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General Physics (PHY 2140)

Lecture 7Lecture 7

Ø Electricity and Magnetism

üMagnetismüMagnetic fields and forceüApplication of magnetic forces

Chapter 19

http://www.physics.wayne.edu/~alan/2140Website/Main.htm

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Lightning ReviewLightning Review

Last lecture:

1.1. DC circuitsDC circuitsüü Series and Parallel ResistorsSeries and Parallel Resistors

üü Kirchoff’sKirchoff’s rulesrules

üü RC circuitRC circuit

Review Problem: The three light bulbs in the circuit all have the same resistance. Given that brightness is proportional to power dissipated, the brightness of bulbs B and C together, compared with the brightness of bulb A, is

1. twice as much.2. the same.3. half as much.

1 1

0, 0n n

i ii i

I V= =

= =∑ ∑( )/

/

1 t RC

t RC

q Q e

q Qe

−

−

= −

=

L

L

321

321

1111RRRR

RRRR

eq

eq

++=

+++=

P=I2R, I=V/2R

P=2x(V2/4R2)R

So P=1/2(V2/R)

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MagnetismMagnetism

Magnetic effects from natural magnets have been known for a longMagnetic effects from natural magnets have been known for a longtime. Recorded observations from the Greeks more than 2500 yeartime. Recorded observations from the Greeks more than 2500 years s ago.ago.

The word magnetism comes from the Greek word for aThe word magnetism comes from the Greek word for a certain type certain type of stone (lodestone) containing iron oxide found in of stone (lodestone) containing iron oxide found in MagnesiaMagnesia, a , a district in northern Greece.district in northern Greece.

Properties of lodestones: could exert forces on similar stones aProperties of lodestones: could exert forces on similar stones and nd could impart this property (magnetize) to a piece of iron it toucould impart this property (magnetize) to a piece of iron it touched.ched.

Small sliver of lodestone suspended with a string will always alSmall sliver of lodestone suspended with a string will always align ign itself in a northitself in a north--south directionsouth direction——it detects the earth’s magnetic field.it detects the earth’s magnetic field.

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Magnetic MaterialsMagnetic Materials(a simple look at an advanced topic)(a simple look at an advanced topic)

• Materials can be classified by how they respond to an applied magnetic field, Bapp.

• Paramagnetic (aluminum, tungsten, oxygen,…)

• Atomic magnetic dipoles (~atomic bar magnets) tend to line up with the field, increasing it. But thermal motion randomizes their directions, so only a small effect persists: Bind ~ Bapp •10-5

• Diamagnetic (gold, copper, water,…)

• The applied field induces an opposing field; again, this is usually very weak; Bind ~ -Bapp •10-5 [Exception: Superconductors exhibit perfect diamagnetism à they exclude all magnetic fields]

• Ferromagnetic (iron, cobalt, nickel,…)• Somewhat like paramagnetic, the dipoles prefer to line up with

the applied field. But there is a complicated collective effectdue to strong interactions between neighboring dipoles à they tend to all line up the same way.

• Very strong enhancement. Bind ~ Bapp •10+5

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Ferromagnets, cont.Ferromagnets, cont.• Even in the absence of an applied B, the dipoles tend to

strongly align over small patches – “domains”. Applying an external field, the domains align to produce a large net magnetization.

• “Soft” ferromagnets• The domains re-randomize when the field is removed

• “Hard” ferromagnets• The domains persist even when the field is removed• “Permanent” magnets

• Domains may be aligned in a different direction by applying a new field

• Domains may be re-randomized by sudden physical shock• If the temperature is raised above the “Curie point” (770° for

iron), the domains will also randomize à paramagnet

MagneticDomains

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1A •Which kind of material would you use in a video tape?

(a) diamagnetic

(b) paramagnetic

(c) “soft” ferromagnetic

1B •How does a magnet attract screws, paper clips, refrigerators, etc., when they are not “magnetic”?

MiniMini--quizquiz

(d) “hard” ferromagnetic

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1A •Which kind of material would you use in a video tape?

(a) diamagnetic

(b) paramagnetic

(c) “soft” ferromagnetic

MiniMini--quizquiz

(d) “hard” ferromagnetic

Diamagnetism and paramagnetism are far too weak to be used for a video tape. Since we want the information to remain on the tape after recording it, we need a “hard” ferromagnet. These are the key to the information age—cassette tapes, hard drives, ZIP disks, credit card strips,…

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1B •How does a magnet attract screws, paper clips, refrigerators, etc., when they are not “magnetic”?

MiniMini--quizquiz

The materials are all “soft” ferromagnets. The external field temporarily aligns the domains so there is a net dipole, which is then attracted to the bar magnet.

