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Essential University Physics
Richard Wolfson
26
Magnetism:
Force and Field
PowerPoint® Lecture prepared by Richard Wolfson
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-1
In this lecture you’ll learn
• To describe magnetism in
relation to electric charge
• To calculate magnetic
forces on charges and
currents
• And to describe the
trajectories of charged
particles in magnetic fields
• To explain the origin of
magnetic fields
• And to calculate the
magnetic fields of simple
current distributions
• To describe the effects of
magnetism in matter
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-2
Magnetic field and magnetic force
• The magnetic field, designated B, exerts a force on
moving electric charges.
• The force depends on the
charge q, the magnetic field B,
the charge velocity v , and on
the orientation between
v and B.
• The magnitude of the force is
given by F = qvBsin, and its
direction follows from the
right-hand rule.
• The magnetic force may be
written in terms of the vector
cross product:
F  qv  B
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-3
Clicker question
•
The figure shows a proton moving in a magnetic field.
What will the direction of the magnetic force on the
proton be in both cases?
A. parallel to B
B. into the page
C. out of the page
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-4
Clicker question
•
The figure shows a proton moving in a magnetic field.
What will the direction of the magnetic force on the
proton be in both cases?
A. parallel to B
B. into the page
C. out of the page
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-5
Charged particles in magnetic fields
• Since the magnetic force is always at right angles to a
charged particle’s velocity…
• A particle moving in a plane
perpendicular to the field
undergoes uniform circular
motion.
• The cyclotron frequency f of
the motion is independent of
the particle’s speed:
f = qB/2p m
(for speeds much less than that of light).
• When the particle has a
component of motion along
the field, its trajectory is a
spiral.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-6
The magnetic force on a current
• An electric current consists of moving charges, so a
current-carrying conductor experiences a magnetic force.
• The force actually involves
both magnetic forces on the
moving charges, and an
electric force associated
with charge separation.
• The force is F  IL  B ,
where L is a vector
describing the length and
orientation of a straight
conductor.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-7
Clicker question
•
The figure shows a flexible conducting wire passing
through a magnetic field that points out of the page.
The wire is deflected upward, as shown. In which
direction is current flowing in the wire?
A. to the left
B. to the right
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-8
Clicker question
•
The figure shows a flexible conducting wire passing
through a magnetic field that points out of the page.
The wire is deflected upward, as shown. In which
direction is current flowing in the wire?
A. to the left
B. to the right
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-9
Origin of the magnetic field
• The magnetic field not only produces forces on moving
electric charges…
• The magnetic field also arises from moving electric
charge.
• The Biot-Savart law gives
the magnetic field arising
from an infinitesimal
current element:
0 I dL  rˆ
dB 
4p r 2
• The field of a finite current
follows by integrating:
0 I dL  rˆ
B   dB  
4p r 2
• Here 0 is the permeability constant, equal to 4p  10–7 N/A2.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-10
Behavior of magnetic field lines
• Magnetic fields originate in moving charge.
• But unlike static electric fields, whose field lines begin
and end on charges, magnetic field lines don’t begin or
end on the moving charges and currents that are their
source.
• Instead, magnetic field lines
generally encircle the
moving charges or currents.
• Their direction follows from
the right-hand rule.
• In special cases, field lines
may extend to infinity in
both directions, but they
don’t begin or end.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-11
Using the Biot-Savart law: a line current
• Integrating the contributions
from current elements along an
infinite line gives a field that
falls off as the inverse of the
distance y from the wire:
B = 0I/2p y
• The field encircles the current.
• Two parallel wires experience
forces from each other’s
magnetic field:
• Parallel currents attract.
• Antiparallel currents repel.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
F
0 I1I 2 L
is the force on
2p d
a length L of either wire when
they're a distance d apart.
Slide 26-12
Clicker question
•
A flexible wire is wound into a flat spiral as shown in
the figure. If a current flows in the direction shown,
will the coil tighten or become looser?
A. The coil will tighten.
B. The coil will become looser.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-13
Clicker question
•
A flexible wire is wound into a flat spiral as shown in
the figure. If a current flows in the direction shown,
will the coil tighten or become looser?
A. The coil will tighten.
B. The coil will become looser.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-14
Using the Biot-Savart law: a current loop
• Integrating the contributions
from current elements along a
circular loop of current gives a
field on the loop axis that
depends on the distance x
along the axis:
B

