(1) I have completed at least 50% of the reading and study-guide assignments associated with the lecture, as indicated on the course schedule. a) True b) False

iClicker Quiz

Hint: This is a good time to read All of the chapter summaries about magnetism. Today: B fields near magnets & the forces they create. A. B field & Units; B. Terms: Poles, magnetic (dipole) moments C. How to draw & understand drawings. D. Figure out how & when to use the right hand rule to get direction of forces from moving charges in magnetic fields. Also qvxB (cross products) to get magnitude.

Tuesday is a late day

b 7 due Friday

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 exert forces on each other

• Like poles repel each other N-N or S-S • Unlike poles attract each other N-S

Introduction

Magnetic Poles, cont. •The poles received their names due to the way a magnet behaves in the Earth’s magnetic field. •If a bar magnet is suspended so that it can move freely, it will rotate.

–The magnetic north pole points toward the Earth’s north geographic pole. • This means the Earth’s north geographic pole is a magnetic south pole.

• Similarly, the Earth’s south geographic pole is a magnetic north pole. Introduction

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 always found in pairs.

–All attempts so far to detect an isolated magnetic pole has been unsuccessful. • No matter how many times a permanent magnetic is

cut in two, each piece always has a north and south pole.

Introduction

Magnetic Fields

•Reminder: an electric field surrounds any electric charge •The region of space surrounding any moving electric charge also contains a magnetic field. •A magnetic field also surrounds a magnetic substance making up a permanent magnet.

Section 29.1

Magnetic Fields, cont.

•A vector quantity •Symbolized by •Direction is given by the direction a north pole of a compass needle points in that location •Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look.

B

Section 29.1

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.

Section 29.1

Magnetic Field Lines, Bar Magnet

•Iron filings are used to show the pattern of the magnetic field lines. •The direction of the field is the direction a north pole would point.

Section 29.1

Magnetic Field Lines, Opposite Poles

•Iron filings are used to show the pattern of the magnetic 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 Section 29.1

Magnetic Field Lines, Like 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 like charges

Section 29.1

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.

Section 29.1

Where is Provo? A on the left or B on the right

Magnetic field of the earth

Magnetic north is geographic south (except for slight angle of declination) Large magnetic dipole moment at the center of the earth.

µ

Earth’s Magnetic Poles •More proper terminology would be that a magnet has “north-seeking” and “south-seeking” poles. •The north-seeking pole points to the north geographic pole.

– This would correspond to the Earth’s south magnetic pole. •The south-seeking pole points to the south geographic pole.

– This would correspond to the Earth’s north magnetic pole. •The configuration of the Earth’s magnetic field is very much like the one that would be achieved by burying a gigantic bar magnet deep in the Earth’s interior. •

Section 29.1

Earth’s Magnetic Field •The source of the Earth’s magnetic field is likely convection currents in the Earth’s core. •There is strong evidence that the magnitude of a planet’s magnetic field is related to its rate of rotation. •The direction of the Earth’s magnetic field reverses occasionally. Time ~105 years.

Section 29.1

Units of Magnetic Field

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

A non-SI commonly used unit is a gauss (G)

1 T = 104 G

Magnetic dipole moments units (J/T)

mAN

m/sCN

mWbT 2 ⋅

=⋅

==

2mATJ

⋅=Really? Its units are amps x Area?

Apparently so.

Protons & Magnetic Poles. pp

A. Are repelled by North Poles & attracted by South Poles.

B. Are repelled by South Poles & attracted by North Poles.

C. Are attracted by the closest pole D. Are neither attracted nor repelled by

Poles

More About Direction

Section 29.1

Lorentz Force law. F = q v x B

BvqF

×=

Positive charge RHR

Negative charge LHR

More About Magnitude of F

•The magnitude of the magnetic force on a charged particle is FB = |q| v B sin θ.

�θ is the smaller angle between v and B –FB is zero when the field and velocity are parallel

or antiparallel �θ = 0 or 180o

–FB is a maximum when the field and velocity are perpendicular �θ = 90o

Section 29.1

B x

z

y a) +y b) −y c) +z d) −z e) zero

What is the direction of the Lorentz force on the proton?

p+

e-

What is the direction of the Lorentz force on the electron?

μ-

What is the direction of the Lorentz force on the muon?

B

At the equator, the B field points north.

a) up (radially outward) b) down (radially inward) c) north d) south e) east f) west

Positive charge moving near the equator.

v

If charge is moving east, which way does the Lorentz force point? v

If charge is moving upward, which way does the Lorentz force point?

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 particle regardless of whether the particle is moving.