- The effect vanishes with no applied B field- It does not matter which pole is used.

End of paper clipS N

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Applications: A “bit” of historyApplications: A “bit” of history

IBM introduced the first IBM introduced the first hard diskhard disk in in 19571957, when , when data usually was stored on data usually was stored on tapes. It consisted of tapes. It consisted of 50 50 plattersplatters, 24 inch diameter, , 24 inch diameter, and was twice the size of and was twice the size of a refrigerator. a refrigerator.

It cost It cost $35,000$35,000 annually in leasing fees (IBM would not annually in leasing fees (IBM would not sell it outright). It’s sell it outright). It’s totaltotal storage capacity was storage capacity was 5 MB5 MB, a , a huge number for its time! huge number for its time!

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Magnetic Field DirectionMagnetic Field Direction

The magnetic field direction (of a magnet bar) can The magnetic field direction (of a magnet bar) can studied with a small compass. studied with a small compass.

N S

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Magnetic Field LinesMagnetic Field Lines

N S

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Bar MagnetBar MagnetBar magnet Bar magnet ... two poles: N and S... two poles: N and S

Like poles repel; Unlike poles attract.Like poles repel; Unlike poles attract.

Magnetic Field lines: Magnetic Field lines: (defined in same way as electric field lines, (defined in same way as electric field lines, direction and density)direction and density)

• Does this remind you of a similar case in electrostatics?

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Magnetic Field Lines of a Magnetic Field Lines of a bar magnetbar magnet

Electric Field LinesElectric Field Linesof an Electric Dipoleof an Electric Dipole

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Magnetic MonopolesMagnetic MonopolesPerhaps there exist Perhaps there exist magnetic chargesmagnetic charges, just like electric charges. Such an , just like electric charges. Such an entity would be called a entity would be called a magnetic monopolemagnetic monopole (having + or (having + or -- magnetic magnetic charge).charge).

How can you isolate this magnetic charge?How can you isolate this magnetic charge?

Try cutting a bar magnet in half:Try cutting a bar magnet in half:

•• Many searches for magnetic monopolesMany searches for magnetic monopoles——the existence of which the existence of which would explain (within framework of QM) the quantization of electwould explain (within framework of QM) the quantization of electric ric charge (argument of charge (argument of DiracDirac))

•• No monopoles have ever been found!No monopoles have ever been found!

NS N NS SEven an individual electron has a magnetic “dipole”!

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Source of Magnetic Fields?Source of Magnetic Fields?

What is the source of magnetic fields, if What is the source of magnetic fields, if notnot magnetic charge?magnetic charge?

Answer: Answer: electric charge in motionelectric charge in motion!!

nn e.g., current in wire surrounding cylinder (solenoid) e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. produces very similar field to that of bar magnet.

Therefore, understanding source of field generated by bar magnetTherefore, understanding source of field generated by bar magnet lies lies in understanding currents at atomic level within bulk matter.in understanding currents at atomic level within bulk matter.

Orbits of electrons about nuclei

Intrinsic “spin” of electrons (more important effect)

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19.2 Magnetic Field of the Earth19.2 Magnetic Field of the Earth

A small magnetic bar should be said to have north and A small magnetic bar should be said to have north and south south seekingseeking poles. The north of the bar points towards poles. The north of the bar points towards the North of the Earth.the North of the Earth.The geographic north corresponds to a south The geographic north corresponds to a south magnetic pole and the geographic south magnetic pole and the geographic south corresponds to a magnetic northcorresponds to a magnetic north..The configuration of the Earth magnetic resemble that of The configuration of the Earth magnetic resemble that of a (big) magnetic bar one would put in its center.a (big) magnetic bar one would put in its center.

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Magnetic Field of the EarthMagnetic Field of the Earth

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Magnetic Field of the EarthMagnetic Field of the Earth

Near the ground, the field is NOT parallel to the surface Near the ground, the field is NOT parallel to the surface of the Earth. of the Earth. nn The angle between the direction of the magnetic field and the The angle between the direction of the magnetic field and the

horizontal is called dip angle. horizontal is called dip angle.

The north and south magnetic pole do not exactly The north and south magnetic pole do not exactly correspond to the south and north geographic north.correspond to the south and north geographic north.nn South magnetic pole found (in 1832) to be just north of Hudson South magnetic pole found (in 1832) to be just north of Hudson

bay in Canada bay in Canada –– 1300 miles from the north geographical pole. 1300 miles from the north geographical pole.

