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Chapter 19Chapter 19

MagnetismMagnetism

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MagnetsMagnets PolesPoles of a magnet are the ends where of a magnet are the ends where

objects are most strongly attractedobjects are most strongly attracted Two poles, called Two poles, called northnorth and and southsouth

Like poles repel each other and unlike poles Like poles repel each other and unlike poles attract each otherattract each other Similar to electric chargesSimilar to electric charges

Magnetic poles cannot be isolatedMagnetic poles cannot be isolated If a permanent magnetic is cut in half repeatedly, you If a permanent magnetic is cut in half repeatedly, you

will still have a north and a south polewill still have a north and a south pole This differs from electric chargesThis differs from electric charges There is some theoretical basis for monopoles, but There is some theoretical basis for monopoles, but

none have been detectednone have been detected

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More About MagnetismMore About Magnetism

An unmagnetized piece of iron can An unmagnetized piece of iron can be magnetized by stroking it with a be magnetized by stroking it with a magnetmagnet Somewhat like stroking an object to Somewhat like stroking an object to

charge an objectcharge an object Magnetism can be inducedMagnetism can be induced

If a piece of iron, for example, is If a piece of iron, for example, is placed near a strong permanent placed near a strong permanent magnet, it will become magnetizedmagnet, it will become magnetized

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Types of Magnetic Types of Magnetic MaterialsMaterials

Soft magnetic Soft magnetic materials, such as materials, such as iron, are easily magnetizediron, are easily magnetized They also tend to lose their They also tend to lose their

magnetism easilymagnetism easily Hard magneticHard magnetic materials, such as materials, such as

cobalt and nickel, are difficult to cobalt and nickel, are difficult to magnetizemagnetize They tend to retain their magnetismThey tend to retain their magnetism

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Magnetic FieldsMagnetic Fields

A vector quantityA vector quantity Symbolized by Symbolized by BB Direction is given by the direction Direction is given by the direction

a a north polenorth pole of a compass needle of a compass needle points in that locationpoints in that location

Magnetic field linesMagnetic field lines can be used to can be used to show how the field lines, as traced show how the field lines, as traced out by a compass, would lookout by a compass, would look

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Magnetic Field Lines, Magnetic Field Lines, sketchsketch

A compass can be used to show the A compass can be used to show the direction of the magnetic field lines (a)direction of the magnetic field lines (a)

A sketch of the magnetic field lines (b)A sketch of the magnetic field lines (b)

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Magnetic Field Lines, Bar Magnetic Field Lines, Bar MagnetMagnet

Iron filings are Iron filings are used to show the used to show the pattern of the pattern of the electric field lineselectric field lines

The direction of The direction of the field is the the field is the direction a north direction a north pole would pointpole would point

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Magnetic Field Lines, Magnetic Field Lines, Unlike PolesUnlike Poles

Iron filings are Iron filings are used to show the used to show the pattern of the pattern of the electric field lineselectric field lines

The direction of The direction of the field is the the field is the direction a north direction a north pole would pointpole would point Compare to the Compare to the

electric field electric field produced by an produced by an electric dipoleelectric dipole

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Magnetic Field Lines, Like Magnetic Field Lines, Like PolesPoles

Iron filings are used Iron filings are used to show the pattern to show the pattern of the electric field of the electric field lineslines

The direction of the The direction of the field is the direction field is the direction a north pole would a north pole would pointpoint Compare to the Compare to the

electric field produced electric field produced by like chargesby like charges

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Magnetic and Electric Magnetic and Electric FieldsFields

An electric field surrounds any An electric field surrounds any stationary electric chargestationary electric charge

A magnetic field surrounds any A magnetic field surrounds any movingmoving electric charge electric charge

A magnetic field surrounds any A magnetic field surrounds any magnetic materialmagnetic material

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Earth’s Magnetic FieldEarth’s Magnetic Field

The Earth’s geographic north pole The Earth’s geographic north pole corresponds to a magnetic south polecorresponds to a magnetic south pole

The Earth’s geographic south pole The Earth’s geographic south pole corresponds to a magnetic north polecorresponds to a magnetic north pole Strictly speaking, a north pole should be Strictly speaking, a north pole should be

a “north-seeking” pole and a south pole a “north-seeking” pole and a south pole a “south-seeking” polea “south-seeking” pole

