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Chapter 26: Magnetism
Permanent magnets
For magnets, like poles repel and
opposite poles attract.
no monopoles!
A permanent magnet
will attract a metal like
iron with either the
north or south pole.
Q
Magnetic force
•  The magnetic field, designated , exerts a force on
moving electric charges.
•  The magnetic force may be
written in terms of the vector
cross product:
Clicker question
•  A proton (charge +e) sits at rest in a uniform magnetic
field, B, that is directed to the right. What is the direction
of the magnetic force on the charge?
1.  to the right
2.  up
3.  into the screen
4.  zero
5.  none of the above
CT 32.3b
A neg. and a pos. particle move with certain
velocities in a constant, uniform magnetic
field (which points to the right.) The (+)
particle moves directly left; the (–) particle
moves directly up.
The force on the (-)
particle due to the Bfield is…
A: into page
C: 0
B
+
-
B: out of page
D: right
E: left
CT 32.5
Here is an event display from a high
energy experiment. (You are seeing
millions of tracks of charged particles
leaving the central region) There is a 1
Tesla uniform magnetic field coming out
of the page. What sign is the electric
charge for the highlighted (yellow)
charged track?
A: +
B: -
positron
Invisible
Highenergy
photon
B
More energetic e
+ e- pair
electron
Scattered atomic
electron
CT 32.24
Charges (+q) flows right through a narrow wire,
which sits in a uniform B field pointing INTO the
page.
V between top and bottom of the tube is
A: + (top is higher)
B: - (top lower)
C: 0
Field lines are not lines of force
The lines tracing the magnetic field crossed through the velocity vector
of a moving charge will give the direction of force by the RHR.
Motion of charge in uniform magnetic field
•  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.
•  Centripetal force:
•  The cyclotron frequency:
v
qB
f=
=
2πr
2πm
•  independent of the particle’s speed
•  useful for particle accelerator
•  If velocity has a component along B
field: spiral motion
Mass spectrometer
velocity selector:
�
�
� + �v × B
� =0
F� = q E
E
v=
B
The magnetic force on a current-currying
wire
•  An electric current consists of moving charges, so a
current-carrying conductor experiences a magnetic force.
�
F� = (nAL)(q�v × B)
�
F� = (qn�v AL) × B
�
�
F� = (JAL)
×B
� ×B
�
F� = I L
Magnetic Fields Exert Torques on Current Loops
τ = IBA sin(θ) = µB sin(θ)
The Electric Motor
Slide 24-43
disk drive motor
toyota hybrid motor
Clicker question
•  In what direction is the net magnetic force on the current
loop (current is counter-clockwise as seen from above)?
S
N
far side
1.  up
2.  down
3.  zero
4.  none of the above
I
near side
Loudspeaker engineering
•  To create music, we need longitudinal pulses in the air. The speaker
cone is a very clever combination of induced and permanent
magnetism arranged to move the cone to create compressions in the
air.
Sources of Magnetic Fields
Electric Currents Create Magnetic Fields
A long, straight wire
A current loop
A solenoid
Slide 24-14
•  Current produces a magnetic field
•  Current produced by moving charges
•  Moving charges produce magnetic field!
•  The Biot-Savart law gives
the magnetic field arising
from an infinitesimal
current element:
•  The field of a finite current
follows by integrating:
Magnetic field of a straight current carrying
conductor:
•  all components of Idl produce
field in same direction (-z axis)
sin(φ) = sin(π − φ) = �
µ0 I
Bz = −
4π
µ0 I
Bz = −
4π
�
�
a
a
a
a
x
x2 + y 2
sin(φ[y])
dy
2
r
x
dy
2
2
3/2
(x + y )
µ0 I
2a
√
Bz = −
4π x x2 + a2
• as a goes to infinity:
µ0 I
Bz = −
2πx
CT 32.14b
What is the direction of the Force acting on
I
I
the Red Wire?
A)  Up
B)  Right
C)  Left
D)  Into the Page
E)  Out of the Page
©University of Colorado, Boulder (2008)
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.
Using the Biot-Savart law: a current loop
•  Only component of dB in x
direction doesn’t cancel out
µ0 I dL
dBx =
cos θ
2
2
4π x + a
•  For large distances (x >> a),
this reduces to
CT 32.16
What is the direction of the B- Field at point P?
A:
B:
C: 0
D: down
E: Other
©University of Colorado, Boulder
(2008)
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 µ = IA, with A
the loop area.
•  For an N-turn loop, µ = 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.
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
•  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.
CT 32.14c
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.
i2
i1
©University of Colorado, Boulder (2008)
Ampère’s law
•  Analogous to Gauss’s
law,
•  For steady currents,
Ampère’s law is
O
where the integral is taken
around any closed loop,
and Iencircled is the current
encircled by that loop.
•  Useful only for current
distributions with
symmetry
•  Just as general as BiotSavart law
Field inside a long cylindrical conductor
µ0 I
outside conductor: Iencl=I
2πBr = µ0 I =⇒ B =
2π r
2
r
inside conductor: 2πBr = µ Jπr 2 = µ I
0
0
2
R
µ0 Ir
=⇒ B =
2π R2
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 = µ0nI, where n is the
number of turns per unit length.
The Magnetic Field of a Solenoid
A short solenoid
A long solenoid
Slide 24-22
Magnetism in matter
•  Magnetism in matter arises from atomic current loops associated
with orbiting and spinning electrons.
Classical picture of
•  In ferromagnetic materials
magnetic dipole moment
like iron, strong interactions
arising from orbiting
electron
among individual magnetic
dipoles result in large-scale
magnetic properties,
including strong attraction
to magnets.
•  Paramagnetic materials
exhibit much weaker
magnetism.
•  Diamagnetic materials
respond oppositely, and are
repelled by magnets.
Origin of a permanent magnet
Electron Magnetic Moments: Ferromagnetism
A nonmagnetic solid (copper)
A ferromagnetic solid (iron)
Inducing a Magnetic Moment in a Piece of Iron
Electromagnet
•  Use current to induce magnetic moment in iron