Clicker Question
Which point A or B has the larger magnitude Magnetic Field?
I
I v d B
=
μ
0 i d v
L
4
π r
×
2 r ˆ
B
A
A B
C : The B-field is the same at A and B.
Answer: Case B has the larger magnetic field. Use the
Biot-Savart Law to get the directions of the B-field due to the two semi-circular portions of the loop. In A the two fields oppose each other; in B they add.
60
What about the B-field at other points?
B=?
z
Calculate the contribution from each piece dL.
d v
B
=
μ
4
0
π i d v
L r 2
× r ˆ i
R
Then integrate around the loop.
v
B
=
μ
0
2 ( z 2 iR 2
+
R 2 ) 3 / 2 z ˆ
61
Clicker Question
In the limit as z >> R, the expression for the B-field becomes?
A)
B)
C) v
B
=
μ
0 iR
2 z
3 z ˆ
2 v
B
=
=
μ
0
2
μ
0
( z
2 iR
2
+
R
2
) i z ˆ
2 R v
B
=
μ
0
2 ( z 2 iR 2
+
R 2 ) 3 / 2 z ˆ i
R
B=?
z
62 v
B
=
μ
0
2 iR
2 z
3 z ˆ v
B
=
μ v
μ
2
π
0
= i
π
R
2 z v i A
3 z ˆ
=
μ
2
π
0 iA z ˆ z
3
* A = area of loop
Magnetic Dipole Moment
R v
B
=
μ
2
π
0 z
3 i
B=?
z
63
Any current loop looks like a Magnetic Dipole far away.
v
B
=
μ
π
0
z 3
The B-field drops as the distance^3
And depends on the Magnetic Dipole
Moment.
μ v
= v i A
64
Current Loop
μ
μ v
= v i A
Area = A i
Direction of A: Curl right hand fingers along current, thumb points along A.
What are the forces and torques on the loop?
65
1

Clicker Question
A square loop of side length a of wire carrying current I is in a uniform magnetic field B. The loop is perpendicular to B (B out of the page). What is the magnitude of the net force on the wire?
A: iaB
B: 4iaB v
F
= i v
L
× v
B
B
C: 2iaB
D: 0
E: None of these
I
Clicker Question
The same loop is now in a non-uniform field. r
B
= $ ⋅ where A is a constant. The direction of the net force is?
y x
B stronger
B weaker
B
A
D
E: net force is zero
C
66 67
In a uniform B-field, regardless of the orientation between the
B and the Magnetic Moment of the loop
μ
, the net force is always zero.
However, that does not mean the net torque is zero!
68
μ
If
μ is not parallel to B, then there is
τ v a net torque.
=
× v
F
τ v
=
r v
×
( i v
L
× v
B )
=
2 [( L / 2 ) iLB sin
θ
]
τ v
τ v
τ v
=
=
= i iL v
A
μ
2
B
×
× v
B sin v
B
θ
69
Basic principle behind many motors!
v
v
B
Torque wants to twist the loop so that
μ and B align.
Remember this point!
70
Interaction between two current carrying wires i i
The two wires may exert forces on each other through Magnetic
Interactions.
1. Think of the Red Wire as creating a B-field .
2. Then think of that B-field creating a force on the moving charges
(current) in the Blue Wire .
71
2