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Earth’s Magnetic FieldEarth’s Magnetic Field

The Earth’s The Earth’s magnetic field magnetic field resembles that resembles that achieved by achieved by burying a huge burying a huge bar magnet deep bar magnet deep in the Earth’s in the Earth’s interiorinterior

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Dip Angle of Earth’s Dip Angle of Earth’s Magnetic FieldMagnetic Field

If a compass is free to rotate vertically as If a compass is free to rotate vertically as well as horizontally, it points to the earth’s well as horizontally, it points to the earth’s surfacesurface

The angle between the horizontal and the The angle between the horizontal and the direction of the magnetic field is called the direction of the magnetic field is called the dip angledip angle The farther north the device is moved, the farther The farther north the device is moved, the farther

from horizontal the compass needle would befrom horizontal the compass needle would be The compass needle would be horizontal at the equator The compass needle would be horizontal at the equator

and the dip angle would be 0°and the dip angle would be 0° The compass needle would point straight down at the The compass needle would point straight down at the

south magnetic pole and the dip angle would be 90°south magnetic pole and the dip angle would be 90°

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More About the Earth’s More About the Earth’s Magnetic PolesMagnetic Poles

The dip angle of 90° is found at a point The dip angle of 90° is found at a point just north of Hudson Bay in Canadajust north of Hudson Bay in Canada This is considered to be the location of the This is considered to be the location of the

south magnetic polesouth magnetic pole The magnetic and geographic poles are The magnetic and geographic poles are

not in the same exact locationnot in the same exact location The difference between true north, at the The difference between true north, at the

geographic north pole, and magnetic north geographic north pole, and magnetic north is called the is called the magnetic declinationmagnetic declination

The amount of declination varies by location on The amount of declination varies by location on the earth’s surfacethe earth’s surface

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Earth’s Magnetic Earth’s Magnetic DeclinationDeclination

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Source of the Earth’s Source of the Earth’s Magnetic FieldMagnetic Field

There cannot be large masses of There cannot be large masses of permanently magnetized materials permanently magnetized materials since the high temperatures of the since the high temperatures of the core prevent materials from retaining core prevent materials from retaining permanent magnetizationpermanent magnetization

The most likely source of the Earth’s The most likely source of the Earth’s magnetic field is believed to be magnetic field is believed to be electric currents in the liquid part of electric currents in the liquid part of the corethe core

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Reversals of the Earth’s Reversals of the Earth’s Magnetic FieldMagnetic Field

The direction of the Earth’s The direction of the Earth’s magnetic field reverses every few magnetic field reverses every few million yearsmillion years Evidence of these reversals are found Evidence of these reversals are found

in basalts resulting from volcanic in basalts resulting from volcanic activityactivity

The origin of the reversals is not The origin of the reversals is not understoodunderstood

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Magnetic FieldsMagnetic Fields

When moving through a magnetic When moving through a magnetic field, a charged particle field, a charged particle experiences a magnetic forceexperiences a magnetic force This force has a maximum value when This force has a maximum value when

the charge moves perpendicularly to the charge moves perpendicularly to the magnetic field linesthe magnetic field lines

This force is zero when the charge This force is zero when the charge moves along the field linesmoves along the field lines

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Magnetic Fields, contMagnetic Fields, cont

One can define a magnetic field in One can define a magnetic field in terms of the magnetic force terms of the magnetic force exerted on a test chargeexerted on a test charge Similar to the way electric fields are Similar to the way electric fields are

defineddefined

sinqv

FB

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Units of Magnetic FieldUnits of Magnetic Field

The SI unit of magnetic field is the The SI unit of magnetic field is the TeslaTesla (T) (T)

Wb is a WeberWb is a Weber The cgs unit is a The cgs unit is a GaussGauss (G) (G)

1 T = 101 T = 1044 G G

mA

N

)s/m(C

N

m

WbT

2

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A Few Typical B ValuesA Few Typical B Values

Conventional laboratory magnetsConventional laboratory magnets 25000 G or 2.5 T25000 G or 2.5 T

Superconducting magnetsSuperconducting magnets 300000 G or 30 T300000 G or 30 T

Earth’s magnetic fieldEarth’s magnetic field 0.5 G or 5 x 100.5 G or 5 x 10-5-5 T T

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Finding the Direction of Finding the Direction of Magnetic ForceMagnetic Force

