Physics 122B
Electricity and Magnetism
Lecture 18 (Knight: 32.2,3,7)
Magnetic Fields and Forces
February 16, 2007 (28 Slides)
Martin Savage
Mid Term 2 Results
7/19/2016
Physics 122B - Lecture 18
2
Lecture 18 Announcements
 Lecture HW Assignment #6 has been posted on
Tycho and is due at 10 PM next wed. .
 Requests for regrades of Exam 2 (see Syllabus for
procedure) will be accepted until noon Friday.
7/19/2016
Physics 122B - Lecture 18
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Experiments with Magnetism:
Experiment 1
Tape a bar magnet to a cork
and allow it to float in a dish of
water.
The magnet turns and aligns
itself with the north-south
direction.
The end of the magnet that
points north is called the
magnet’s north-seeking pole, or
simply its north pole. The
other end is the south pole.
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Experiments with Magnetism:
Experiment 2
Bring the north poles of two bar magnets near to each other. Then
bring the north pole of one bar magnet near the south pole of another
bar magnet.
When the two north poles are brought near, a repulsive force
between them is observed. When the a north and a south pole are
brought near, an attractive force between them is observed.
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Experiments with Magnetism:
Experiment 3
Bring the north pole of a bar magnet near a compass needle.
When the north pole is brought near, the north-seeking pole of the
compass needle points away from the magnet’s north pole. Apparently
the compass needle is itself a little bar magnet.
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Experiments with Magnetism:
Experiment 4
Use a hacksaw to cut a bar magnet in half. Can you isolate the north
pole and the south pole on separate pieces?
No. When the bar is cut in half two new (but weaker) bar magnets
are formed, each with a north pole and a south pole. The same result
would be found, even if the magnet was sub-divided down to the
microscopic level.
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Experiments with Magnetism:
Experiment 5
Bring a bar magnet near an assortment of objects.
Some of the objects, e.g. paper clips, will be attracted to
the magnet. Other objects, e.g., glass beads, aluminum foil,
copper tacks, will be unaffected. The objects that are
attracted to the magnet are equally attracted by the north
and south poles of the bar magnet
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Experiments with Magnetism:
Experiment 6
Bring a magnet near the
electrode of an electroscope.
There is no observed effect,
whether the electroscope is
charged or discharged and
whether the north or the south
pole of the magnet is used.
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Conclusions from Experiments
1.
2.
3.
4.
5.
Magnetism is not the same as electricity. Magnetic poles
are similar to charges but have important differences.
Magnetism is a long range force. The compass needle
responds to the bar magnet from some distance away.
Magnets have two poles, “north” (N) and “south” (S). Like
poles repel and opposite poles attract.
Poles of a magnet can be identified with a compass. A
north magnet pole (N) attracts the south-seeking end of
the compass needle (which is a south pole).
Some materials (e.g., iron) stick to magnets and others do
not. The materials that are attracted are called magnetic
materials. Magnetic materials are attracted by either
pole of a magnet. This is similar in some ways to the
attraction of neutral objects by an electrically charged
rod by induced polarization.
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Compasses and Geomagnetism
The north (N) pole of a compass
needle is attracted to the
geographical north pole of the
Earth and repelled by its
geographic south pole.
Apparently, the Earth is a large
magnet, and one with a magnetic
south pole near the Earth’s
geographic north pole.
The reasons for the Earth’s magnetism
are complex, involving standing currents in
the Earth’s molten interior. The Earth’s magnetic poles are
separated somewhat from the geographic poles, and move
around some as a function of time.
There is evidence in the geological record that the
Earth’s field has reversed directions at varying intervals
spaced by millions of years. This is not well understood.
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The Discovery
of the Magnetic Field
In April of 1820, the Danish physicist Hans
Christian Oersted was giving a evening lecture in
which he was demonstrating the heating of a
wire when an electrical current passed through
it. He noticed that a compass that was nearby
on the table deflected each time he made the
current flow.
Until that time, physicists had considered
electricity and magnetism to be unrelated
phenomena. Oersted discovered that the
“missing link” between electricity and magnetism
was the electric current.
Hans Christian Oersted
(1777- 1851)
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Current Effect on Compass
Place some compasses around a wire. When no current is flowing in the
wire, all compasses point north.
When current flows in the wire, the compasses point in a ring around
the wire.
The Right-Hand Rule: Grip the wire so that the thumb of your right
hand points in the direction of the current. Then your fingers will point in
the direction that the compasses point. This is the direction of the
magnetic field created by the current flow.
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Vector Conventions
Example: field around a current.
For discussions of magnetism, we will need a three-dimensional
perspective. , but we will use two-dimensional diagrams when we can. To
get the 3rd dimension into a two-dimensional diagram, we will indicate
vectors into and out of a diagram by using crosses and dots, respectively.
Rule: a dot () means you are looking at the point of an arrow coming
toward you; a cross () means you are looking at the tail feathers of an
arrow going away from you.
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The Magnetic Field
Definition of the magnetic field:
(1) The magnetic field at each point is a
vector, with both a magnitude, which
we call the magnetic field strength
B, and a 3-D direction.
(2) A magnetic field is created at all
points in the space surrounding a
current carrying wire.
(3) The magnetic field exerts a force on
magnetic poles. The force on a
north pole is parallel to B, and the
force on a south pole is anti-parallel
to B.
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Magnetic Field Lines
The magnetic field can be graphically
represented as magnetic field lines, with
the tangent to a given field line at any
point indicating the local field direction
and the spacing of field lines indicating the
local field strength. The field line
direction indicates the direction of force
on an isolated north magnetic pole.
B-field lines
never cross.
B-field line spacing
indicates field
strength
Field Map
Field Lines
weak
strong
B-field lines always
form closed loops.
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Two Kinds of Magnetism?
We jumped from a discussion of the
magnetic effects of permanent bar magnets to
the magnetic fields produced by current
carrying wires.
Are there two distinct kinds of magnetism?
No. As we will see in the lectures that
follow, the two manifestations of magnetism are
actually two aspects of the same fundamental
magnetic force.
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Question
The magnetic field at point P points
a) Up.
b) Down.
c) Right.
d) Into the page.
e) Out of the page.
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Moving Charges
Ba vxr
Moving charges are the source of the magnetic field:
  qv sin 