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More on the Magnetic Field of the EarthMore on the Magnetic Field of the Earth

The difference between the geographical north and the direction The difference between the geographical north and the direction pointed at by a compass changes from point to point and is callepointed at by a compass changes from point to point and is called d the magnetic declination.the magnetic declination.Source of the fieldSource of the field : charge: charge--carrying convection currents in the core carrying convection currents in the core of the earth. of the earth. nn In part related to the rotation of the earthIn part related to the rotation of the earth

The orientation of the field “flips” and changes over time The orientation of the field “flips” and changes over time –– every few every few million years…million years…nn Basalt rocks (iron content)Basalt rocks (iron content)

Other planets (e.g. Jupiter) are found to have a magnetic field.Other planets (e.g. Jupiter) are found to have a magnetic field.

http://www.nasa.gov/vision/earth/lookingatearth/29dec_magneticfihttp://www.nasa.gov/vision/earth/lookingatearth/29dec_magneticfield.htmleld.html

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Magnetic Field of the Earth Magnetic Field of the Earth -- normalnormal

G.A. Glatzmaier and P.H. Roberts

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Magnetic Field of the Earth During a Field Magnetic Field of the Earth During a Field ReversalReversal

G.A. Glatzmaier and P.H. Roberts

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MiniMini--quizquiz

You travel to Australia for a business trip and bring You travel to Australia for a business trip and bring along your Americanalong your American--made compass. Does the made compass. Does the compass work correctly in Australia???compass work correctly in Australia???

• No problem using the compass in Australia. • North pole of the compass will be attracted to the

South magnetic pole…i.e. the North geo. pole• The vertical component of the field is different

(opposite) but that cannot be detected with normal operation of the compass.

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19.3 Magnetic Fields 19.3 Magnetic Fields

Stationary charged particles do NOT interact with a Stationary charged particles do NOT interact with a magnetic field.magnetic field.Charge Charge movingmoving through a magnetic field experience a through a magnetic field experience a magnetic forcemagnetic force..Value of the force is maximumValue of the force is maximum when the charge moves when the charge moves perpendicularly to the field linesperpendicularly to the field lines. . Value of the force is zeroValue of the force is zero when the charge moves when the charge moves parallel to the field linesparallel to the field lines..

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Magnetic Fields in analogy with Electric FieldsMagnetic Fields in analogy with Electric Fields

Electric FieldElectric Field::nn Distribution of chargeDistribution of charge creates an creates an electricelectric field field EE((rr) )

in the surrounding space.in the surrounding space.nn Field exerts a force Field exerts a force FF=q =q EE((rr)) on a charge on a charge qq at at rr

Magnetic FieldMagnetic Field::nn MovingMoving chargecharge or current creates a or current creates a magneticmagnetic field field

BB((rr) ) in the surrounding space.in the surrounding space.nn Field exerts a force Field exerts a force FF on a charge on a charge movingmoving qq at at rr

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Strength of the Magnetic FieldStrength of the Magnetic Field

Define the Define the magnetic field, Bmagnetic field, B, at a given point in space in , at a given point in space in terms of the terms of the magnetic forcemagnetic force imparted on a moving imparted on a moving charge at that point.charge at that point.Observations show that the force is proportional toObservations show that the force is proportional tonn The fieldThe fieldnn The chargeThe chargenn The velocity of the particleThe velocity of the particlenn The sine of the angle between the field and the direction of theThe sine of the angle between the field and the direction of the

particle’s motion.particle’s motion.

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Strength and direction of the Strength and direction of the Magnetic ForceMagnetic Force on a charge on a charge in motionin motion

sinF qvB θ=

+q

F

B

v

θ

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Magnetic Field MagnitudeMagnetic Field Magnitude

sinF

Bqv θ

=

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Magnetic Field UnitsMagnetic Field Units

[F] = [F] = newtonnewton[v] = m/s[v] = m/s[q] = C[q] = C[B] = [B] = teslatesla (T).(T).nn Also called Also called weberweber ((WbWb) per square meter.) per square meter.nn 1 T = 1 Wb/m1 T = 1 Wb/m22..nn 1 T = 1 N s m1 T = 1 N s m--11 CC--11..nn 1 T = 1 N A1 T = 1 N A--11 mm--11..

CGS unit is the Gauss (G)CGS unit is the Gauss (G)nn 1 T = 101 T = 1044 G.G. (Earth’s field ~ 0.5 G)(Earth’s field ~ 0.5 G)

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Right Hand RuleRight Hand RuleProvides a convenient trick to remember the Provides a convenient trick to remember the spatial relationship between F, v, and B.spatial relationship between F, v, and B.Consider the motion of positive chargeConsider the motion of positive chargeDirection of force reversed if negative charge.Direction of force reversed if negative charge.