0 Ia 2
2 x a
2
2

32
• For large distances (x >> a),
this reduces to
B
0 Ia 2
2x 3
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-15
Magnetic dipoles
• The 1/x3 dependence of the current-loop’s
magnetic field is the same as the inverse-cube
dependence of the electric field of an electric
dipole.
• In fact, a current loop constitutes a
magnetic dipole.
• Its dipole moment is  = IA, with A
the loop area.
• For an N-turn loop,  = NIA.
• The direction of the dipole moment
vector is perpendicular to the loop
area.
• The fields of electric and magnetic
dipoles are similar far from their
sources, but differ close to the
sources.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-16
Dipoles and monopoles:
Gauss’s law for magnetism
• There do not appear to be any magnetic analogs of electric charge.
• Such magnetic monopoles, if they existed, would be the source of radial
magnetic field lines beginning on the monopoles, just as electric field lines
begin on point charges.
• Instead, the dipole is the simplest magnetic configuration.
• The absence of magnetic monopoles is expressed in Gauss’s law for
magnetism:
O
 B  dA  0
• Gauss’s law for magnetism is one of the four fundamental laws of
electromagnetism.
• Gauss’s law ensures that magnetic field
lines have no beginnings or endings,
but generally form closed loops.
• If monopoles are ever discovered, the
right-hand side of Gauss’s law for
magnetism would be nonzero.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-17
Clicker question
•
The figure shows two sets of field lines. Which set
could be a magnetic field?
A. (a)
B. (b)
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-18
Clicker question
•
The figure shows two sets of field lines. Which set
could be a magnetic field?
A. (a)
B. (b)
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-19
Electric motors
• The electric motor is a vital technological application of
the torque on a current loop.
• A current loop
spins between
magnet poles.
• In a DC motor,
the commutator
keeps reversing
the current
direction to keep
the loop spinning
in the same
direction.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-20
Ampère’s law
• Gauss’s law for electricity provides a global description
of the electric field in relation to charge that is equivalent
to Coulomb’s law.
• Analogously, Ampère’s law provides a global description
of the magnetic field in relation to moving charge that is
equivalent to the Biot-Savart law.
• But where Gauss’s law involves a surface integral over a closed
surface, Ampère’s law involves a line integral around a closed
loop.
• For steady currents, Ampère’s law says

O B  dr  0 I encircled
where the integral is taken around any closed loop, and Iencircled
is the current encircled by that loop.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-21
Ampère’s law and current
• Ampère’s law says wherever the integral of the magnetic field
around a closed loop is nonzero, then there must be current flowing
through the area bounded by the loop.
•
The oppositely directed
magnetic fields in these
structures in the solar corona
necessarily involve currents
flowing perpendicular to the
image, as application of
Ampère’s law to the rectangular
amperian loop shows.
•
Currents in the three wires
shown are the same, but one is
opposite the other two. If
O
 B  dr  0 around loop 2,
which current is the
opposite one?
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-22
Using Ampère’s law
• Ampère’s law is always true, but it can be used to
calculate magnetic fields only in cases with sufficient
symmetry.
• Then it’s possible to
choose an amperian loop
around which O B  dr
can be evaluated in terms
of the unknown B.
• An example: Ampère’s
law quickly gives the 1/r
field of a line current—or
outside any current
distribution with line
symmetry.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Cross section of a long cylindrical
wire. Any field line can serve as
an amperian loop, for evaluating
the field both outside and inside
the wire.
Slide 26-23
A current sheet
• An infinite current sheet is an idealization of a wide,
flat distribution of current.
• Application of Ampère’s
law shows that the magnetic
field outside the sheet is
uniform and has magnitude
where Js is the current per
unit width: B  12 0 J s .
• However, the field direction
reverses across the current
sheet.
• Far from a finite current
sheet, the field begins to
resemble that of a line
current.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-24
Solenoids
• A solenoid is a long, tightly wound
coil of wire.
• When a solenoid’s length is much
greater than its diameter, the
magnetic field inside is nearly
uniform except near the ends, and
the field outside is very small.
• In the ideal limit of an infinitely
long solenoid, the field inside the
solenoid is uniform everywhere,
and the field outside is zero.
• Application of Ampère’s law
shows that the field of an infinite
solenoid is B = 0nI, where n is the
number of turns per unit length.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-25
Electric and magnetic fields of common
charge and current distributions
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-26
Summary
• Magnetism involves moving electric charge.
• Magnetic fields exert forces on moving electric charges:
• The magnetic force on a charge q moving with velocity v
in a magnetic field B is F  qv  B.
• The magnetic force on a length L of current-carrying conductor is
F  IL  B.
• Magnetic fields arise from moving electric charge, as described
by
• The Biot-Savart law: B   dB  
0 I dL  rˆ
4p r 2
• Ampère’s law: O
 B  dr  0 I encircled
• Magnetic fields encircle the currents and moving charges that
are their sources.
• Unlike static electric fields, magnetic field lines don’t begin or end.
• This fact is expressed in Gauss’s law for magnetism: O
 B  dA  0.
Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley
Slide 26-27