PC Chapter 30
Transcript of PC Chapter 30
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Chapter 30
Sources of the Magnetic Field
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Biot-Savart Law Introduction Biot and Savart conducted experiments
on the force exerted by an electric
current on a nearby magnet They arrived at a mathematical
expression that gives the magnetic field
at some point in space due to a current
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Biot-Savart Law Set-Up The magnetic field is
dB at some point P
The length elementis ds
The wire is carrying
a steady current ofI
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Biot-Savart Law
Observations The vectordB is perpendicular to both
ds and to the unit vector directed from
ds toward P The magnitude ofdB is inversely
proportional to r2, where ris the
distance from ds to P
r
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Biot-Savart Law
Observations, cont The magnitude ofdB is proportional to
the current and to the magnitude ds of
the length element ds The magnitude ofdB is proportional to
sin , where is the angle between the
vectors ds and r
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The observations are summarized inthe mathematical equation called the
Biot-Savart law:
The magnetic field described by the lawis the field due to the current-carryingconductor
Biot-Savart Law Equation
24
Io dd r
s rB
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Permeability of Free Space The constant o is called the
permeability of free space
o = 4 x 10-7 T. m / A
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Total Magnetic Field dB is the field created by the current in the
length segment ds
To find the total field, sum up thecontributions from all the current elementsI
ds
The integral is over the entire current distribution
24
Io d
r
s r
B
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Biot-Savart Law Final Notes The law is also valid for a current
consisting of charges flowing through
space ds represents the length of a small
segment of space in which the charges
flow For example, this could apply to the
electron beam in a TV set
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B Compared to E Distance
The magnitude of the magnetic field varies
as the inverse square of the distance fromthe source
The electric field due to a point charge also
varies as the inverse square of thedistance from the charge
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B Compared to E, 2 Direction
The electric field created by a point charge
is radial in direction The magnetic field created by a current
element is perpendicular to both the length
element ds and the unit vectorr
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Source An electric field is established by an
isolated electric charge The current element that produces a
magnetic field must be part of an extended
current distribution Therefore you must integrate over the entire
current distribution
B Compared to E, 3
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B Compared to E, 4 Ends
Magnetic field lines have no beginning and
no end They form continuous circles
Electric field lines begin on positive
charges and end on negative charges
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B for a Long, Straight
Conductor The thin, straight wire
is carrying a constant
current
Integrating over all the
current elements gives
2
1
1 2
4
4
Isin
Icos cos
o
o
B d
a
a
sin d dx s r k
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B for a Long, Straight
Conductor, Special Case If the conductor is
an infinitely long,
straight wire, 1= 0and 2=
The field becomes
2
Io
B a
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B for a Long, Straight
Conductor, Direction The magnetic field lines
are circles concentricwith the wire
The field lines lie inplanes perpendicular toto wire
The magnitude ofB is
constant on any circle ofradius a The right-hand rule for
determining thedirection ofB is shown
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B for a Curved Wire Segment Find the field at point O
due to the wire
segment Iand Rare constants
will be in radians4
IoB
R
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B for a Circular Loop of Wire Consider the previous result, with = 2
This is the field at the centerof the loop
24 4 2
I I I o o o
B R R R
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B for a Circular Current Loop The loop has a
radius ofRand
carries a steadycurrent ofI
Find B at point P
2
32 2 22
Iox
RB
x R
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Comparison of Loops Consider the field at the center of the
current loop
At this special point,x= 0 Then,
This is exactly the same result as from thecurved wire
2 2
3 32 2 2 2
2
I Io ox
R R B
xx R
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Magnetic Field Lines for a
Loop
Figure (a) shows the magnetic field lines surroundinga current loop Figure (b) shows the field lines in the iron filings Figure (c) compares the field lines to that of a bar
magnet
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Magnetic Force Between Two
Parallel Conductors Two parallel wires
each carry a steady
current The field B2 due to
the current in wire 2
exerts a force on
wire 1 ofF1 =I1B2
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Active Figure 30.8
(SLIDESHOW MODE ONLY)
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Magnetic Force Between Two
