of the current
Fingers curl in the direction of B
Magnetism
17
Current Carrying Wire
• B field is created at ALL POINTS in space
surrounding the wire.
• The B field had magnitude and direction.
• Force on a magnet increases with the
current.
• Force is found to vary as ~(1/d) from the
wire.
Magnetism
18
Compass and B Field
• Observations
– North Pole of magnets
tend to move toward
the direction of B while
S pole goes the other
way.
– Field exerts a
TORQUE on a
compass needle.
– Compass needle is a
magnetic dipole.
– North Pole of
compass points
toward the NORTH.
Magnetism
19
Planet Earth
Magnetism
20
Inside it all.
8000
Miles
Magnetism
21
On the surface it looks like this..
Magnetism
22
Inside: Warmer than Floriduh
Magnetism
23
Much Warmer than Floriduh
Magnetism
24
Finally
Magnetism
25
In Between






The molten iron core exists in a magnetic
field that had been created from other
sources (sun…).
The fluid is rotating in this field.
This motion causes a current in the molten
metal.
The current causes a magnetic field.
The process is self-sustaining.
The driving force is the heat (energy) that
is generated in the core of the planet.
Magnetism
26
After molten lava emerges from a volcano, it solidifies to a rock. In
most cases it is a black rock known as basalt, which is faintly
magnetic, like iron emerging from a melt. Its magnetization is in the
direction of the local magnetic force at the time when it cools down.
Instruments can measure the magnetization of basalt. Therefore, if
a volcano has produced many lava flows over a past period, scientists
can analyze the magnetizations of the various flows and from them
get an idea on how the direction of the local Earth's field varied in
the past. Surprisingly, this procedure suggested that times existed
when the magnetization had the opposite direction from today's. All
sorts of explanation were proposed, but in the end the only one
which passed all tests was that in the distant past, indeed, the
magnetic polarity of the Earth was sometimes reversed.
Magnetism
27
Ancient Navigation
Magnetism
28
This planet is really screwed up!
NORTH
POLE
Magnetism
SOUTH POLE
29
Repeat
Navigation
DIRECTION
N
S
If N direction
is pointed to by
the NORTH pole
of the Compass
Needle, then the
pole at the NORTH
of our planet must
be a SOUTH MAGNETIC
POLE!
Compass
Direction
Navigation
DIRECTION
S
N
And it REVERSES from time to time.
Magnetism
30
Magnetism
31
Rowland’s Experiment
Field is created by
any moving charge.
Rotating
INSULATING
Disk
which is
CHARGED
+ or –
on exterior.
++
Magnetism
+ +
++
xxx
xxx B
xxx
Increases with
charge on the
disk.
Increases with
angular velocity of
the disk.
Electrical curent is a
moving charge.
32
A Look at the Physics

B
q

v

q B
There is NO force on
a charge placed into a
magnetic field if the
charge is NOT moving.
There is no force if the charge
moves parallel to the field.
• If the charge is moving, there
is a force on the charge,
perpendicular to both v and B.
F=qvxB
Magnetism
33
WHAT THE HECK IS
THAT???
• A WHAT PRODUCT?
• A CROSS PRODUCT – Like an
angry one??
• Alas, yes ….
• F=qv X B
Magnetism
34
The Lorentz Force
This can be summarized as:

 
F  qv  B
F
or:
F  qvBsin 
v
B
mq
 is the angle between B and V
Magnetism
35
Note
B is sort of the Force per unit
(charge-velocity)
Whatever that is!!
Magnetism
36
Practice
B and v are parallel.
Crossproduct is zero.
So is the force.
Which way is the Force???
Magnetism
37
Units
F  Bqv Sin(θ )
Units :

F
N
N
B


qv Cm / s Amp  m
Magnetism
1 tesla  1 T  1 N/(A - m)
38
teslas are
At the Surface of the Earth
3 x 10-5 T
Typical Refrigerator Magnet
5 x 10-3 T
Laboratory Magnet
0.1 T
Large Superconducting Magnet
10 T
Magnetism
39
The Magnetic Force is Different
From the Electric Force.
Whereas the electric force
acts in the same direction as
the field:
The magnetic force acts in a
direction orthogonal to the
field:


