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Magnetic Field
Magnetic Fields and Forces
In our study of electricity, we described the interactions between charged objects
in terms of electric fields. Recall that an electric field surrounds any electric charge. In
addition to containing an electric field, the region of space surrounding any moving
electric charge also contains a magnetic field. A magnetic field also surrounds a magnetic
substance making up a permanent magnet.
Historically, the symbol B has been used to represent a magnetic field, and this is
the notation we use in this text. The direction of the magnetic field B at any location is
the direction in which a compass needle points at that location. As with the electric field,
we can represent the magnetic field by means of drawings with magnetic field lines.
Fig.1
Figure .1 shows how the magnetic field lines of a bar magnet can be traced with
the aid of a compass. Note that the magnetic field lines outside the magnet point away
from north poles and toward south poles. One can display magnetic field patterns of a bar
magnet using small iron filings.
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We can define a magnetic field B at some point in space in terms of the magnetic
force FB that the field exerts on a charged particle moving with a velocity v, which we
call the test object. For the time being, let us assume that no electric or gravitational fields
are present at the location of the test object. Experiments on various charged particles
moving in a magnetic field give the following results:
 The magnitude FB of the magnetic force exerted on the particle is proportional
to the charge q and to the speed v of the particle.
 The magnitude and direction of FB depend on the velocity of the particle and
on the magnitude and direction of the magnetic field B.
 When a charged particle moves parallel to the magnetic field vector, the
magnetic force acting on the particle is zero.
 When the particle’s velocity vector makes any angle  ≠ 0 with the magnetic
field, the magnetic force acts in a direction perpendicular to both v and B; that
is, FB is perpendicular to the plane formed by v and B (Fig. 2-a).
 The magnetic force exerted on a positive charge is in the direction opposite
the direction of the magnetic force exerted on a negative charge moving in the
same direction (Fig. 2-b).
 The magnitude of the magnetic force exerted on the moving particle is
proportional to sin , where  is the angle the particle’s velocity vector makes
with the direction of B.
We can summarize these observations by writing the magnetic force in the form
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Fig.2
In the SI unit of magnetic field is the newton per coulomb-meter per second, which
is called the tesla (T):
or
Example An electron in a television picture tube moves toward the front of the tube
with a speed of 8.0 x 106 m/s along the x axis (Fig. 3). Surrounding the neck of the
tube are coils of wire that create a magnetic field of magnitude 0.025 T, directed at
an angle of 60° to the x axis and lying in the xy plane.
(A) Calculate the magnetic force on the electron
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(B) Find a vector expression for the magnetic force on the electron
Fig. 3
(A)
(B)
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Fig. 4
Motion of a Charged Particle in a Uniform Magnetic Field
The magnetic force acting on a charged particle moving in a magnetic field is
perpendicular to the velocity of the particle and that consequently the work done by the
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magnetic force on the particle is zero. Now consider the special case of a positively
charged particle moving in a uniform magnetic field with the initial velocity vector of the
particle perpendicular to the field. Let us assume that the direction of the magnetic field is
into the page, as in Figure 4. As the particle changes the direction of its velocity in
response to the magnetic force, the magnetic force remains perpendicular to the velocity.
If the force is always perpendicular to the velocity, the path of the particle is a circle!
Figure 4 shows the particle moving in a circle in a plane perpendicular to the magnetic
field.
The particle moves in a circle because the magnetic force FB is perpendicular to v
and B and has a constant magnitude q v B. As Figure 4 illustrates, the rotation is
counterclockwise for a positive charge. If q were negative, the rotation would be
clockwise. We can equate the magnetic force to the product of the particle mass and the
centripetal acceleration as the following:
That is, the radius of the path is proportional to the linear momentum m v of the particle
and inversely proportional to the magnitude of the charge on the particle and to the
magnitude of the magnetic field. The angular speed of the particle is
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The period of the motion (the time interval the particle requires to complete one
revolution) is equal to the circumference of the circle divided by the linear speed of the
particle:
Example A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35-T
magnetic field perpendicular to the velocity of the proton. Find the linear speed of
the proton.
References
This lecture is a part of chapter 27 from the following book
Physics for Scientists and Engineers (with Physics NOW and InfoTrac),
Raymond A. Serway - Emeritus, James Madison University , Thomson
Brooks/Cole © 2004, 6th Edition, 1296 pages
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