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Chapter 19
Magnetism
Review – Magnetic Fields
ELECTRIC FIELDS
 From (+) to (–) charges
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Field lines (electric flux)
Start / End at charges
NO loops! (cons. energy)
Force Law:

MAGNETIC FIELDS
 From (N) to (S) poles
F=qE
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(does work)
 = d × E (elec. dipole)
(× = sin )
F = q v × B (deflection)
F=BIL
(wire)
=×B
(mag. dipole)
Force Law:

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Field lines (magnetic flux)
NO monopoles! (Start/End)
Loop (S) to (N) inside

General
Physics
Review – Right-hand rule
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Essence of a cross product
F=qv×B
v B sin 
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Force is perpendicular to
both velocity and field
Need right-hand rule to
decide which direction
Deflection doesn’t do work
General
Physics
Magnetic Fields II
Sections 6–10
General
Physics
Motion of a Charged Particle in
a Uniform Magnetic Field

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Consider a particle moving in
an external magnetic field so
that its velocity is
perpendicular to the field
The force is always directed
toward the center of the
circular path
The magnetic force causes a
centripetal acceleration,
changing the direction of the
velocity of the particle
General
Physics
Motion of a Charged Particle in
a Uniform Magnetic Field, cont


Equating the magnetic and centripetal forces:
mv 2
F  qvB 
r
Solving for the radius r:
mv
r
qB

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r is proportional to the momentum mv of the particle and
inversely proportional to the magnetic field
Sometimes called the cyclotron equation
Active Figure: Motion of a Charged Particle in a Uniform Magnetic Field
General
Physics
The Mass Spectrometer:
Separating Isotopes

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mv
The cyclotron equation r 
qB
can be applied to the
process of separating isotopes
Singly ionized isotopes are
injected into a velocity selector
Only those isotopes with velocity
v = E/B pass into the deflection
chamber—Why?
Isotopes travel in different circular paths governed by the
cyclotron equation—therefore different mass isotopes separate
Active Figure: The Mass Spectrometer
General
Physics
Magnetic Spectrometer
with Drift (Ion) Chambers

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8-coil toroid
electromagnet
0.3 T maximum field
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2 sectors × 3 drift chambers
954 sense wires
resolution 200 μm
signal to noise 20:1
General
Physics
Particle Moving in an External
Magnetic Field

If the particle’s velocity
is not perpendicular to
the magnetic field, the
path followed by the
particle is a spiral

The spiral path is called
a helix
Active Figure: A Charged Particle with a Helical Path
General
Physics
Charged Particles Trapped in the
Earth’s Magnetic Field—Auroras

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Charged particles from the Sun enter
the Earth’s magnetic field
These particles move in spirals
around the lines of magnetic field
This causes them to become trapped
in the Earth’s magnetic field

An aurora is caused by these
trapped charged particles
colliding with atoms in the upper
atmosphere—producing beautiful
displays of light
General
Physics
Charged Particles Trapped in the
Earth’s Magnetic Field—Auroras
General
Physics
Charged Particles Trapped in the
Earth’s Magnetic Field—Auroras
General
Physics
Charged Particles Trapped in the
Earth’s Magnetic Field—Auroras
General
Physics
Hans Christian Oersted

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1777 – 1851
Best known for
observing that a
compass needle
deflects when placed
near a wire carrying a
current

First evidence of a
connection between
electric and magnetic
phenomena
General
Physics
Magnetic Fields –
Long Straight Wire
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A current-carrying wire
produces a magnetic field
The compass needle
deflects in directions
tangent to the circle

The compass needle points in
the direction of the magnetic
field produced by the current
Active Figure: Magnetic Field Due to a Long Straight Wire
General
Physics
Direction of the Field of a Long
Straight Wire

Right Hand Rule #2
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Grasp the wire in your
right hand
Point your thumb in
the direction of the
current
Your fingers will curl in
the direction of the
field
General
Physics
Magnitude of the Field of a Long
Straight Wire

The magnitude of the field at
a distance r from a wire
carrying a current of I is
o I
B
2 r

µo = 4  x 10-7 T.m / A

µo is called the permeability of
free space
General
Physics
André-Marie Ampère
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1775 – 1836
Credited with the
discovery of
electromagnetism
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Relationship between
electric currents and
magnetic fields
Mathematical genius
evident by age 12
General
Physics
Ampère’s Law

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André-Marie Ampère found a
procedure for deriving the
relationship between the
current in a wire and the
magnetic field produced by
the wire
Ampère’s Circuital Law
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B|| Δℓ = µo I
Sum over the closed path
around the current I
Choose an arbitrary closed path around the current
Sum all the products of B|| Δℓ around the closed path
General
Physics
Ampère’s Law to Find B for a
Long Straight Wire