Experiments show Experiments show that the direction of that the direction of the magnetic force the magnetic force is always is always perpendicular to perpendicular to both both vv and and BB

FFmaxmax occurs when v occurs when v is perpendicular to is perpendicular to BB

F = 0 when v is F = 0 when v is parallel to Bparallel to B

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Right Hand Rule #1Right Hand Rule #1 Hold your right hand Hold your right hand

openopen Place your fingers in the Place your fingers in the

direction of Bdirection of B Place your thumb in the Place your thumb in the

direction of vdirection of v The direction of the The direction of the

force on a positive force on a positive charge is directed out of charge is directed out of your palmyour palm If the charge is negative, If the charge is negative,

the force is opposite that the force is opposite that determined by the right determined by the right hand rulehand rule

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QUICK QUIZ 19.1

A charged particle moves in a straight line through a certain region of space. The magnetic field in that region (a) has a magnitude of zero, (b) has a zero component perpendicular to the particle's velocity, or (c) has a zero component parallel to the particle's velocity.

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QUICK QUIZ 19.1 ANSWER(b). The force that a magnetic field

exerts on a charged particle moving through it is given by F = qvB sin θ = qvB , where B is the component of the field perpendicular to the particle’s velocity. Since the particle moves in a straight line, the magnetic force (and hence B , since qv ≠ 0) must be zero.

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QUICK QUIZ 19.2

The north-pole end of a bar magnet is held near a stationary positively charged piece of plastic. Is the plastic (a) attracted, (b) repelled, or (c) unaffected by the magnet?

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QUICK QUIZ 19.2 ANSWER

(c). The magnetic force exerted by a magnetic field on a charge is proportional to the charge’s velocity relative to the field. If the charge is stationary, as in this situation, there is no magnetic force.

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Magnetic Force on a Magnetic Force on a Current Carrying Current Carrying ConductorConductor

A force is exerted on a current-A force is exerted on a current-carrying wire placed in a magnetic carrying wire placed in a magnetic fieldfield The current is a collection of many The current is a collection of many

charged particles in motioncharged particles in motion The direction of the force is given The direction of the force is given

by right hand rule #1by right hand rule #1

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Force on a WireForce on a Wire The blue x’s indicate The blue x’s indicate

the magnetic field is the magnetic field is directed directed intointo the page the page The x represents the tail The x represents the tail

of the arrowof the arrow Blue dots would be Blue dots would be

used to represent the used to represent the field directed field directed out of out of the the pagepage The The • represents the • represents the

head of the arrowhead of the arrow In this case, there is no In this case, there is no

current, so there is no current, so there is no forceforce

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Force on a Wire, contForce on a Wire, cont

B is into the pageB is into the page Point your fingers into Point your fingers into

the pagethe page The current is up the The current is up the

pagepage Point your thumb up Point your thumb up

the pagethe page The force is to the The force is to the

leftleft Your palm should be Your palm should be

pointing to the leftpointing to the left

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Force on a Wire, finalForce on a Wire, final

B is into the pageB is into the page Point your fingers into Point your fingers into

the pagethe page The current is down The current is down

the pagethe page Point your thumb Point your thumb

down the pagedown the page The force is to the The force is to the

rightright Your palm should be Your palm should be

pointing to the rightpointing to the right

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Force on a Wire, equationForce on a Wire, equation

The magnetic force is exerted on each The magnetic force is exerted on each moving charge in the wiremoving charge in the wire

The total force is the sum of all the The total force is the sum of all the magnetic forces on all the individual magnetic forces on all the individual charges producing the currentcharges producing the current

F = B I F = B I ℓ ℓ sin sin θθ θ is the angle between B and Iθ is the angle between B and I The direction is found by the right hand The direction is found by the right hand

rule, pointing your thumb in the direction of rule, pointing your thumb in the direction of I instead of vI instead of v

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Torque on a Current LoopTorque on a Current Loop

Applies to any Applies to any shape loopshape loop

N is the number of N is the number of turns in the coil turns in the coil

sin NBIA

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Electric MotorElectric Motor An electric motor An electric motor

converts electrical converts electrical energy to mechanical energy to mechanical energyenergy The mechanical energy The mechanical energy

is in the form of is in the form of rotational kinetic energyrotational kinetic energy