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Example: Proton traveling in Earth’s Example: Proton traveling in Earth’s magnetic field.magnetic field.A proton moves with a speed of 1.0 x 10A proton moves with a speed of 1.0 x 1055 m/s through the Earth’s magnetic field which m/s through the Earth’s magnetic field which has a value of 55 has a value of 55 µµTT a particular location. When the proton moves eastward, the a particular location. When the proton moves eastward, the magnetic force acting on it is a maximum, and when it moves nortmagnetic force acting on it is a maximum, and when it moves northward, no magnetic hward, no magnetic force acts on it. What is the strength of the force acts on it. What is the strength of the magnetic forcemagnetic force? And what is the ? And what is the directiondirection of of the magnetic field?the magnetic field?

V = 1.0 x 105 m/s B = 55 µT

sinF qvB θ=Northward or southward.

N

proton

Nx

TxsmxCxF19

6519

108.8

)90)(sin1055)(/100.1)(106.1(−

−−

=

°=

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19.4 Magnetic Force on Current19.4 Magnetic Force on Current--carrying carrying conductor.conductor.

A magnetic force is exerted on a single charge in motion A magnetic force is exerted on a single charge in motion through a magnetic field.through a magnetic field.That implies a force should also be exerted on a That implies a force should also be exerted on a collection of charges in motion through a conductor i.e. a collection of charges in motion through a conductor i.e. a current.current.And it does!!!And it does!!!The force on a current is the sum of all elementary The force on a current is the sum of all elementary forces exerted on all charge carriers in motion.forces exerted on all charge carriers in motion.

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19.4 Magnetic Force on Current:19.4 Magnetic Force on Current:some notation conventionssome notation conventions

If B is directed into the page If B is directed into the page we use blue crosses we use blue crosses representing the tail of arrows representing the tail of arrows indicating the direction of the indicating the direction of the field,field,If B is directed out of the page, If B is directed out of the page, we use dots.we use dots.If B is in the page, we use lines If B is in the page, we use lines with arrow heads. with arrow heads.

x x x xx x x x x

x x x x x xx x x x xx x x x

. . . .. . . . .

. . . . . .. . . . .. . . .

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Force on a wire carrying current Force on a wire carrying current in a magnetic field.in a magnetic field.

x x x xx x x x x

x x x x x xx x x x xx x x x

x x x xx x x x x

x x x x x xx x x x xx x x x

I = 0 I

Bin Bin x x x xx x x x x

x x x x x xx x x x xx x x x

I

Bin

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x x x x x x x x x

x x x x x x x x x x x x x xx x x x x x xx x x x x x x

x x x x x

Force on a wire carrying current in a magnetic field.

Avdq

( ) ( )max dF qv B nAl=

dI nqv A=

maxF BIl=Magnetic Field and Current at right angle from each other.

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Force on a wire carrying current in a magnetic field.Force on a wire carrying current in a magnetic field.

General Case: field at angle General Case: field at angle θθ relative to current.relative to current.

max sinF BIl θ=

I

B

θB sin θ

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Voice CoilVoice Coil

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MiniMini--QuizQuiz

In a lightning strike, there is a rapid flow of negative In a lightning strike, there is a rapid flow of negative charges from a cloud to the ground. In what direction is charges from a cloud to the ground. In what direction is a lightning strike deflected by the Earth’s magnetic a lightning strike deflected by the Earth’s magnetic field?field?

Reasoning:Negative charge flow down.Positive Current upward.B field direction Geo South to Geo NorthAnswer:Force towards the west.

I

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Example: Wire in Earth’s B FieldExample: Wire in Earth’s B Field

A wire carries a current of 22 A from east to west. Assume that A wire carries a current of 22 A from east to west. Assume that at this location at this location the magnetic field of the earth is horizontal and directed from the magnetic field of the earth is horizontal and directed from south to north, south to north, and has a magnitude of 0.50 x 10and has a magnitude of 0.50 x 10--44 T. Find the magnetic force on a 36T. Find the magnetic force on a 36--m length m length of wire. What happens if the direction of the current is reverseof wire. What happens if the direction of the current is reversed?d?