Clicker Question i i
What is the direction of the Force acting on the Blue Wire ?
v
F
total
= i v
L
× v
B
A) Up
B) Right
C)Left
D)Into the Page
E) Out of the Page
72 i
1
R
X i
2
The B-field from the Red Wire at the location of the Blue Wire is into the page.
v
B
1
=
μ
0 i
1
2
π
R
( into the page)
Then the Force on the Blue Wire is to the left.
v
F
2
= i
2 v
L
× v
B
1
= i
2
L
μ
0 i
1
π
R
73
Clicker Question i i
What is the direction of the Force acting on the Red Wire ?
v
F
total
= i v
L
× v
B
A) Up
B) Right
C)Left
D)Into the Page
E) Out of the Page
74
.
i
1
R i
2
The B-field from the Blue Wire at the location of the Red Wire is out of the page.
v
B
2
=
μ
0 i
2
2
π
R
( out of the page)
Then the Force on the Red Wire is to the left.
v
F
1
= i
1 v
L
× v
B
2
= i
1
L
μ
0 i
2
π
R
75 i i
Wires with parallel currents attract each other.
i i
What happens if we flip the direction of one current?
Wires with antiparallel currents repel each other.
Try following the procedure we just outlined to confirm this for yourselves.
Clicker Question
Two loops of wire have current going around in the same direction.
The forces between the loops is:
A)Attractive
B)Repulsive
C)Net force is zero.
76 i
2 i
1
77
3

Good time to take stock of what we have learned.
v
B
=
μ q v v
π
0
4 r
×
2 r ˆ v d B
=
μ
0 i d v
L
4
π r 2
× r ˆ
Moving charges (currents) create B-fields.
Where are the moving charges?
Atoms have moving electric charges.
78 79
Consider a single electron moving in a circular orbit around the nucleus.
i
= ev
2
π
R
=
( 1 .
6
×
10
2
−
19
π
C )( 3
( 10
−
10
×
10
6 m ) m / s )
=
7 .
6
×
10
−
4
A
What is the B-field at the center (at the nucleus)?
v
B
=
μ
2
0
R i
≈
5Tesla
Electron in orbit, but current in opposite direction.
In most materials all the electrons in orbit have random orientations.
Superposition of B-field vectors over many atoms gives B=0
80 81
In Ferromagnetic materials (Fe, Ni, Cr, some alloys containing these metals too), the atomic currents can all orient the same way ( domains ), making a net B-field.
B B
B B
Sometimes the material is fragmented into many domains (top) and is thus unmagnetized. If the domains align (bottom) there is a net magnetic field (magnetized).
82 83
4

Magnetic Stripes
"Magnetic domains in a ferrimagnetic garnet film are studies under a polarizing microscope. These regions of 'up' or 'down' magnetization within the material form a labyrinth of dark and light stripes, each about 44/100,000th of an inch wide. Each cluster is color-coded according to the orientation of its line segments." "Magnetic domains in a ferromagnetic garnet film as seen in a polarizing microscope. Up and down domains form a labyrinth of dark and light stripes (11 microns wide). Digital pattern analysis identifies clusters of straight-line segments as a fundamental structural motif. Each cluster is color-coded according to the (average) orientation of its line segments."
84
85
In all permanent magnets (just like a current loop) the B-field points away from one end ( North ) and towards the other end ( South ).
Very important convention.
Thus, these magnets act just like a current loop!
i
86
Clicker Question
What is the current direction in this loop? And which side of the loop is the north pole?
A. Current clockwise; north pole on top
B. Current counterclockwise, north pole on top
C. Current clockwise; north pole on bottom
D. Current counterclockwise, north pole on bottom i
87
Atoms have a Magnetic Dipole Moments from the orbit of the electrons.
μ v
= v i A
But, amazingly electrons themselves also have a Magnetic
Dipole Moment.
μ v
88
Is the electron really a fundamental point-like particle?
Or does it have little charges moving around inside it?
Or could the electron be a rotating sphere of charge?
89
5

