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Chapter 29

Magnetic Fields

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A Brief History of Magnetism 13th century BC

Chinese used a compass Uses a magnetic needle

Probably an invention of Arabic or Indian origin

800 BC

Greeks Discovered magnetite (Fe3O4) attracts pieces of

iron

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A Brief History of Magnetism, 2 1269

Pierre de Maricourt found that the direction

of a needle near a spherical naturalmagnet formed lines that encircled the

sphere

The lines also passed through two pointsdiametrically opposed to each other

He called the pointspoles

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William Gilbert

Expanded experiments with magnetism

to a variety of materials

Suggested the Earth itself was a large

permanent magnet

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Hans Christian Oersted

Discovered therelationship between

electricity and

magnetism

An electric current ina wire deflected a

nearby compass

needle

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A Brief History of Magnetism,

final 1820s

Faraday and Henry

Further connections between electricityand magnetism

A changing magnetic field creates anelectric field

Maxwell A changing electric field produces a

magnetic field

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Magnetic Poles Every magnet, regardless of its shape,

has two poles

Called north and south poles Poles exert forces on one another

Similar to the way electric charges exertforces on each other

Like poles repel each other

N-N or S-S Unlike poles attract each other

N-S

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Magnetic Poles, cont. The poles received their names due to the way

a magnet behaves in the Earths magnetic field

If a bar magnet is suspended so that it canmove freely, it will rotate The magnetic north pole points toward the Earths

north geographic pole

This means the Earths north geographic pole is a

magnetic south pole

Similarly, the Earths south geographic pole is a

magnetic north pole

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Magnetic Poles, final The force between two poles varies as

the inverse square of the distance

between them A single magnetic pole has never been

isolated

In other words, magnetic poles are alwaysfound in pairs There is some theoretical basis for the

existence ofmonopoles single poles

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Magnetic Fields A vector quantity Symbolized by B Direction is given by the direction a

north pole of a compass needle pointsin that location

Magnetic field lines can be used toshow how the field lines, as traced outby a compass, would look

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Magnetic Field Lines, Bar

Magnet Example The compass can

be used to trace the

field lines The lines outside the

magnet point from

the North pole to the

South pole

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Active Figure 29.1

(SLIDESHOW MODE ONLY)

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Magnetic Field Lines, Bar

Magnet Iron filings are used

to show the pattern

of the electric fieldlines

The direction of the

field is the direction

a north pole wouldpoint

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Magnetic Field Lines, Unlike

Poles Iron filings are used to

show the pattern of

the electric field lines The direction of the

field is the direction a

north pole would point

Compare to the

electric field

produced by an

electric dipole

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Magnetic Field Lines, Like

Poles Iron filings are used to

show the pattern of theelectric field lines

The direction of thefield is the direction anorth pole would point

Compare to theelectric fieldproduced by likecharges

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Definition of Magnetic Field The magnetic field at some point in

space can be defined in terms of the

magnetic force, FB The magnetic force will be exerted on a

charged particle moving with a velocity,

v Assume (for now) there are no gravitational

or electric fields present

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FB on a Charge Moving in a

Magnetic Field The magnitude FB of the force exerted

on the particle is proportional to the

charge, q, and to the speed, v, of theparticle

The magnitude and the direction of the

force depend on the velocity of theparticle and on the magnitude and the

direction of the magnetic field B

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Magnetic Field, cont. When a charged particle moves parallel

to the magnetic field vector, the

magnetic force acting on the particle iszero

When the particles velocity vector

makes any angle 0 with B, FB acts ina direction perpendicular to both v and

B

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Magnetic Field, final FB exerted on a positive charge is in the

direction opposite the direction of the

magnetic force exerted on a negativecharge moving in the same direction

The magnitude ofFB is proportional to

sin , where is the angle the particles

velocity makes with the direction ofB

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More About

Direction

FB is perpendicular to the plane formed by vand B

Oppositely directed forces exerted on

oppositely charged particles will cause the

particles to move in opposite directions

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Magnetic Field, Formula The properties can be summarized in a

vector equation:

FB = q v x B

FB is the magnetic force

q is the charge

v is the velocity of the moving charge

B is the magnetic field

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Direction: Right-Hand Rule #1 The fingers point in the

direction ofv

B comes out of yourpalm Curl your fingers in the

direction ofB

The thumb points in the

direction ofv x B which

is the direction ofFB

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Direction: Right-Hand Rule #2 Alternative to Rule #1 Thumb is in the

direction ofv Fingers are in the

direction ofB Palm is in the direction

ofFB On a positive particle You can think of this as

your hand pushing the

particle

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More About Magnitude ofF The magnitude of the magnetic force on

a charged particle is FB = |q| vB sin

is the smaller angle between v and B

FB is zero when v and B are parallel or

antiparallel= 0 or 180o

FB is a maximum when v and B are

perpendicular

= 90o

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Differences Between Electric

and Magnetic Fields Direction of force

The electric force acts along the direction of the

electric field The magnetic force acts perpendicular to the

magnetic field

Motion

The electric force acts on a charged particleregardless of whether the particle is moving