An electric motor An electric motor consists of a rigid consists of a rigid current-carrying loop current-carrying loop that rotates when that rotates when placed in a magnetic placed in a magnetic fieldfield

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Electric Motor, 2Electric Motor, 2

The torque acting on the loop will The torque acting on the loop will tend to rotate the loop to smaller tend to rotate the loop to smaller values of values of θ until the torque becomes 0 θ until the torque becomes 0 at θ = 0°at θ = 0°

If the loop turns past this point and If the loop turns past this point and the current remains in the same the current remains in the same direction, the torque reverses and direction, the torque reverses and turns the loop in the opposite turns the loop in the opposite directiondirection

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Electric Motor, 3Electric Motor, 3

To provide continuous rotation in To provide continuous rotation in one direction, the current in the one direction, the current in the loop must periodically reverseloop must periodically reverse In ac motors, this reversal naturally In ac motors, this reversal naturally

occursoccurs In dc motors, a In dc motors, a split-ring commutatorsplit-ring commutator

and brushes are usedand brushes are used Actual motors would contain many Actual motors would contain many

current loops and commutatorscurrent loops and commutators

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Electric Motor, finalElectric Motor, final

Just as the loop becomes Just as the loop becomes perpendicular to the magnetic field perpendicular to the magnetic field and the torque becomes 0, inertia and the torque becomes 0, inertia carries the loop forward and the carries the loop forward and the brushes cross the gaps in the ring, brushes cross the gaps in the ring, causing the current loop to reverse causing the current loop to reverse its directionits direction This provides more torque to continue This provides more torque to continue

the rotationthe rotation The process repeats itselfThe process repeats itself

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Force on a Charged Force on a Charged Particle in a Magnetic FieldParticle in a Magnetic Field

Consider a particle Consider a particle moving in an external moving in an external magnetic field so that its magnetic field so that its velocity is perpendicular velocity is perpendicular to the fieldto the field

The force is always The force is always directed toward the directed toward the center of the circular center of the circular pathpath

The magnetic force The magnetic force causes a centripetal causes a centripetal acceleration, changing acceleration, changing the direction of the the direction of the velocity of the particlevelocity of the particle

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Force on a Charged Force on a Charged ParticleParticle

Equating the magnetic and Equating the magnetic and centripetal forces:centripetal forces:

Solving for r:Solving for r:

r is proportional to the momentum of r is proportional to the momentum of the particle and inversely proportional the particle and inversely proportional to the magnetic fieldto the magnetic field

r

mvqvBF

2

qB

mvr

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Bending an Electron Beam Bending an Electron Beam in an External Magnetic in an External Magnetic FieldField

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Particle Moving in an Particle Moving in an External Magnetic Field, 2External Magnetic Field, 2

If the particle’s If the particle’s velocity is velocity is notnot perpendicular to perpendicular to the field, the path the field, the path followed by the followed by the particle is a spiralparticle is a spiral The spiral path is The spiral path is

called a called a helixhelix

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Magnetic Fields – Magnetic Fields – Long Straight WireLong Straight Wire

A current-carrying A current-carrying wire produces a wire produces a magnetic fieldmagnetic field

The compass needle The compass needle deflects in directions deflects in directions tangent to the circletangent to the circle The compass needle The compass needle

points in the direction points in the direction of the magnetic field of the magnetic field produced by the produced by the currentcurrent

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Direction of the Field of a Direction of the Field of a Long Straight WireLong Straight Wire

Right Hand Rule Right Hand Rule #2#2 Grasp the wire in Grasp the wire in

your right handyour right hand Point your thumb in Point your thumb in

the direction of the the direction of the currentcurrent

Your fingers will Your fingers will curl in the direction curl in the direction of the fieldof the field

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Magnitude of the Field of a Magnitude of the Field of a Long Straight WireLong Straight Wire

The magnitude of the field at a The magnitude of the field at a distance r from a wire carrying a distance r from a wire carrying a current of I iscurrent of I is

µµoo = 4 = 4 x 10 x 10-7 -7 T m / AT m / A µµo o is called the is called the permeability of free permeability of free

spacespace

r2

IB o

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AmpAmpère’s Lawère’s Law

AndrAndré-Marie é-Marie AmpAmpère found a ère found a procedure for deriving the procedure for deriving the relationship between the current in relationship between the current in a arbitrarily shaped wire and the a arbitrarily shaped wire and the magnetic field produced by the magnetic field produced by the wirewire