Parallel Conductors, cont. Substituting the equation for B2 gives
Parallel conductors carrying currents in the
same direction attract each other
Parallel conductors carrying current in
opposite directions repel each other
1 2
1 2
I Io
F a l
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Magnetic Force Between Two
Parallel Conductors, final The result is often expressed as the
magnetic force between the two wires, FB This can also be given as the force per
unit length:
1 2
2
I IB oF
al
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Definition of the Ampere The force between two parallel wires
can be used to define the ampere
When the magnitude of the force per
unit length between two long parallel
wires that carry identical currents and
are separated by 1 m is 2 x 10-7N/m,the current in each wire is defined to be
1 A
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Definition of the Coulomb The SI unit of charge, the coulomb, is
defined in terms of the ampere
When a conductor carries a steadycurrent of 1 A, the quantity of charge
that flows through a cross section of the
conductor in 1 s is 1 C
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Magnetic Field of a Wire A compass can be used
to detect the magneticfield
When there is no currentin the wire, there is nofield due to the current
The compass needles allpoint toward the Earthsnorth pole Due to the Earths
magnetic field
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Magnetic Field of a Wire, 2 Here the wire carries
a strong current
The compassneedles deflect in adirection tangent tothe circle
This shows thedirection of themagnetic fieldproduced by the wire
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Active Figure 30.9
(SLIDESHOW MODE ONLY)
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Magnetic Field of a Wire, 3 The circular
magnetic field
around the wire isshown by the iron
filings
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Amperes Law The product ofB.ds can be evaluated for
small length elements ds on the circular path
defined by the compass needles for the longstraight wire
Amperes law states that the line integral of
B.ds around any closed path equals oI
whereIis the total steady current passing
through any surface bounded by the closed
path.s I
o
d
B
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Amperes Law, cont. Amperes law describes the creation of
magnetic fields by all continuous current
configurations Most useful for this course if the current
configuration has a high degree of symmetry
Put the thumb of your right hand in the
direction of the current through the amperianloop and your fingers curl in the direction you
should integrate around the loop
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Field Due to a Long Straight
Wire From Amperes Law Want to calculate
the magnetic field ata distance rfrom thecenter of a wirecarrying a steadycurrentI
The current isuniformly distributedthrough the crosssection of the wire
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Field Due to a Long Straight Wire
Results From Amperes Law Outside of the wire, r> R
Inside the wire, we needI, the current
inside the amperian circle
2
2
( ) I
I
o
o
d Br
Br
B s
2
2
2
2
2
( ) I ' I ' I
I
o
o
rd Br
R
B r
R
B s
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Field Due to a Long Straight Wire
Results Summary The field is
proportional to r
inside the wire The field varies as
1/routside the wire
Both equations are
equal at r= R
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Magnetic Field of a Toroid Find the field at a
point at distance r
from the center ofthe toroid
The toroid has N
turns of wire
2
2
( ) I
I
o
o
d Br N
NB
r
B s
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Magnetic Field of an Infinite
Sheet Assume a thin,
infinitely large sheet
Carries a current oflinear current densityJs
The current is in the y
direction Js represents the
current per unit lengthalong the zdirection
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Magnetic Field of an Infinite
Sheet, cont. Use a rectangular amperian surface
The wsides of the rectangle do not
contribute to the field
The twosides (parallel to the surface)contribute to the field
2
Io o s
so
d J
JB
B s l
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A solenoid is a long
wire wound in the
form of a helix A reasonably uniform
magnetic field can be
produced in the
space surrounded bythe turns of the wire The interiorof the
solenoid
Magnetic Field of a Solenoid
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Magnetic Field of a Solenoid,
Description The field lines in the interior are
approximately parallel to each other
uniformly distributed close together
This indicates the field is strong and
almost uniform
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Magnetic Field of a Tightly
Wound Solenoid The field distribution
is similar to that of abar magnet
As the length of thesolenoid increases the interior field
becomes more
uniform the exterior field
becomes weaker
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Ideal Solenoid
Characteristics An ideal solenoidis
approached when:
the turns are closelyspaced
the length is much
greater than the
radius of the turns
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Amperes Law Applied to a
Solenoid Amperes law can be used to find the
interior magnetic field of the solenoid
Consider a rectangle with side parallelto the interior field and side wperpendicular to the field