The magnetic force acts on a charged particle only

when the particle is in motion

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More Differences Between

Electric and Magnetic Fields Work

The electric force does work in displacing a

charged particle The magnetic force associated with a

steady magnetic field does no work when a

particle is displaced This is because the force is

perpendicular to the displacement

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Work in Fields, cont. The kinetic energy of a charged particle

moving through a magnetic field cannot

be altered by the magnetic field alone When a charged particle moves with a

velocity v through a magnetic field, the

field can alter the direction of thevelocity, but not the speed or the kinetic

energy

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Units of Magnetic Field The SI unit of magnetic field is the tesla (T)

Wb is a weber

The cgs unit is a gauss (G) 1 T = 104 G

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Typical Magnetic Field Values

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Magnetic Force on a Current

Carrying Conductor A force is exerted on a current-carrying

wire placed in a magnetic field The current is a collection of many charged

particles in motion

The direction of the force is given by the

right-hand rule

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Notation Note The dots indicate the

direction is out of the page

The dots represent thetips of the arrowscoming toward you

The crosses indicate the

direction is into the page The crosses represent

the feathered tails of thearrows

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Force on a Wire In this case, there is

no current, so there

is no force Therefore, the wire

remains vertical

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Force on a Wire (2) B is into the page

The current is up the page The force is to the left

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Force on a Wire, (3) B is into the page

The current is down

the page The force is to the

right

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Force on a Wire, equation The magnetic force is

exerted on each

moving charge in thewire F = qvd x B

The total force is the

product of the force onone charge and thenumber of charges F = (qvd x B)nAL

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Force on a Wire, (4) In terms of the current, this becomes

F = IL x B L is a vector that points in the direction of

the current Its magnitude is the length L of the segment

Iis the current

B is the magnetic field

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Force on a Wire,

Arbitrary Shape Consider a small

segment of the wire,

ds The force exerted on

this segment is F =

Ids x B

The total force is

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Force on a Wire, Case 1 L is the vector sum of

all the length elementsfrom a to b

The magnetic force ona curved current-carrying wire in auniform field is equal tothat on a straight wireconnecting the endpoints and carrying thesame current

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Force on a Wire, Case 2 An arbitrary shaped closed

loop carrying current I in a

uniform magnetic field

The length elements form

a closed loop, so the

vector sum is zero

The net magnetic force

acting on any closedcurrent loop in a uniform

magnetic field is zero

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Torque on a Current Loop The rectangular loop

carries a current I in

a uniform magneticfield

No magnetic force

acts on sides 1 & 3 The wires are

parallel to the field

and L x B = 0

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Torque on a Current Loop, 2 There is a force on sides 2

& 4 -> perpendicular to thefield

The magnitude of themagnetic force on thesesides will be:

F2 = F4 =I

aB The direction ofF2 is out of

the page The direction ofF4 is into

the page

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Torque on a Current Loop, 3 The forces are equal

and in opposite

directions, but notalong the same line

of action

The forces produce

a torque aroundpoint O

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Torque on a Current Loop,

Equation The maximum torque is found by:

The area enclosed by the loop is ab, so

max = IAB This maximum value occurs only when the

field is parallel to the plane of the loop

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General Assume the

magnetic field

makes an angle of< 90o with a line

perpendicular to the

plane of the loop

The net torque

about point O will be

= IAB sin

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Active Figure 29.14


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Summary The torque has a maximum value when the

field is perpendicular to the normal to the

plane of the loop The torque is zero when the field is parallel to

the normal to the plane of the loop

= IAxB whereA is perpendicular to the

plane of the loop and has a magnitude equal

to the area of the loop

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Direction ofA The right-hand rule can

be used to determine

the direction ofA Curl your fingers in the

direction of the current

in the loop

Your thumb points inthe direction ofA

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Magnetic Dipole Moment The product IA is defined as the

magnetic dipole moment, , of theloop Often called the magnetic moment

SI units: A m2

Torque in terms of magnetic moment: = x B Analogous to = px E for electric dipole

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Galvanometer Revisited Another look at the DArsonval

galvanometer

is the tensional spring constant

The angle of deflection of the

pointer is directly proportional to

the current in the loop

The deflection also depends on

the design of the meter

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Charged Particle in a

Magnetic Field Consider a particle moving

in an external magnetic

field with its velocity

perpendicular to the field The force is always

directed toward the center

of the circular path

The magnetic force causesa centripetal acceleration,

changing the direction of

the velocity of the particle

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Active Figure 29.18

http://../Active_Figures/Active%20Figures%20Media/AF_2918.html

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Force on a Charged Particle Equating the magnetic and centripetal

forces:

Solving forr:

ris proportional to the momentum of theparticle and inversely proportional to themagnetic field

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More About Motion of

Charged Particle The angular speed of the particle is

The angular speed, , is also referred to as

the cyclotron frequency

The period of the motion is

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Motion of a Particle, General If a charged particle

moves in a magnetic

field at somearbitrary angle with

respect to B, its path

is a helix

Same equationsapply, with

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Active Figure 29.19


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Bending of an Electron Beam Electrons are

accelerated from

rest through apotential difference

Conservation of

energy will give v

Other parameterscan be found

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Particle in a Nonuniform

Magnetic Field The motion is

complex For example, the

particles canoscillate back andforth between twopositions

This configuration isknown as amagnetic bottle

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Van Allen Radiation Belts The Van Allen radiation

belts consist of charged

particles surrounding the

Earth in doughnut-shapedregions

The particles are trapped

by the Earths magnetic

field The particles spiral from

pole to pole May result in Auroras

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Charged Particles Moving in

Electric and Magnetic Fields

In many applications, charged particles

will move in the presence of both

magnetic and electric fields In that case, the total force is the sum of

the forces due to the individual fields

In general: F = qE + qv x B

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Velocity Selector

Used when all the

particles need to

move with the same

velocity

A uniform electric

field is perpendicular

to a uniformmagnetic field

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Velocity Selector, cont.

When the force due tothe electric field isequal but opposite to

the force due to themagnetic field, theparticle moves in astraight line

This occurs forvelocities of value

v= E/ B

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Active Figure 29.23


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Velocity Selector, final

Only those particles with the givenspeed will pass through the two fields

undeflected The magnetic force exerted on particles

moving at speed greater than this isstronger than the electric field and the

particles will be deflected upward Those moving more slowly will be

deflected downward

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Mass Spectrometer

A mass spectrometer

separates ions

according to theirmass-to-charge ratio

A beam of ions passes

through a velocityselector and enters a

second magnetic field

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Active Figure 29.24


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Mass Spectrometer, cont.

After entering the second magnetic

field, the ions move in a semicircle of

radius rbefore striking a detector at P If the ions are positively charged, they

deflect upward

If the ions are negatively charged, theydeflect downward

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Thomsons e/m Experiment

Electrons are

accelerated from the

cathode They are deflected

by electric and

magnetic fields The beam of

electrons strikes a

fluorescent screen

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Thomsons e/m Experiment, 2

In general, m/q can be determined bymeasuring the radius of curvature and

knowing the magnitudes of the fields

Thomsons variation found e/me bymeasuring the deflection of the beamand the fields

om rB B

q E

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Cyclotron

A cyclotron is a device that canaccelerate charged particles to very

high speeds The energetic particles produced are

used to bombard atomic nuclei andthereby produce reactions

These reactions can be analyzed byresearchers

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Cyclotron, 2

D1 and D2 are called

dees because oftheir shape

A high frequencyalternating potentialis applied to thedees

A uniform magneticfield is perpendicularto them

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Cyclotron, 3

A positive ion is released near thecenter and moves in a semicircular path

The potential difference is adjusted sothat the polarity of the dees is reversedin the same time interval as the particletravels around one dee

This ensures the kinetic energy of theparticle increases each trip

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Cyclotron, final

The cyclotrons operation is based on the fact

that Tis independent of the speed of the

particles and of the radius of their path

When the energy of the ions in a cyclotron

exceeds about 20 MeV, relativistic effects

come into play

2 2 2

21

2 2

q B RK mv

m

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Hall Effect

When a current carrying conductor is placed

in a magnetic field, a potential difference is

generated in a direction perpendicular to both

the current and the magnetic field

This phenomena is known as the Hall effect

It arises from the deflection of charge carriers

to one side of the conductor as a result of themagnetic forces they experience

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Hall Effect, cont.

The Hall effect gives information

regarding the sign of the charge carriers

and their density It can also be used to measure

magnetic fields

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Hall Voltage

This shows an

arrangement for

observing the Hall

effect

The Hall voltage is

measured between

points a and c

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Hall Voltage, cont

When the charge carriers are negative, the

upper edge of the conductor becomesnegatively charged

When the charge carriers are positive, theupper edge becomes positively charged

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Hall Voltage, final

VH = EHd= vd B d

dis the width of the conductor

vd is the drift velocity IfB and dare known, vd can be found

RH = 1 / nq is called the Hall coefficient

A properly calibrated conductor can be used to

H

H

I IB R BV

nqt t

pc chapter 29

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