– The magnetic force acts on a charged particle only when the particle is in motion.

Section 29.1

Work in Fields •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 of its point of application.

•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 given velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy.

Section 29.1

Units of Magnetic Field

•The SI unit of magnetic field is the tesla (T). –Wb is a weber

•A non-SI commonly used unit is a gauss (G). –1 T = 104 G

2 ( / )Wb N NTm C m s A m

= = =⋅ ⋅

Section 29.1

Notation Notes •When vectors are perpendicular to the page, dots and crosses are used.

– The dots represent the arrows coming out of the page.

– The crosses represent the arrows going into the page.

•The same notation applies to other vectors.

Section 29.2

Magnetic force on a current-carrying wire

The force exerted on a small segment is

BsF

×= dId

Which way does the force on this ds point?

∫ ×= BsF

dI

The total force on exerted on the wire is

Dots (out): Crosses (in):

Total force is path independent for uniform fields.

( ) BLIBdIdI

×=×=×= ∫∫ sBsF

Which wire experiences the greatest force?

No net force. But there is a torque.

A current loop in a uniform field

If the field is not uniform, a net force is possible.

Aμ NI=

IabNIA ==µ

If I = 1 Amp, a = 3 m and b = 2 m, then µ = 6 A⋅m2.

Electric dipoles

Points from (−) to (+).

+−= rp Q

p

30

ˆ)ˆ(34

1r

prrpE −⋅=

πε

Idealized as a “point” dipole in the limit that r+− → 0 and Q → ∞ as p stays constant.

Magnetic dipoles

Points from (S) to (N).

30 ˆ)ˆ(3

4 rB μrrμ −⋅

=π

µ

Idealized as a “point” dipole in the limit that A → 0 and I → ∞ as µ stays constant.

Aμ I=

µ

Magnetic dipole moments of subatomic particles

1836J/T109.27

224 B

Ne

B me µµµ ≈×== −Bohr magneton Nuclear magneton

NnNpBe µµµµµµ 913.1973.2 −==−=electron neutron proton

µ

L

Subatomic particles have angular momentum (L), and can also have a magnetic dipole moment µ = γL, suggesting that their moments arise due to spinning internal electric charge. Though this analogy is very limited, we often refer to magnetic dipole moments as “spins”.

B µ µ µ

θ

BABμτ ×=×= )(NI

BABμ ⋅−=⋅−= )(NIU

Which current-carrying loop has the lowest potential energy?

Which loop experiences the greatest torque? Direction?

Charge-particle motion in a magnetic field Magnetic force is always ⊥ to direction of motion, driving the charge in circles.

rmvqvBFB

2

==

Thus, the magnetic force is a centripetal force.

Bv×= qFB

qBmvr =Radius:

Angular frequency:

Bmq

rv

==ω

constant and || ==⊥ vm

qBrv

Only v⊥ leads to circular motion, as v|| creates no force.

Together, both components result in a spiral trajectory.

Charge-particle motion in a magnetic field

Charge-particle motion in a magnetic field

Magnetic fields do no work on electrically-charged particles, and kinetic energy is conserved!

0)( =⋅×=⋅= sBvs dqdFdW B

Because v and ds are parallel, the FB and ds are perpendicular.

mqBrmvK2

)( 22

21 ==

If B increases, then r decreases.

B

+

Charged particles trapped in the magnetosphere

http://en.wikipedia.org/wiki/Aurora_%28astronomy%29

http://en.wikipedia.org/wiki/Aurora_%28astronomy%29

Which of these trajectories belongs to a negatively charged particle?

Velocity Selector

BEv

qEqvBFF EB

=

=

=

Mass spectrometer

Brv

mq

=

Separately measure v and r.

E

B FE

FE

e- ? BEv

qEqvBFF EB

=⇒

=⇒=

Cathode ray tube

Doesn’t matter if current carriers are positive or negative. Either way, the force direction is the same.

Current flowing through conducting object in the presence of ⊥ B field. Positive and negative carriers would be forced in opposite directions.

For a metallic object, ∆V = Vtop – Vbottom should be (1) positive (2) negative (3) zero?

Case #1: Electric current flow in a wire.

B

(1) A (2) B (3) ∆V = 0

Which terminal is more positive? Case #2: Fluid mixture of positive and negative ions flowing through a tube.

Hall Effect

dEdBvV

EqBvqF

effdH

effcdc

][)(

][)(

==

==

dtqnIv

dtvqnAJI

ccd

dcc

=⇒

==

tqnIBBd

dtqnIqV

cccccH =

=

The electric field diverges outward from a point charge.