The side of length
inside the solenoidcontributes to the field This is path 1 in the diagram
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Amperes Law Applied to a
Solenoid, cont. Applying Amperes Law gives
The total current through the
rectangular path equals the current
through each turn multiplied by the
number of turns
BdsBdd1path 1path
= == sBsB
NIBdo
== sB
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Magnetic Field of a Solenoid,
final Solving Amperes law for the magnetic
field is
n = N/ is the number of turns per unitlength
This is valid only at points near thecenter of a very long solenoid
I Io oNB n
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Magnetic Flux The magnetic flux
associated with amagnetic field isdefined in a waysimilar to electricflux
Consider an areaelement dA on anarbitrarily shapedsurface
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Magnetic Flux Through a
Plane, 1 A special case is
when a plane of
areaA makes anangle with dA
The magnetic flux is
B = BA cos
In this case, the field
is parallel to the
plane and = 0
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Magnetic Flux Through A
Plane, 2 The magnetic flux is
B = BA cos
In this case, the fieldis perpendicular to the
plane and
= BA This will be the
maximum value of the
flux
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Active Figure 30.21
(SLIDESHOW MODE ONLY)
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Gauss Law in Magnetism Magnetic fields do not begin or end at
any point The number of lines entering a surface
equals the number of lines leaving the
surface
Gauss law in magnetism says:
0d B A
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Displacement Current Amperes law in the original form is
valid only if any electric fields present
are constant in time Maxwell modified the law to include time-
saving electric fields
Maxwell added an additional term whichincludes a factor called the
displacement current,Id
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Displacement Current, cont. The displacement current is notthe
current in the conductor
Conduction current will be used to refer tocurrent carried by a wire or other conductor
The displacement current is defined as
E = E. dA is the electric flux and o is the
permittivity of free space
IE
d o
d
dt
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Amperes Law General
Form Also known as the Ampere-Maxwell
law
I I I Eo d o o od
d dt
B s
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Amperes Law General
Form, Example The electric flux
through S2 is EA
A is the area of thecapacitor plates
Eis the electric field
between the plates
Ifq is the charge onthe plate at any
time, E = EA = q/o
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Amperes Law General
Form, Example, cont. Therefore, the displacement current is
The displacement current is the sameas the conduction current through S1
The displacement current on S2 is thesource of the magnetic field on thesurface boundary
E
d o
d dqI
dt dt
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Ampere-Maxwell Law, final Magnetic fields are produced both by
conduction currents and by time-varying
electric fields This theoretical work by Maxwell
contributed to major advances in the
understanding of electromagnetism
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Magnetic Moments
In general, any current loop has amagnetic field and thus has a magnetic
dipole moment This includes atomic-level current loops
described in some models of the atom This will help explain why some
materials exhibit strong magneticproperties
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Magnetic Moments Classical
Atom
The electrons move in
circular orbits
The orbiting electron
constitutes a tiny currentloop
The magnetic moment of
the electron is associated
with this orbital motion L is the angular momentum
is magnetic moment
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Magnetic Moments Classical
Atom, 2
This model assumes the electron
moves
with constant speed v in a circular orbit of radius r
travels a distance 2rin a time interval T
The orbital speed is2r
vT
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Magnetic Moments Classical
Atom, 3
The current is
The magnetic moment is
The magnetic moment can also be
expressed in terms of the angular
momentum
2I
e ev
Tr
12
I A evr
2 e
e L
m
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Magnetic Moments Classical
Atom, final
The magnetic moment of the electron is
proportional to its orbital angular
momentum The vectors L andpoint in opposite
directions
Quantum physics indicates that angularmomentum is quantized
f
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Magnetic Moments of Multiple
Electrons
In most substances, the magnetic
moment of one electron is canceled by
that of another electron orbiting in thesame direction
The net result is that the magnetic effect
produced by the orbital motion of theelectrons is either zero or very small
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Electron Spin
Electrons (and other particles) have an
intrinsic property called spin that also
contributes to their magnetic moment The electron is not physically spinning
It has an intrinsic angular momentum as if
it were spinning Spin angular momentum is actually a
relativistic effect
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Electron Spin, cont.