F  qE



F  qv  B
(Use “Right-Hand” Rule to
determine direction of F)
And
--the
charge
must
be
moving
!!
Magnetism
40
So…
A moving charge can create a magnetic
field.
A moving charge is acted upon by a
magnetic field.
In Magnetism, things move.
In the Electric Field, forces and the
field can be created by stationary
charges.
Magnetism
41
Trajectory of Charged Particles
in a Magnetic Field
(B field points into plane of paper.)
+
+B
+
v+
+
+
+
+
+
+
+
+ F
+
+
+
+ F +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
B
+
+
+
+
+
Magnetism
v
42
Trajectory of Charged Particles
in a Magnetic Field
(B field points into plane of paper.)
+
+B
+
v+
+
+
+
+
+
+
+
+ F
+
+
+
+ F +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
B
+
+
+
+
+
Magnetism
v
Magnetic Force is a centripetal force
43
Review of Rotational Motion

 = s / r  s =  r  ds/dt = d/dt r  v =  r
s
r
 = angle,  = angular speed,  = angular acceleration

at
ar
at = r 
tangential acceleration
ar = v2 / r radial acceleration
The radial acceleration changes the direction of motion,
while the tangential acceleration changes the speed.
Uniform Circular Motion
ar
 = constant  v and ar constant but direction changes

Magnetism
v
ar = v2/r = 2 r
KE = ½ mv2 = ½ mw2r2
F = mar = mv2/r = m2r
44
Magnetism
45
Radius of a Charged Particle
Orbit in a Magnetic Field
+B
+
+
v+
+
+
+
+
+
+
+
+
+
r
+
+
+
+
+
+
F
+
Magnetism
Centripetal
Force
=
Magnetic
Force
mv 2

 qvB
r

mv
r
qB
 
Note: as Fv , the magnetic
46
force does no work!
Cyclotron Frequency
+B
+
+
v+
+
+
+
+
+
+
+
+
+
r
+
+
+
+
+
+
F
+
Magnetism
The time taken to complete one
orbit is:
2r
v
2 mv

v qB
T 
1
qB
f  
T 2 m
qB
 c  2f 
m
47
More Circular Type Motion in a
Magnetic Field
Magnetism
48
Mass Spectrometer
Smaller Mass
Magnetism
49
Magnetism
50
An Example
A beam of electrons whose kinetic energy is K emerges from a thin-foil
“window” at the end of an accelerator tube. There is a metal plate a distance d
from this window and perpendicular to the direction of the emerging beam. Show
that we can prevent the beam from hitting the plate if we apply a uniform
magnetic field B such that
2mK
B
2 2
ed
Magnetism
51
Problem Continued
From Before
r
mv
r
qB
1 2
2K
K  mv so v 
2
m
m 2K
2mK
r

d
2 2
eB m
e B
Solve for B :
Magnetism
2mK
B
e2d 2
52
Let’s Look at the effect of crossed E and B Fields:
x x x B
E
x x x
v
q , m
Magnetism
•
53
What is the relation between the intensities of the electric and
magnetic fields for the particle to move in a straight line ?.
x x x B
E
x x x
v
q• m
FE = q E and FB = q v B
If FE = FB the particle will move
following a straight line trajectory
qE=qvB
v=E/B
FB FE
•
Magnetism
54
What does this mean??
v=E/B
Magnetism
This equation only
contains the E and
B fields in it.
Mass is missing!
Charge is missing!
This configuration
is a velocity filter!
55
“Real” Mass Spectrometer

Create ions from injected species.
This will contain various masses, charges
and velocities.
 These are usually accelerated to a certain
ENERGY (KeV) by an applied electric
field.
 The crossed field will only allow a
selected velocity to go forward into the
MS.
 From before: R=mv/Bq

Magnetism
56
Components of MS:
Accelerate the ions through a known potential difference .
1 2
mv  qVapplied
2
So
q 1 2 1
 v
m 2 Vapplied
The velocity can be selected via an E x B field and the MS will
separate by:
mv
R
Bq
Magnetism
Unknown is mass to charge ratio
which can be sorted from the spectrum
57
Magnetism
58
PHYSICS NEWS
• Exam will be available Friday or Monday
(depending on grader).
• QUIZ on Friday – Magnetism through
today’s work.
• Exam #3 is 4 weeks from today!
• FINAL EXAM 1.5 weeks after that.
• ADVICE: Start to study old material
NOW!
Magnetism
59
Magnetism
60
REMEMBER The Lorentz Force
This can be summarized as:

 
F  qv  B
F
or:
F  qvBsin 
v
B
mq
 is the angle between B and V
Magnetism
61
VECTOR CALCULATIONS
i
a  b  ax
bx
Magnetism
j
ay
by
k
az
bz
62
Wires
• A wire with a current
contains moving charges.
• A magnetic field will
apply a force to those
moving charges.
• This results in a force
on the wire itself.
– The electron’s sort of
PUSH on the side of the
wire.
F
Remember: Electrons go the “other way”.
Magnetism
63
The Wire in More Detail
Assume all electrons are moving
with the same velocity vd.
q  it  i
L
vd
F  qvd B  i
L
vd B  iLB
vd
vector :
F  iL  B
L in the direction of the motion
of POSITIVE charge (i).
B out of plane of the paper
Magnetism
64
Magnetic Levitation
Magnetic Force
mg
Where does B point????
Into the paper.
Current = i
iLB  mg
mg
B
iL
m/l is a constant for the wire.
Magnetism
65
MagLev
Magnetism
66
Magnetic Repulsion
Magnetism
67
Detail
Magnetism
68
Moving Right Along ….
Magnetism
69
Acceleration
Magnetism
70
Don’t Buy A Ticket Quite Yet..
This is still experimental.
Much development still required.
Some of these attempts have been
abandoned because of the high cost
of building a MagLev train.
Probably 10-20 years out.
Or More.
Magnetism
71
Current Loop
What is force
on the ends??
Loop will tend to rotate due to the torque the field applies to the loop.
Magnetism
72
The Loop
OBSERVATION
Force on Side 2 is out
of the paper and that on
the opposite side is into
the paper. No net force
tending to rotate the loop
due to either of these forces.
The net force on the loop is
also zero,
pivot
Magnetism
73
An Application
The Galvanometer
Magnetism
74
The other sides
t1=F1 (b/2)Sin()
=(B i a) x (b/2)Sin()
total torque on
the loop is: 2t1
Total torque:
t=(iaB) bSin()
=iABSin()
(A=Area)
Magnetism
75
A Coil
For a COIL of N turns, the net
torque on the coil is therefore :
τ  NiABSin(θ )
Normal to the
coil
RIGHT HAND RULE TO FIND NORMAL
TO THE COIL:
“Point or curl you’re the fingers of your right
hand in the direction of the current and your
thumb will point in the direction of the normal
to the coil.
Magnetism
76
Dipole Moment Definition
Define the magnetic
dipole moment of
the coil m as:
m=NiA
Magnetism
We can convert this
to a vector with A
as defined as being
normal to the area as
in the previous slide.
77
Current Loop
t  iAB sin 
Consider a coil with N turns of wire.
Define Magnetic Moment
μ  NiA
and
t  μB
Magnetism
78
Verify the Direction
τ  μB
m
Magnetism
B
79
A length L of wire carries a current i. Show that if the wire is
formed into a circular coil, then the maximum torque in a given
magnetic field is developed when the coil has one turn only, and
that maximum torque has the magnitude … well, let’s see.
Circumference = L/N
Magnetism
L
2r 
N
L
r
2N
80
Problem continued…
t  NiAB since sin( m , B) is maximum
when the angle is 90o
A  r 2
 L 
t  NiB 

 2N 
2
2
L N 
L2
  iB 
t  iB 
....... (BiA)
2

4 N
 2N 
Maximum when N  1 and
Magnetism
iBL2
t
4
81
So Far
 Moving Charges experience forces due
to magnetic field.
 We showed in the demo that a compass
placed near a current carrying wire was
affected.
 The current carrying wire therefore
creates a magnetic field.
 Current is affected by a B field and at the
same time creates a B field of its own!
Magnetism
82
Try to remember…
1
rdq
 r  dq
dE 
  2 
3


40  r  r
40 r
1
r
 UNIT VECTOR
r
Magnetism
83
The “Coulomb’s Law of Magnetism
A Vector Equation … duck
Magnetism
84
For the Magnetic Field,
current “elements” create the
field. This is the Law of
Biot-Savart
In a similar fashion to E field :
m 0 ids  runit m 0 ids  r
B