Could the electron be a rotating sphere of charge?
We know from the world’s best microscopes
(particle physics accelerators) that the electron is less than 10 -18 meters across.
Thus, knowing the Magnetic Dipole Moment, we can calculate how fast the outside of the sphere must be rotating. v
=
2 .
5
×
10
14 meters/sec ond
90
This velocity is faster than the speed of light, and thus not allowed in Einstein’s Theory of Relativity.
v
=
2 .
5
×
10 14 m/s
>>
3
×
10 8 m / s (speed of light)
Thus, it may be that electrons just have an intrinsic property of Magnetic Dipole Moment.
This is not understood at any more fundamental level today.
91
It turns out in permanent magnets, the atoms have many electrons in orbits with different orientations.
Thus, it is often the intrinsic magnetic dipole moment of the electrons that is important and tends to be aligned into domains.
Clicker Question
Consider the wire loop shown.
What is the direction of the net
Magnetic Field at point P?
A) Zero
B) Into the Board
C)Out of the Board
D)Left
E) None of the Above v
B
= ∫ d v
B
= ∫
μ
0 i
4
π
| v d L |
=
R 2
μ
0 i
4
π
R 2
∫
| v d L |
Integrate only around half a circle in this case.
94 i
92
Quick Review of the three right hand rules to date….
1. Vector Cross Product i
2. Magnetic Field Direction (fingers curling) from a wire with current i (thumb)
3. Fingers along current loop, thumb in direction of the Magnetic Field
93
How does a rail gun work?
v
F
= v i L
× v
B
Conducting
Gas
Projectile i
X B
95
6

Rail Guns are by far the most spectacular type of electromagnetic accelerators ever developed. They hold the record for fastest object accelerated of a significant mass, for the 16000m/s firing of a .1 gram object by Sandia
National Research Laboratories' 6mm Hypervelocity
Launcher, and they can also propel objects of very sizeable masses to equally impressive velocities, such as in the picture to the left, where Maxwell Laboratories' 32Megajoule gun fires a 1.6kilogram projectile at 3300m/s (that's
9megajoules of kinetic energy!) at Green Farm research facility.
96
Now that we know what goes on inside these magnets, we should be able to predict how they behave!
What happens if we break an iron magnet into two pieces?
97
Clicker Question
A permanent bar magnet is broken in half. Do the pieces attract or repel?
1 2
A: Attract
B: Repel
C: Neither! There is no net force.
i i
Answer: They attract. The bar magnet can be thought of as a coil of current. Parallel currents attract. The currents on the ends are parallel so there is an attraction.
98
Clicker Question
A permanent bar magnet is broken in half. The two pieces are interchanged, keeping their orientations fixed, as shown below. Do the pieces attract or repel?
1 2
2 1
A: Attract
B: Repel
C: Neither! There is no net force.
Answer: They attract. The bar magnet can be thought of as a coil of current. Parallel currents attract. When the magnet is broken and rearranged, the currents on the ends are parallel so there is an attraction.
99
Opposite Poles attract.
Similar Poles repel.
This is not new physics. It is identical to the physics of our current loops.
Now one can see why we can never break a magnet and get just one pole!
Why does a compass align a certain way in magnetic fields?
100 101
7

Clicker Question
Uniform B Field
If the permanent magnet in the compass has a Magnetic Dipole Moment pointing along the RED arrow (North End of the
Magnet), what is the direction of the torque for the compass as shown in the uniform B field?
A)Clockwise
v
v
B)Counter Clockwise v
B
C)There is no net torque
What we call the Earth’s Magnetic North Pole is really the south end of a giant magnet. It is the direction a compass’ north pole points.
102 103
Why do magnets stick to the fridge (but not stainless) and attract some metal objects?
104 105
Remember back to Induced Electric Dipoles….
Wooden 2x4 Demonstration
+
+
+
+
-
-
-
+
-
-
-
-
Induced charge always produces an attractive force!
+
-
-
-
The opposite charge is always closest, thus resulting in attraction.
Now think about induced Magnetic Dipole Moment Alignment
Randomly Oriented
Magnetic Dipoles
μ
Magnetic Dipoles
Aligned with
External B-field.
Induced
μ
.
106 107
8

Demonstration i i
Just like two loops of wire with current going in the same direction – attractive force!
108 109
i i
Then induced magnetic dipoles in metal are rotated the opposite way.
Both current loops are opposite to before, thus still attractive!
110
Some materials, like stainless steel, do not have very large induced Magnetic
Dipole Moments.
Thus, there is not enough attractive force to hold the magnet to the fridge
To save on cost, the sides of such fridges are not stainless, and thus support magnets ☺
111
112
9