2

ˆr

dQkd erE =

The magnetic field circulates around an electric current (RHR).

?=B

d

Biot-Savart Law 2

ˆ)(r

dQkd mrvB ×

=

2

ˆ)(r

Idkmrs ×

=

AmT10

470 ⋅

== −

πµ

mk

∫∫×

=×

= 30

20

4ˆ

4 rdI

rdI rsrsB

πµ

πµ

dB is proportional to sin(θ).

Biot-Savart example: long straight wire z runs from −∞ to +∞ as θ runs from −π/2 to +π/2.

razazrdzd ρzrzs

ˆˆˆˆ 22 +−=+==

φφφrs ˆ)cos(ˆ)cos(

)cos(ˆˆ 3

32 adda

aa

rdza

rd θθ

θθθ

=

==

×

)(cos)cos( 222 θ

θθ dadzaz

ara

=⇒+

==

zρ

points along current direction.

points radially away from wire.

ρzφ ˆˆˆ ×= circulates around wire.

.ˆˆˆˆ then ,ˆˆ If yxzφxρ =×==z

z

ρ

(out) φ

Biot-Savart example: long straight wire z runs from −∞ to +∞ as θ runs from −π/2 to +π/2.

φφφBB ˆ2

ˆ)sin(4

)cos(4

ˆ 0

2/

2/

02/

2/0

aI

aId

aId

πµθ

πµθθ

πµ

π

π

π

π====

−−∫∫

θθπ

µπ

µ da

Ir

dId )cos(4

ˆˆ4

02

0 φrsB =×

=

z

z

ρ

(out) φ φrs ˆ)cos(ˆ2 a

dr

d θθ=

×

Biot-Savart example: finite wire segment z runs from z1 to z2 as θ runs from θ1 to θ2.

[ ]φφBB ˆ)sin()sin(4

)cos(4

ˆ12

00 2

1

θθπ

µθθπ

µ θ

θ−=== ∫∫ a

Ida

Id

z

z

ρ

(out) φ

a

I

P

[ ] [ ])4/sin()4/sin(4

)sin()sin(4

012

01 ππ

πµθθ

πµ

−−=−=aI

aIB

θ1 θ2

21

21

−

aIπ

µ220=

1

2

3

4

B-field at P due to current in wire segment #1 is directed (1) into screen (2) out of screen (3) left (4) right (5) up (6) down?

Wire segment #2?

All four segments contribute equally to the magnitude.

aIBBBBBB

)2/(4 0

14321 πµ

==+++=

Biot-Savart example: square loop

Biot-Savart example: circular loop

z

r

points perp. to the loop (left).

points radially inward.

φ circulates CCW with current. P

a rz

φs ˆdsd =

zzrφrsB ˆ4

ˆ4

ˆˆ4

ˆ4

02

02

02

0 φπ

µπ

µπ

µπ

µ daI

adsI

adsI

rdId ==

×=

×=

zzBB ˆ24

ˆ 02

00

aId

aId µφ

πµ π

=== ∫∫

Biot-Savart example: partial loop

B-field at P due to current in wire segment 1 is directed (1) into screen (2) out of screen (3) some other direction?

B-field at P due to current in wire segment 2 is directed (1) into screen (2) out of screen (3) some other direction?

022

0 0 +

+=

aIB µ

πθ

1

2

3

P

Let θ = 45º

aI

aI

16224/ 00 µµ

ππ

=

=

0ˆ =×rsd for segments 1 and 3!

z

x

z

dBρ

dBz

ρ

φ

Biot-Savart example: loop axis ρzrφs ˆˆˆ 22 azazradd −=+== φ

zρρφzφrφ ˆˆ)ˆˆ()ˆˆ(ˆ azaz +=×−×=×

3

20

30

30

4)ˆ(

4)(

4 rdaI

radI

rdIdB zz

zφ

πµφ

πµ

πµ

=×

=×

=rφrs

2/322

20

3

202

03

20

)(224 azaI

raId

raIdBB zz +

==== ∫∫µµφ

πµ π

dBρ integrates to zero

z

x

z

dBρ

dBz

ρ

φ

Biot-Savart example: loop axis (z >> a)

3

20

2/322

20

2)(2 zaI

azaIB azz

µµ →

+= >>

zzzzzzzμrrμB ˆ2

ˆ4

2ˆ2ˆˆ)ˆˆ(3ˆ)ˆ(33

20

30

333 zaI

zNIA

zk

zk

rk m

mmµ

πµµµµ

===−⋅

=−⋅

=

Use the point-dipole formula:

Parallel current-carrying wires The first wire produces a field that is experienced by the second wire.

aIB

πµ2

021 =

aIIBIF

πµ

2210

21221

==B21

aIIF

πµ2

210=

To achieve a mutually attractive force, the two currents should be (1) parallel (2) antiparallel?