B 0
,
direction
of
right-hand
rule

2
 4 r

This is the Biot-Savart Law. It is an
inverse-square law, like Coulomb’s Law, but it
is more complicated because it depends on
the angle between the velocity v and radius
vector r from the moving charge to the point
of observation.
The force constant 0/4 = 10-7 T m/A,
where Tesla or T are the magnetic field
units. 1 T = 1 N A-1m-1.
Note that B = 0 when  = 00 or 1800 and is
maximum at 900.
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Field of a Moving Charge
0 qv sin 
B
4 r 2
All charges create E Fields, but only moving charges create B fields.
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Example: The Magnetic Field
of a Proton
A proton moves on the x axis with
velocity vx = 1.0 x 107 m/s.
As it passes the origin, what are the
magnetic fields at Cartesian coordinate
locations (1,0,0), (0,1,0), and (1,1,0) in mm?
Position (1,0,0) is along the x axis,
so =0 and Bx=0.
Position (0,1,0) is along the y axis,
so =900 and r=1 mm.
 qv sin 
B(0,1, 0)  0
4 r 2
(1.6 1019 C)(1.0 107 m/s)(1)
7
 (1.0 10 N/A/m)
(1.0 103 m)2
 1.60 1013 kˆ T
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B(1,1, 0)  B(0,1, 0) sin(45) / 2
 0.57 1013 kˆ T
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Superposition
Magnetic fields obey the principle of superposition. The magnetic
fields from a number of moving charges add vectorially to produce a
net magnetic field. In particular, the magnetic fields produced by
individual electrons moving in a current-carrying wire add to produce
a magnetic field that is characteristic of the net current.
Btotal  B1  B2 
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 Bn
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Multiplying Vectors
Given two vectors:
A  Axiˆ  Ay ˆj  Az k
B  Bxiˆ  By ˆj  Bz kˆ
There are two ways of
mutiplying these vectors:
Dot Product (Scalar Product)
Cross Product (Vector Product)
A B  Ax Bx  Ay By  Az Bz
A  B   Ay Bz  Az By  iˆ
 A B Cos  AB
  Az Bx  Ax Bz  ˆj
  Ax By  Ay Bx  kˆ
A·B is B times the projection
of A on B, or vice versa.
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 A B Sin  AB (aˆ  bˆ)
iˆ
 Ax
Bx
ˆj
Ay
By
kˆ
Az
Bz
(determinant)
iˆ
Ax
ˆj
Ay
kˆ
Az
Bx
By
Bz
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Cross Product Form
of the Biot-Savart Law
The Biot-Savart Law can be represented
more compactly using a vector cross
product. This automatically gives a B field
that is perpendicular to the plane of the
charge velocity and radius vector to the
point at which the field is being evaluated.
0 q  v  rˆ 
B
4
r2
Note that the r in the numerator
is ^
r (unit vector), not r (vector)!
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Example: The Magnetic Field
of a Electron
An electron is moving to the right, as
shown in the diagram.
What is the direction of the electron’s
magnetic field at the position indicated by
the black dot?
B