The classical model of
electron spin is the
electron spinning on itsaxis
The magnitude of the spin
angular momentum is
is Plancks constant
32
S h
El t S i d M ti
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Electron Spin and Magnetic
Moment
The magnetic moment characteristicallyassociated with the spin of an electron
has the value
This combination of constants is calledthe Bohr magneton B = 9.27 x 10-24J/T
spin2 e
e
m
h
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Magnetization Vector
The magnetic state of a substance isdescribed by a quantity called themagnetization vector, M
The magnitude of the vector is defined as themagnetic moment per unit volume of thesubstance
The magnetization vector is related to themagnetic field by Bm = oM When a substance is placed in an magnetic field,
the total magnetic field is B = Bo + oM
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Magnetic Field Strength
The magnetic field strength, H, is themagnetic moment per unit volume due
to currents H is defined as Bo/o The total magnetic field in a region can
be expressed as B = o(H + M) H and M have the same units
SI units are amperes per meter
Cl ifi ti f M ti
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Classification of Magnetic
Substances
Paramagneticand ferromagnetic
materials are made of atoms that have
permanent magnetic moments Diamagneticmaterials are those made
of atoms that do not have permanent
magnetic moments
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Magnetic Susceptibility
The dimensionless factor can be considered
a measure of how susceptible a material is to
being magnetized M =XH
For paramagnetic substances,Xis positive
and M is in the same direction as H
For diamagnetic substances,Xis negativeand M is opposite H
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Table of Susceptibilities
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Magnetic Permeability
The total magnetic field can be
expressed as
B = o(H + M) = o(1 +X)H = mH
m is called the magnetic permeability
of the substance and is related to the
susceptibility by m= o(1 +X)
Cl if i M t i l b
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Classifying Materials by
Permeability
Materials can be classified by how their
permeability compares with the
permeability of free space (o) Paramagnetic: m > o
Diamagnetic: m < o BecauseXis very small for
paramagnetic and diamagnetic
substances, m ~ o for those substances
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Ferromagnetic Materials
For a ferromagnetic material, M is not a
linear function ofH
The value ofm is not only acharacteristic of the substance, but
depends on the previous state of the
substance and the process it underwentas it moved from its previous state to its
present state
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Ferromagnetic Materials, cont
Some examples of ferromagnetic materials are: iron
cobalt nickel
gadolinium
dysprosium
They contain permanent atomic magnetic
moments that tend to align parallel to each other
even in a weak external magnetic field
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Domains
All ferromagnetic materials are made up
of microscopic regions called domains
The domain is an area within which allmagnetic moments are aligned
The boundaries between various
domains having different orientationsare called domain walls
Domains Unmagnetized
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Domains, Unmagnetized
Material
The magnetic
moments in the
domains arerandomly aligned
The net magnetic
moment is zero
Domains External Field
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Domains, External Field
Applied
A sample is placed inan external magneticfield
The size of thedomains withmagnetic momentsaligned with the field
grows The sample is
magnetized
Domains External Field
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Domains, External Field
Applied, cont.