0=⋅−=∆ ∫ sE dV Kirchoff’s voltage rule

Gauss’s law of electrostatics: E-field lines start on positive charges and end on negative charges. No loops!

Static electric fields are “conservative”.

B

Oersted’s B field loops around a current-carrying wire.

Static B fields are not “conservative”. 0≠⋅∫ sB d

Ampere’s Law Id 0µ=⋅∫ sB

ds

B

Andre-Marie Ampere

The magnetic circulation around a closed loop is proportional to the electric current flowing through the loop, where the positive directions of electric current (I) and magnetic circulation (C ) are related by the RHR.

≡CMagnetic circulation

Magnetic circulation quiz

Highest magnetic circulation (consider the sign)?

Largest magnetic circulation (magnitude only)?

Highest magnetic circulation (consider the sign)?

Largest magnetic circulation (magnitude only)?

Magnetic circulation quiz

Inverting Ampere’s law to determine B requires a high symmetry situation.

Ampere’s Law example: long straight wire

rIB

πµ2

0=

IrBdsBd 0)2( µπ∫∫ ===⋅ sB

2

enc

===

RrI

AAIJAI enc

enc

Ampere’s Law example: inside a thick wire

enc0)2( IrBdsBd µπ∫∫ ===⋅ sB

Inside a wire, the enclosed current increases linearly with r.

200

22 RrI

rIB enc

πµ

πµ

==

InINB 00 µµ ==

Ampere’s Law example: inside a solenoid

NIBlllBd 00)0(0 µ==+++=⋅∫ sB

rINB

NIrBdsBd

πµ

µπ

2

)()2(

0

0

=

===⋅ ∫∫ sB

Ampere’s Law example: inside a toroid

Use Ampere’s law to determine B.

There’s not enough symmetry.

1

2

?1

0

=⋅∫ sB dµ

−1 A −3 A

(1) 0 A (2) −1 A (1) −4 A (2) +26 A

Current density J flowing uniformly along +z direction. What’s the magnetic circulation around this loop?

This tube or radius r is coated with uniform positive surface charge density σ and rotates around its axis with angular velocity ω, so as to produce a uniform magnetic field B inside the tube? Find the magnitude of B?

00 ==⋅=Φ ∫ BB Qd µAB

Magnetic flux and Gauss’s law of magnetism

AB dd B ⋅=Φ

Computed just like electric flux

The total magnetic flux exiting a closed surface is zero. We’ve never observed a magnetic monopole.

∫ ⋅=Φ AB d

∫⋅= AB d

0)( =⋅= 0B

Uniform magnetic field

B B B

What is the sign of the magnetic flux through the flat top surface? a. positive b. negative

dA ∫ ⋅=Φ AB dB

By “entering”, we mean that the dA vector points inward. By “exiting”, we mean that the dA vector points outward

B

θ θ

Which shaded surface has the greatest total magnetic flux (magnitude only)? a. hemisphere b. cube face c. both the same

In the left-hand hemispherical figure, what will be the sign of the “entering” magnetic flux through (1) the flat top cap and (2) the bowl. a. positive b. negative c. zero

Assorted regions of a dipole field

(a) (b) (c)

Which closed surface has the largest entering flux?

0)(sin

)cos()sin(2

0

2210

2

00

=

=

=⋅=Φ ∫∫∫

π

ππ

θµµ

φθθθµ

r

ddr

kd mAB

rA

kjirˆ)sin(

ˆ)cos(ˆ)sin()sin(ˆ)cos()sin(ˆ2 φθθ

θφθφθ

ddrd =

++=

dA

zθ

33

ˆˆ)ˆˆ(3ˆ)ˆ(3r

kr

k mmzrrzμrrμB µµ −⋅

=−⋅

=

φθθθµ

θµµ

µµ

ddr

k

dAr

kdAr

k

dAr

kd

m

mm

m

)sin()cos(2

)cos(2ˆˆˆˆ3

ˆˆˆˆ)ˆˆ(3

33

3

=

=⋅−⋅

=

⋅−⋅⋅=⋅

rzrz

rzrrrzABzµ

Point-dipole flux

Review of E & M concepts to date

Magnetism Electricity

BId =⋅∫ sE0

1µ

EQd =⋅∫ AE0εClosed surfaces

Closed loops

BQd =⋅∫ AB0

1µ

EId =⋅∫ sB0

1µ

20 4

1r

QE E

πε=Point-charge fields

20 4 rQB B

πµ=

Gauss’s law of elec. Gauss’s law of mag.