B=0.50 x 10-4 T. I = 22 Al = 36 mFmax = BIl

( )( )( )max

4

2

0.50 10 22 36

4.0 10

F BIl

T A m

N−

−

= ×

=

= ×

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19.5 Torque on a Current Loop19.5 Torque on a Current Loop

Imagine a current loop in a magnetic field as follows:Imagine a current loop in a magnetic field as follows:

BI

b

a

Ba/2

F

F

F

F

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1 2F F BIb= =

( ) ( )max 1 22 2 2 2a a a aF F BIb BIbτ = + = +

max BIba BIAτ = =

sinBIAτ θ=

BI

b

a

Ba/2

F

F

F

F

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sinNBIAτ θ=

In a motor, one has “N” loops of current

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Example: Torque on a circular loop in a Example: Torque on a circular loop in a magnetic fieldmagnetic field

A circular loop of radius 50.0 cm is oriented at an A circular loop of radius 50.0 cm is oriented at an angle of 30.0angle of 30.0oo to a magnetic field of 0.50 T. The to a magnetic field of 0.50 T. The current in the loop is 2.0 A. Find the magnitude of the current in the loop is 2.0 A. Find the magnitude of the torque.torque.

r = 0.500 mθ = 30o

B = 0.50 TI = 2.0 AN = 1

( )( ) ( )2

sin

0.50 2.0 0.5

0.39

0 sin30.0 o

NBIA

T A m

Nm

τ θ

π

τ

=

= =

B

30.0o

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Galvanometer/ApplicationsGalvanometer/Applications

Device used in the construction Device used in the construction of ammeters and voltmeters.of ammeters and voltmeters.

Magnet

Current loop or coil

Spring

Scale

sinNBIAτ θ=

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Galvanometer used as AmmeterGalvanometer used as Ammeter

Typical galvanometer have an internal resistance of the order ofTypical galvanometer have an internal resistance of the order of60 60 ΩΩ -- that could significantly disturb (reduce) a current that could significantly disturb (reduce) a current measurement.measurement.Built to have full scale for small Built to have full scale for small current ~ 1 current ~ 1 mAmA or less. or less. Must therefore be mounted in parallel with a small resistor or Must therefore be mounted in parallel with a small resistor or shunt resistor.shunt resistor.

Galvanometer 60 Ω

Rp

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Galvanometer 60 Ω

Rp

• Let’s convert a 60 Ω, 1 mA full scale galvanometer to an ammeter that can measure up to 2 A current.

• Rp must be selected such that when 2 A passes through the ammeter, only 0.001 A goes through the galvanometer.

( ) ( ) ( )0.001 60 1.999

0.03002p

p

A A R

R

Ω =

= Ω

• Rp is rather small!• The equivalent resistance of the circuit is also small!

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Galvanometer used as Voltmeter• Finite internal resistance of a galvanometer must also

addressed if one wishes to use it as voltmeter. • Must mounted a large resistor in series to limit the current

going though the voltmeter to 1 mA.• Must also have a large resistance to avoid disturbing

circuit when measured in parallel.

Galvanometer 60 ΩRs

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Galvanometer 60 ΩRs

Maximum voltage across galvanometer:

( )( )max 0.001 60 0.06V A V∆ = Ω =

Suppose one wish to have a voltmeter that can measure voltage difference up to 100 V:

( ) ( )100 0.001 60

99940p

p

V A R

R

= + Ω

= Ω Large resistance

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19.6 Motion of Charged Particle in magnetic field19.6 Motion of Charged Particle in magnetic field

Consider positively charge Consider positively charge particle moving in a uniform particle moving in a uniform magnetic field.magnetic field.Suppose the initial velocity of Suppose the initial velocity of the particle is perpendicular to the particle is perpendicular to the direction of the field.the direction of the field.Then a magnetic force will be Then a magnetic force will be exerted on the particle and exerted on the particle and make follow a circular path.make follow a circular path.

× × × × × × ×

× × × × × × ×

× × × × × × ×

× × × × × × ×

Fv

qr

Bin

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The magnetic force produces a centripetal acceleration.2mv

F qvBr

= =

mvr

qB=

The particle travels on a circular trajectory with a radius:

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Example: Proton moving in uniform magnetic fieldExample: Proton moving in uniform magnetic field

A proton is moving in a circular orbit of radius 14 cm in a unifA proton is moving in a circular orbit of radius 14 cm in a uniform orm magnetic field of magnitude 0.35 T, directed perpendicular to thmagnetic field of magnitude 0.35 T, directed perpendicular to the e velocity of the proton. Find the orbital speed of the proton.velocity of the proton. Find the orbital speed of the proton.

r = 0.14 mB = 0.35 Tm = 1.67x10-27 kgq = 1.6 x 10-19 C

mvr

qB=

( ) ( )( )( )19 2

27

6

1.6 10 0.35 14 10

1.67 10

4.7 10 ms

qBrv

m

C T m

kg

− −

−

=

× ×=

×

= ×

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Application: Mass SpectrometerApplication: Mass Spectrometer

mvr

qB=

See prob. 30 in text