2

4
r
4  r 3
permeabili ty
m 0  4 10 7 Tm / A  1.26 10 7 Tm
BY DEFINITION
Magnetism
85
Magnetic Field of a Straight
Wire
• We intimated via magnets that the
Magnetic field associated with a
straight wire seemed to vary with 1/d.
• We can now PROVE this!
Magnetism
86
From the Past
Using Magnets
Magnetism
87
Right-hand rule: Grasp the
element in your right hand with
your extended thumb pointing
in the direction of the current.
Your fingers will then naturally
curl around in the direction of
the magnetic field lines due to
that element.
Magnetism
88
Let’s Calculate the FIELD
Note:
For ALL current elements
ds X r
is into the page
Magnetism
89
The Details
m 0 ids sin(  )
dB 
4
r2
Negative portion of the wire
contribute s an equal amount so we
integrate from 0 to  and DOUBLE it.

m 0i sin(  )ds
B
2 0
r2
Magnetism
90
Moving right along
r  s R
2
2
sin   sin(    ) 
R
s2  R2
So

m 0i
m 0i
Rds
B

3
/
2

2 0 s 2  R 2 
2R
Magnetism
1/d
91
A bit more complicated
A finite wire
Magnetism
92
P1
NOTE : sin(  )  sin(    )
ds  r  ds r sin(  )
r


ds
R
sin(  ) 
r
m 0i ds sin(  )
dB 
2
4
r

r s R
Magnetism
2

2 1/ 2
93
More P1
L/2
m 0i
ds
B
3/ 2

2
2
4  L / 2 s  R 
and
m 0i
L
B
2R L2  4 R 2
when L  ,
Magnetism
m 0i
B
2R
94
P2
m 0iR
ds
B
4 L s 2  R 2 3 / 2
0
or
m 0i
L
B
4R s 2  R 2
Magnetism
95
APPLICATION:
Find the magnetic field B at point P in for i = 10 A and a = 8.0
cm.
Magnetism
96
Circular Arc of Wire
Magnetism
97
More arc…
ds
ds  Rd 
m 0 ids m 0 iRd 
dB 

2
4 R
4 R 2


m 0 iRd  m 0i
B   dB  

d
2

4 R
4R 0
0
m 0 i
B
at point C
4R
Magnetism
98
Howya Do Dat??
ds r  0
No Field at C
Magnetism
99
Force Between Two Current
Carrying Straight Parallel
Conductors
Wire “a” creates
a field at wire “b”
Magnetism
Current in wire “b” sees a
force because it is moving
in the magnetic field of “a”.
100
The Calculation
The FIELD at wire " b" due to
wire " a" is what we just calculated :
m 0ia
Bat "b" 
2d
Fon "b"  ib L  B
Since L and B are at right angles...
m 0 Lia ib
F
2d
Magnetism
101
Definition of the Ampere
The force acting between currents in parallel
wires is the basis for the definition of the
ampere, which is one of the seven SI base
units. The definition, adopted in 1946, is
this: The ampere is that constant current
which, if maintained in two straight, parallel
conductors of infinite length, of negligible
circular cross section, and placed 1 m apart
in vacuum, would produce on each of these
conductors a force of magnitude 2 x 10-7
newton per meter of length.
Magnetism
102
TRANSITION
AMPERE
Magnetism
103
Welcome to
Andre’ Marie Ampere’s Law
Normally written as a “circulation” vector
equation.
We will look at another form, but first…
Magnetism
104
Remember GAUSS’S LAW??
E

d
A


Surface
Integral
Magnetism
qenclosed
0
105
Gauss’s Law
• Made calculations easier than
integration over a charge distribution.
• Applied to situations of HIGH
SYMMETRY.
• Gaussian SURFACE had to be defined
which was consistent with the geometry.
• AMPERE’S Law is the Gauss’ Law of
Magnetism! (Sorry)
Magnetism
106
The next few slides have been
lifted from Seb Oliver
on the internet
Whoever he is!
Magnetism
107
Biot-Savart
• The “Coulombs Law of Magnetism”
 m 0  ids  rˆ
dB   
2
 4  r
Magnetism
108
Invisible Summary
• Biot-Savart Law
 m 0  ids  rˆ
dB   
2
 4  r
m0 I
– (Field produced by wires)
B
2R
– Centre of a wire loop radius R
m NI
B 0
– Centre of a tight Wire Coil with N turns
2R
– Distance a from long straight wire
• Force between two wires
• Definition of Ampere
Magnetism
m0 I
B
2a
F m 0 I1 I 2