Sum over a closed circular path
around current I
B|| Δℓ = µo I

Sum all products B|| Δℓ around
the closed path
B·2r = µo I

The magnitude of the magnetic
field a distance r from the wire
o I
B
2 r
General
Physics
Magnetic Field of a Current
Loop
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The strength of a magnetic field
produced by a wire can be
enhanced by forming the wire into
a loop
All the segments, Δx, contribute to
the field, increasing its strength
The magnitude of the magnetic
field at the center of a circular
loop with a radius R
B
o I
2R
General
Physics
Magnetic Field of a Current
Loop – Total Field
General
Physics
Magnetic Field of a Solenoid
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If a long straight wire is
bent into a coil of several
closely spaced loops, the
resulting device is called a
solenoid
It is also known as an
electromagnet since it
acts like a magnet only
when it carries a current
General
Physics
Magnetic Field of a Solenoid, 2

The field lines inside the solenoid are
nearly parallel, uniformly spaced, and
close together
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This indicates that the field inside the solenoid
is nearly uniform and strong
The exterior field is nonuniform, much
weaker, and in the opposite direction to
the field inside the solenoid
General
Physics
Magnetic Field in a Solenoid, 3

The field lines of the solenoid resemble those of
a bar magnet – dipole magnetic field
General
Physics
Magnetic Field in a Solenoid from
Ampère’s Law
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A cross-sectional view of a tightly
wound solenoid
If the solenoid is long compared to
its radius, we assume the field
inside is uniform and outside is zero
Apply Ampère’s Law to the blue
dashed rectangle
The magnitude of the field inside a
solenoid is constant at all points far
from its ends
B  0 nI
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n is the number of turns per unit length
n=N/ℓ
General
Physics
Magnetic Force Between Two
Parallel Conductors
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The force on wire 1 is due
to the current in wire 1
and the magnetic field
produced by wire 2
The force per unit length
is: F
 I I

o
1
2 d
2
General
Physics
Force Between Two
Conductors, cont
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Parallel conductors carrying
currents in the same direction
attract each other
Parallel conductors carrying
currents in the opposite
directions repel each other
Active Figure: Force Between Long Parallel Wires
General
Physics
Defining Ampere and Coulomb

The force between parallel conductors can be
used to define the Ampere (A)
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If two long, parallel wires 1 m apart carry the same
current, and the magnitude of the magnetic force per
unit length is 2 x 10-7 N/m, then the current is
defined to be 1 A
The SI unit of charge, the Coulomb (C), can be
defined in terms of the Ampere

If a conductor carries a steady current of 1 A, then
the quantity of charge that flows through any cross
section in 1 second is 1 C
General
Physics
Magnetic Effects of Electrons –
Orbits

An individual atom should act like a magnet because of
the motion of the electrons about the nucleus
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Each electron circles the atom once in about every 10-16 seconds
This would produce a current of 1.6 mA and a magnetic field of
about 20 T at the center of the circular path
However, the magnetic field produced by one electron in
an atom is often canceled by an oppositely revolving
electron in the same atom
The net result is that the magnetic effect produced by
electrons orbiting the nucleus is either zero or very small
for most materials
General
Physics
Magnetic Effects of Electrons –
Spins

Electrons also have spin

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The classical model is to
consider the electrons to
spin like tops
It is actually a quantum
effect
General
Physics
Magnetic Effects of Electrons –
Spins, cont
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The field due to the spinning is generally
stronger than the field due to the orbital
motion
Electrons usually pair up with their spins
opposite each other, so their fields cancel
each other

That is why most materials are not naturally
magnetic
General
Physics
Magnetic Effects of Electrons –
Domains

In some materials, the spins do not naturally
cancel
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Such materials are called ferromagnetic
Large groups of atoms in which the spins are
aligned are called domains
When an external field is applied, the domains
that are aligned with the field tend to grow at
the expense of the others

This causes the material to become magnetized
General
Physics
Domains, cont
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Random alignment (left) shows an
unmagnetized material
When an external field is applied, the domains
aligned with B grow (right)
General
Physics
Domains and Permanent
Magnets
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In hard magnetic materials, the domains remain aligned
after the external field is removed
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The result is a permanent magnet
In soft magnetic materials, once the external field is
removed, thermal agitation causes the materials to
quickly return to an unmagnetized state
When a ferromagnetic core is placed
inside a current-carrying loop, the
magnetic field is enhanced since the
domains in the core material align,
increasing the magnetic field
General
Physics