AmpAmpère’s Circuital Lawère’s Circuital Law BB|| || Δℓ = µΔℓ = µoo I I Sum over the closed pathSum over the closed path

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AmpAmpère’s Law, contère’s Law, cont

Choose an Choose an arbitrary closed arbitrary closed path around the path around the currentcurrent

Sum all the Sum all the products of products of BB|| || Δℓ Δℓ around the closed around the closed pathpath

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AmpAmpère’s Law to Find B for ère’s Law to Find B for a Long Straight Wirea Long Straight Wire

Use a closed circular Use a closed circular pathpath

The circumference of The circumference of the circle is 2 the circle is 2 r r

This is identical to the This is identical to the result previously result previously obtainedobtained

r2

IB o

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Magnetic Force Between Magnetic Force Between Two Parallel ConductorsTwo Parallel Conductors

The force on wire The force on wire 1 is due to the 1 is due to the current in wire 1 current in wire 1 and the magnetic and the magnetic field produced by field produced by wire 2wire 2

The force per unit The force per unit length is:length is:

d2

IIF 21o

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Force Between Two Force Between Two Conductors, contConductors, cont

Parallel conductors carrying Parallel conductors carrying currents in the same direction currents in the same direction attract each other attract each other

Parallel conductors carrying Parallel conductors carrying currents in the opposite directions currents in the opposite directions repel each otherrepel each other

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Defining Ampere and Defining Ampere and CoulombCoulomb

The force between parallel conductors The force between parallel conductors can be used to define the Ampere (A)can be used to define the Ampere (A) If two long, parallel wires 1 m apart carry If two long, parallel wires 1 m apart carry

the same current, and the magnitude of the the same current, and the magnitude of the magnetic force per unit length is 2 x 10magnetic force per unit length is 2 x 10-7-7 N/m, then the current is defined to be 1 AN/m, then the current is defined to be 1 A

The SI unit of charge, the Coulomb (C), The SI unit of charge, the Coulomb (C), can be defined in terms of the Amperecan be defined in terms of the Ampere If a conductor carries a steady current of 1 If a conductor carries a steady current of 1

A, then the quantity of charge that flows A, then the quantity of charge that flows through any cross section in 1 second is 1 Cthrough any cross section in 1 second is 1 C

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QUICK QUIZ 19.5

If I1 = 2 A and I2 = 6 A in the figure below, which of the following is true:

(a) F1 = 3F2, (b) F1 = F2, or (c) F1 = F2/3?

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QUICK QUIZ 19.5 ANSWER

(b). The two forces are an action-reaction pair. They act on different wires, and have equal magnitudes but opposite directions.

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Magnetic Field of a Magnetic Field of a Current LoopCurrent Loop

The strength of a The strength of a magnetic field magnetic field produced by a wire produced by a wire can be enhanced can be enhanced by forming the wire by forming the wire into a loopinto a loop

All the segments, All the segments, Δx, contribute to Δx, contribute to the field, increasing the field, increasing its strengthits strength

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Magnetic Field of a Magnetic Field of a Current Loop – Total FieldCurrent Loop – Total Field

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Magnetic Field of a Magnetic Field of a SolenoidSolenoid

If a long straight wire If a long straight wire is bent into a coil of is bent into a coil of several closely several closely spaced loops, the spaced loops, the resulting device is resulting device is called a solenoidcalled a solenoid

It is also known as It is also known as an electromagnet an electromagnet since it acts like a since it acts like a magnet only when it magnet only when it carries a currentcarries a current

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Magnetic Field of a Magnetic Field of a Solenoid, 2Solenoid, 2

The field lines inside the solenoid are The field lines inside the solenoid are nearly parallel, uniformly spaced, and nearly parallel, uniformly spaced, and close togetherclose together This indicates that the field inside the This indicates that the field inside the

solenoid is nearly uniform and strongsolenoid is nearly uniform and strong The exterior field is nonuniform, The exterior field is nonuniform,

much weaker, and in the opposite much weaker, and in the opposite direction to the field inside the direction to the field inside the solenoidsolenoid

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Magnetic Field in a Magnetic Field in a Solenoid, 3Solenoid, 3

The field lines of the solenoid resemble The field lines of the solenoid resemble those of a bar magnetthose of a bar magnet

