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Chapter 24 Magnetic Fields and Forces
Thursday, March 11, 2010
8:26 PM
It seems that microscopic electric currents are the ultimate
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It seems that microscopic electric currents are the ultimate
cause of magnetism. For example, each neutron has a little
bit of internal magnetism; in technical language, we say
that each neutron has a non-zero magnetic dipole
moment. In other words, part of the nature of a neutron is
that it acts like a very tiny bar magnet.
The cause of the magnetism of a neutron is thought to be
circulating charges within the neutron; remember that
neutrons are composed of three quarks, all of which have
electric charge.
Each proton also has a magnetic dipole moment,
presumably for the same reason as neutrons: Protons are
also composed of circulating quarks. Each electron also
has a magnetic dipole moment, which is mysterious,
because electrons have no known internal structure, as far
as our best experimenters can tell so far. If the electron
truly is a "point" particle, then this becomes even more
mysterious; how on earth can the electron have
magnetism, if magnetism is due to circulating electric
current within the electron, when the electron has no
room inside it for any circulating anything?
Recognizing that electric currents create magnetic fields
raises the opposite question: Do magnetic fields have any
influence on charged particles? The answer is yes, but in a
much more complicated way than the way charged
particles exert forces on each other. A magnetic field
exerts NO force whatsoever on a charged particle at rest.
Furthermore, a magnetic field exerts NO force on a
charged particle if the charged particle moves parallel to a
magnetic field line. The only way a magnetic field exerts a
force on a charged particle is if the charged particle has a nonzero component of its velocity perpendicular to the magnetic
field. Furthermore, the magnitude of the force exerted by the
magnetic field on the charged particle depends on the strength
of the magnetic field, the magnitude of the particle's charge, and
the magnitude of the component of the particle's velocity that is
perpendicular to the field. We'll discuss this in more detail later.
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All this strange interaction of charged particles and magnetic
fields becomes a bit less mysterious when you approach the
subject from the perspective of special relativity.
• the theme of unification in physics; Maxwell, Einstein
Example: Here is the magnetic field pattern for a magnetic
dipole; note the similarity to the electric field pattern for an
electric dipole.
Here is the magnetic field pattern for two bar magnets placed with
unlike poles nearby:
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Here is the magnetic field pattern for two bar magnets
placed with like poles nearby:
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Earth's magnetic field:
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"Refrigerator" magnets
Magnetic hard-disk drive
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Example: The two insulated wires in the figure cross at a 30° angle but do
not make electrical contact. Each wire carries a 5.0 A current. Points 1 and
2 are each 4.0 cm from the intersection and are equally distant from both
wires. Determine the magnitudes and directions of the magnetic fields at
points 1 and 2.
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In using the right-hand rule, you are free to change the order
of F, B, and v, as long as you keep them in "cyclic order."
(That is, FBv, BvF, and vFB are all in the same cyclic order.)
Different textbooks use different orders that nevertheless
have the same cyclic order, so you may have learned a
different version of the rule in a previous course. Just use
whichever cyclic order is convenient for you, as they all work
just fine.
Some people like to use a variant of the right-hand rule
where they curl their fingers through the acute angle
between vectors v and B; then their thumb points in the
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between vectors v and B; then their thumb points in the
direction of the cross product of v and B. This is why I like to
place my index finger in the direction of v and my middle
finger in the direction of B; then either version of the righthand rule leads to my thumb pointing in the direction of the
cross product.
Also note carefully that to get the direction of the force F,
you have to flip the direction of the cross product only if the
charge is negative; if the charge is positive, the direction of
the force is the same as the direction of the cross product.
For this reason, some teachers use a left-hand rule, where
they assume that negatively-charged particles are moving,
rather than positively-charged particles. I won't do this, but
will stick to the usual "conventional current" convention,
and always use right-hand rules.
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Path of a charged particle in a uniform magnetic field: could
be a circle, a helix, or a straight line, depending on the
initial direction of the charged particle. There are three
possibilities:
• if the initial velocity of the charged particle is parallel to the
magnetic field, the magnetic field does not exert a force on
the charged particle, so the particle's trajectory is a straight
line, by Newton's first law of motion
• if the initial velocity of the charged particle is perpendicular
to the magnetic field, the particle moves in a circle:
• if the initial velocity of the charged particle has nonzero components both parallel and perpendicular to
the magnetic field, then the particle's path will be a
combination of the two types of motion just discussed,
and the path ends up being a helix:
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A natural example of this type of motion of a charged
particle in a magnetic field is aurorae in Earth's
atmosphere:
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The bending of a charged particle in a magnetic field
also provides the idea for a mass spectrometer, which
separates samples of particles by mass:
How does one ensure that the velocities of the
injected particles are the same? For one possibility,
see the example a little further below.
The same phenomenon is used to accelerate charged
particles in the particle accelerators used for highenergy physics experiments. The accelerator in the
following diagram is called a cyclotron:
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The bending of a charged particle in a magnetic field also
provides the idea for an electromagnetic flow meter:
Example: Velocity selector
How does one arrange for the charged particles in a
mass spectrometer to enter the region of the magnetic
field with the same velocities? There are a number of
different "velocity selectors" that can be used; one
such is based on using a region of space with crossed
electric and magnetic fields, illustrated below. Derive a
formula for the speed of a particle that will go straight
through the velocity selector without deflection in
terms of the strengths of the fields.
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Solution: For a positively charged particle injected
into the velocity selector, the electric field exerts a
downward force of magnitude qE and the magnetic
field exerts an upward force of magnitude qvB. The
net force on the charged particle is zero when these
two forces balance:
Thus, if you choose the electric and magnetic fields
just right, you can get zero deflection for just the
desired speed. Notice that the result is independent
of the charge of the particle, so particles with
different charges but the same speed can be
injected into the mass spectrometer.
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http://electronics.howstuffworks.com/speaker5.htm
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Magnetic Fields Exert Torques on Current Loops
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison -W esley.
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Slide 24-42
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CP 24 Problem 24.23 describes two particles that orbit
the earth's magnetic field lines. Calculate the frequency of
the circular orbit for (a) an electron with speed 1.0 × 106
m/s, and (b) a proton with speed 5.0 × 104 m/s. (The
strength of the earth's magnetic field is approximately
5.0 × 10-5 T.)
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CP 26 A mass spectrometer similar to the one in Figure 24.36
is designed to separate protein fragments. The fragments are
ionized by the removal of a single electron, then they enter a
0.80 T uniform magnetic field at a speed of 2.3 × 105 m/s. If a
fragment has a mass that is 85 times the mass of the proton,
determine the distance between the points where the ion
enters and exits the magnetic field.
CP 45 The two 10-cm-long parallel wires in the figure are
separated by 5.0 mm. For what value of the resistor R will the
force between the two wires be 5.4 × 10-5 N?
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CP 47 An electron travels with a speed of 1.0 × 107 m/s
between two parallel charged plates, as shown in the figure.
The plates are separated by 1.0 cm and are charged by a 200 V
battery. What magnetic field strength and direction will allow
the electron to pass between the plates without being
deflected?
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CP 56 A 1.0-m-long, 1.0-mm-diameter copper wire carries a
current of 50.0 A towards the East. Suppose we create a
magnetic field that produces an upward force on the wire
exactly equal in magnitude to the wire's weight, causing the
wire to "levitate." What are the magnetic field's magnitude
and direction?
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