The material is placed ina stronger field
The domains not alignedwith the field becomevery small
When the external fieldis removed, the materialmay retain a netmagnetization in thedirection of the originalfield
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Magnetization Curve
The total field
increases with
increasing current
(O to a)
As the external field
increases, more
domains are alignedand Hreaches a
maximum at point a
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Magnetization Curve, cont
At point a, the material is approachingsaturation
Saturation is when all magnetic moments arealigned
If the current is reduced to 0, the externalfield is eliminated
The curve follows the path from a to b At point b, B is not zero even though the
external field Bo = 0
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Magnetization Curve, final
B total is not zero because there ismagnetism due to the alignment of thedomains
The material is said to have a remanentmagnetization
If the current is reversed, the magneticmoments reorient until the field is zero
Increasing this reverse current will cause thematerial to be magnetized in the oppositedirection
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Magnetic Hysteresis
This effect is called magnetic hysteresis
It shows that the magnetization of a
ferromagnetic substance depends on thehistory of the substance as well as on the
magnetic field applied
The closed loop on the graph is referred to as a
hysteresis loop Its shape and size depend on the properties of the
ferromagnetic substance and on the strength of the
maximum applied field
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Example Hysteresis Curves
Curve (a) is for a hard ferromagnetic material
The width of the curve indicates a great amount of remanentmagnetism
They cannot be easily demagnetized by an external field
Curve (b) is for a soft ferromagnetic material
These materials are easily magnetized and demagnetized
More About Magnetization
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More About Magnetization
Curves
The area enclosed by a magnetization curve
represents the energy input required to take
the material through the hysteresis cycle
When the magnetization cycle is repeated,
dissipative processes within the material due
to realignment of the magnetic moments
result in an increase in internal energy This is made evident by an increase in the
temperature of the substance
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Curie Temperature
The Curie temperature is
the critical temperature
above which a
ferromagnetic materialloses its residual
magnetism and becomes
paramagnetic
Above the Curietemperature, the thermal
agitation is great enough to
cause a random orientation
of the moments
Table of Some Curie
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Table of Some Curie
Temperatures
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Curies Law
Curies law gives the relationship
between the applied magnetic field and
the absolute temperature of aparamagnetic material:
Cis called Curies constant
oB
M C
T
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Diamagnetism
When an external magnetic field is
applied to a diamagnetic substance, a
weak magnetic moment is induced inthe direction opposite the applied field
Diamagnetic substances are weakly
repelled by a magnet
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Meissner Effect
Certain types ofsuperconductors alsoexhibit perfect
diamagnetism This is called the
Meissner effect
If a permanent magnet
is brought near asuperconductor, thetwo objects repel eachother
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Earths Magnetic Field
The Earths magnetic field
resembles that achieved by
burying a huge bar magnet
deep in the Earths interior
The Earths south magnetic
pole is located near the north
geographic pole
The Earths north magnetic
pole is located near the
south geographic pole
Dip Angle of Earths Magnetic
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Dip Angle of Earth s Magnetic
Field
If a compass is free to rotate vertically as well
as horizontally, it points to the Earths surface
The angle between the horizontal and thedirection of the magnetic field is called the dip
angle The farther north the device is moved, the farther
from horizontal the compass needle would be The compass needle would be horizontal at the equator and
the dip angle would be 0
The compass needle would point straight down at the south
magnetic pole and the dip angle would be 90
More About the Earths
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More About the Earth s
Magnetic Poles
The dip angle of 90 is found at a point just
north of Hudson Bay in Canada This is considered to be the location of the south
magnetic pole
The magnetic and geographic poles are not in
the same exact location
The difference between true north, at the geographicnorth pole, and magnetic north is called the
magnetic declination The amount of declination varies by location on the Earths
surface
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Earths Magnetic Declination
Source of the Earths
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Source of the Earth s
Magnetic Field
There cannot be large masses ofpermanently magnetized materials since thehigh temperatures of the core prevent
materials from retaining permanentmagnetization
The most likely source of the Earthsmagnetic field is believed to be convection
currents in the liquid part of the core There is also evidence that the planets
magnetic field is related to its rate of rotation
Reversals of the Earths
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Reversals of the Earth s
Magnetic Field
The direction of the Earths magnetic
field reverses every few million years
Evidence of these reversals are found inbasalts resulting from volcanic activity
The origin of the reversals is not
understood