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Magnetic Field in a Magnetic Field in a Solenoid, MagnitudeSolenoid, Magnitude

The magnitude of the field inside a The magnitude of the field inside a solenoid is constant at all points far solenoid is constant at all points far from its endsfrom its ends

B = µB = µoo n I n I n is the number of turns per unit lengthn is the number of turns per unit length n = N / n = N / ℓℓ

The same result can be obtained by The same result can be obtained by applying Ampapplying Ampère’s Law to the ère’s Law to the solenoidsolenoid

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Magnetic Field in a Magnetic Field in a Solenoid from AmpSolenoid from Ampère’s ère’s LawLaw

A cross-sectional A cross-sectional view of a tightly view of a tightly wound solenoidwound solenoid

If the solenoid is long If the solenoid is long compared to its compared to its radius, we assume radius, we assume the field inside is the field inside is uniform and outside uniform and outside is zerois zero

Apply AmpApply Ampère’s Law ère’s Law to the red dashed to the red dashed rectanglerectangle

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Magnetic Effects of Magnetic Effects of Electrons -- OrbitsElectrons -- Orbits

An individual atom should act like a magnet An individual atom should act like a magnet because of the motion of the electrons because of the motion of the electrons about the nucleusabout the nucleus Each electron circles the atom once in about Each electron circles the atom once in about

every 10every 10-16-16 seconds seconds This would produce a current of 1.6 mA and a This would produce a current of 1.6 mA and a

magnetic field of about 20 T at the center of the magnetic field of about 20 T at the center of the circular pathcircular path

However, the magnetic field produced by However, the magnetic field produced by one electron in an atom is often canceled by one electron in an atom is often canceled by an oppositely revolving electron in the same an oppositely revolving electron in the same atomatom

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Magnetic Effects of Magnetic Effects of Electrons – Orbits, contElectrons – Orbits, cont

The net result is that the magnetic The net result is that the magnetic effect produced by electrons effect produced by electrons orbiting the nucleus is either zero orbiting the nucleus is either zero or very small for most materialsor very small for most materials

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Magnetic Effects of Magnetic Effects of Electrons -- SpinsElectrons -- Spins

Electrons also Electrons also have spinhave spin The classical The classical

model is to model is to consider the consider the electrons to spin electrons to spin like topslike tops

It is actually a It is actually a relativistic and relativistic and quantum effectquantum effect

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Magnetic Effects of Magnetic Effects of Electrons – Spins, contElectrons – Spins, cont

The field due to the spinning is The field due to the spinning is generally stronger than the field generally stronger than the field due to the orbital motiondue to the orbital motion

Electrons usually pair up with their Electrons usually pair up with their spins opposite each other, so their spins opposite each other, so their fields cancel each otherfields cancel each other That is why most materials are not That is why most materials are not

naturally magneticnaturally magnetic

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Magnetic Effects of Magnetic Effects of Electrons -- DomainsElectrons -- Domains

In some materials, the spins do not In some materials, the spins do not naturally cancelnaturally cancel Such materials are called Such materials are called ferromagneticferromagnetic

Large groups of atoms in which the spins Large groups of atoms in which the spins are aligned are called are aligned are called domainsdomains

When an external field is applied, the When an external field is applied, the domains that are aligned with the field domains that are aligned with the field tend to grow at the expense of the otherstend to grow at the expense of the others This causes the material to become This causes the material to become

magnetizedmagnetized

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Domains, contDomains, cont

Random alignment, a, shows an Random alignment, a, shows an unmagnetized materialunmagnetized material

When an external field is applied, the When an external field is applied, the domains aligned with B grow, bdomains aligned with B grow, b

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Domains and Permanent Domains and Permanent MagnetsMagnets

In hard magnetic materials, the In hard magnetic materials, the domains remain aligned after the domains remain aligned after the external field is removedexternal field is removed The result is a permanent magnetThe result is a permanent magnet

In soft magnetic materials, once the external field In soft magnetic materials, once the external field is removed, thermal agitation cause the materials is removed, thermal agitation cause the materials to quickly return to an unmagnetized stateto quickly return to an unmagnetized state

With a core in a loop, the magnetic field With a core in a loop, the magnetic field is enhanced since the domains in the is enhanced since the domains in the core material align, increasing the core material align, increasing the magnetic fieldmagnetic field