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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