l
2a 109
Magnetic Field from a long wire
Using Biot-Savart Law
r
I
Take a short vector
on a circle, ds
B
ds
Magnetism
B  ds  B  ds cos
  0  cos  1
Thus the dot product of B &
the short vector ds is:
m0 I
B 
2r
B  ds  B ds
m0 I
B  ds 
ds
2r
110
Sum
.
B ds
around a circular path
m0 I
B  ds 
ds
2r
r
I
B
Sum this around the whole ring
ds
Circumference
of circle
Magnetism
 B  ds
 ds  2r
m0 I

ds
2r
m0 I

ds
2r
m0 I
  B  ds 
2r  m 0 Ι
2r
111
Consider a different path
B ds  0
i
Magnetism
• Field goes as
1/r
• Path goes as r.
• Integral
independent of
r
112
SO, AMPERE’S LAW
by SUPERPOSITION:
We will do a LINE INTEGRATION
Around a closed path or LOOP.
Magnetism
113
Ampere’s Law
B

d
s

m
i
0
enclosed

USE THE RIGHT HAND RULE IN THESE CALCULATIONS
Magnetism
114
The Right Hand Rule
Magnetism
115
Another Right Hand Rule
Magnetism
116
COMPARE
B

d
s

m
i
0
enclosed

Line Integral
E

d
A


Surface Integral
Magnetism
qenclosed
0
117
Simple Example
Magnetism
118
Field Around a Long Straight Wire
B

d
s

m
i
0
enclosed

B  2r  m 0i
m 0i
B
2r
Magnetism
119
Field INSIDE a Wire
Carrying UNIFORM Current
Magnetism
120
The Calculation
 B  ds  B  ds  2rB  m i
0 enclosed
ienclosed
r 2
i 2
R
and
 m 0i 
B
r
2 
 2R 
Magnetism
121
B
m 0i
2R
R
Magnetism
r
122
Procedure
• Apply Ampere’s law only to highly symmetrical
situations.
• Superposition works.
– Two wires can be treated separately and the
results added (VECTORIALLY!)
• The individual parts of the calculation can be
handled (usually) without the use of vector
calculations because of the symmetry.
• THIS IS SORT OF LIKE GAUSS’s LAW
WITH AN ATTITUDE!
Magnetism
123
The figure below shows a cross section of an infinite conducting sheet
carrying a current per unit x-length of l; the current emerges
perpendicularly out of the page. (a) Use the Biot–Savart law and
symmetry to show that for all points P above the sheet, and all points
P´ below it, the magnetic field B is parallel to the sheet and directed
as shown. (b) Use Ampere's law to find B at all points P and P´.
Magnetism
124
FIRST PART
Vertical Components
Cancel
Magnetism
125
Apply Ampere to Circuit
L
B
Infinite Extent
B
  current per unit length
Current inside the loop is therefore :
i  L
Magnetism
126
The “Math”
B
Infinite Extent
B
 B  ds  m i
0 enclosed
BL  BL  m 0L
B
Magnetism
m 0
2
127
A Physical Solenoid
Magnetism
128
Inside the Solenoid
For an “INFINITE” (long) solenoid the previous
problem and SUPERPOSITION suggests that the
field OUTSIDE this solenoid is ZERO!
Magnetism
129
More on Long Solenoid
Field is ZERO!
Field looks UNIFORM
Field is ZERO
Magnetism
130
The real thing…..
Finite Length
Weak Field
Stronger - Leakage
Magnetism
131
Another Way
Ampere :
 B  ds  m i
0 enclosed
0h  Bh  m 0 nih
B  m 0 ni
Magnetism
132
Application
• Creation of Uniform Magnetic Field
Region
• Minimal field outside
– except at the ends!
Magnetism
133
Two Coils
Magnetism
134
“Real” Helmholtz Coils
Used for experiments.
Can be aligned to cancel
out the Earth’s magnetic
field for critical measurements.
Magnetism
135
The Toroid
Slightly less
dense than
inner portion
Magnetism
136
The Toroid
Ampere again. We need only worry
about the INNER coil contained in
the path of integratio n :
 B  ds  B  2r  m Ni (N  total # turns)
0
so
m 0 Ni
B
2r
Magnetism
137