Since the electron has a negative
charge, the direction of B should be:
B ~ (v  rˆ),
which is out of the diagram.
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Question 2
·v
The positive charge is moving straight out of the
page. What is the magnetic field at the position
of the dot?
a) Up.
b) Down.
c) Left
d) Right.
e) Out of the page.
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Ampere’s Law
From Biot-Savart Law :

r r
B.dl = 0 I
Current penetrating the surface enclosed
by closed loop integration
Magnetic analogue of Gauss’s Law
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B-field of Infinite Straight
Current Carryng Wire
Cylindrical symmetry, magnitude of B is same at
any azimuthal angle

r r
B.dl = |B| 2  r
|B| =
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=
0 I
r
dl
B
0 I
2r
I
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Ampere’s Experiment
When Ampere heard of Oersted’s results, he
reasoned that if a current produced a magnetic
effect, it might respond to a magnetic effect.
Therefore, he measured the force between two
parallel current-carrying wires.
He found that parallel currents create an
attraction between the wires, while anti-parallel
currents create repulsion.
André Marie Ampère
(1775 – 1836)
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Magnetic Force
A current consists of moving charges. Ampere’s experiment
implies that a magnetic field exerts a force on a moving charge.
This is true, although the exact form of the force relation was not
discovered until later in the 19th century. The force depends on the
relative directions of the magnetic field and the velocity of the
moving charge, and is perpendicular to both..

F  q vB
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
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Magnetic Force
on Moving Charges
Properties of the magnetic force:
1. Only moving charges experience
the magnetic force. There is no
magnetic force on a charge at rest
(v=0) in a magnetic field.
2. There is no magnetic force on a
charge moving parallel (a=00) or
anti-parallel (a=1800) to a
magnetic field.
3. When there is a magnetic force, it
is perpendicular to both v and B.
4. The force on a negative charge is
in the direction opposite to vxB.
5. For a charge moving perpendicular
to B (a=900) , the magnitude of
the force is F=|q|vB.
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
F  q vB

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Example: Magnetic Force on an
Electron
A long straight wire carries a
10 A current from left to right.
An electron 10 cm above the
wire is traveling to the right
with a speed of 1.0x107 m/s.
What is the magnitude and
direction of the force on the
electron.
Bwire 
0 2 I
4 d
F  evB
 (1.0 107 Tm/A)(20 A)/(0.1 m)
 (1.6  10-19 C)(1.0  107 m/s)(2.0  10-4 T)
 2.0 104 T
=3.2  10-16 N
Direction of F is up. Electron will
move away from the wire.
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Cyclotron Motion
Consider a positive charged particle
with mass m and charge q moving at
velocity v perpendicular to a uniform
magnetic field B. The particle will
move in a circular path of radius rcyc
because of the force F on the particle,
which is:
mv 2
F  qvB 
rcyc
f cyc 
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rcyc 
mv
qB
v
q B

(independent of rcyc and v)
2 rcyc m 2
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Example: The Radius of
Cyclotron Motion
An electron is accelerated from rest
through a potential of 500 V, then
injected into a uniform magnetic field B.
Once in the magnetic field, it completes
a half revolution in 2.0 ns.
What is the radius of the orbit?
1
2
v
mv2  (e)V  0  0
f cyc 
B
2eV
 1.33 107 m/s
m
1
1

 2.5 108 Hz
9
T 2(2.0 10 s)
2 mf cyc
e
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3
 8.94 10 T
rcyc 
mv
 8.47 103 m  8.47 mm
qB
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Lecture 18 Announcements
 Lecture HW Assignment #6 has been posted on
Tycho and is due at 10 PM next wed. .
 Requests for regrades of Exam 2 (see Syllabus for
procedure) will be accepted until noon Friday.
7/19/2016
Physics 122B - Lecture 18
35