Kirchoff’s voltage law Ampere’s law

rIE B

πµ2

0=Line-current fields r

IB E

πµ2

0=

0

0

0

0

Magnetic Units Heaviside-Lorentz MKS Units

Id 0µ=⋅∫ sB

0εQd∫ =⋅ AEGuass’s Law

Ampere’s Law

Qd =⋅∫ AE

cId =⋅∫ sB

20

21

4 rQQFεπ

=

rIIF

πµ2

210=

221

4 rQQF

π=

rIIF

π221=

Point-charge force

Parallel wire force

The values of ε0 and µ0 were designed to make the MKS units work out.

2μ

2μ

2112 BμBμ ⋅−=⋅−=U

A) U < 0 B) U > 0 C) U = 0

How do we lower U? • Increase separation • Decrease separation

1μ

Dipole-dipole force: attractive or repulsive?

J/T109.272

24−×==e

B meµ

Moment per unpaired electron (Bohr magneton)

µ

Atomic magnetic dipole moments

Magnetization:

If the atomic moments are non-zero but randomly directed, then the total magnetization is still zero. To have a non-zero M, there must be at least partial allignment of the atomic moments.

∑=i

iVmM 1

The magnetization vector (M) of a magnetic material is the volume density of magnetic-dipole vectors (mi).

M

Saturation magnetization of Iron (Fe)

Under saturation conditions, all moments (2.3 µB/atom) are alligned.

m/A 1081.1mJ/T 1081.1

mg 1086.7

g 5.855mol

molatoms 10.026

atom3.2J/T1027.9

636

3

6-2324-

×=⋅×=

×

×

×=

=

=

B

B

volmass

massmol

molatoms

atommagnetons

magnetonmoment

volmomentM

µµ

T 2.2m

Amp 108.1Amp

mT 104 :field Saturation6-7

0 =

×

⋅×=

πµ M

Magnetism in materials

Loosely speaking, we can think of H (magnetic intensity) as the externally-applied field, M (magnetization) as the internal field that arises within the material due to the alignment of its atomic dipole moments, and B as the total magnetic field.

MHB +=0

1µ

Magnetic constitutive equation:

M and H both have Amp/meter units, which are different from the Tesla units of B due to the factor of µ0. This is stupid, a bit like saying that 23 ft = 4 m + 3 m. But we deal with it.

Susceptibility and permeability

Think of M as the material’s response to an applied H field, so that M = χH. The proportionality coefficient χ (magnetic susceptibility) and determines how easy it is to magnetize a material.

HHHMHB )1(1

0

χχµ

+=+=+=

00 )1(where)1( µχµµµχ +≡=+= HHB

In free space, χ = 0 and µ = µ0.

We can also define the magnetic permeability (µ) of a material.

0κεQd =⋅∫ AE Id 0)1( µχ+=⋅∫ sB

AQE

0κε=

Gauss’s law of electricity Ampere’s law

InB 0)1( µχ+=

Parallel-plate capacitor Solenoid

Substitute ε0 → ε = κε0 for a dielectric material.

Substitute µ0 → µ = (1+χ)µ0 for a magnetic material.

Accomodating magnetic materials

Paramagnetic materials HBHM 0)1( µχχ +==

Weakly attracted to a magnetic field

Linear response (χ independent of H): no permanent M when H = 0.

Susceptibility is small (weak) and positive: χ > 0 → µ > µ0.

Susceptibility decreases at high-T due to thermal disorder. Curie’s law: M ∝ B/T.

All materials with atomic moments are paramagnetic at high T.

Diamagnetic materials HBHM 0)1( µχχ +==

Weakly repelled from a magnetic field.

Linear response (χ independent of H): no permanent M when H = 0.

Susceptibility is small (weak) and negative: χ < 0 → µ < µ0.

Susceptibility (χ) smaller at high-T due to thermal disorder.

All materials are weakly diamagnetic if they lack atomic moments. When moments are present, diamagnetism isn’t noticeable.

http://www.physics.ucla.edu/marty/diamag/

The Meissner effect for YBa2Cu3O7

Perfect diamagnetism: χ = −1

0)11()1( 00 =−=+= HHB µµχ

When fields from the magnet attempt to enter the SC material, surface currents in the SC do whatever it takes to create exactly the opposite field. The magnet levitates in the opposing fields from the SC.

Superconductor (T < Tc)

Permanent magnet

Ferromagnetic materials

2/1

0

−

C

C

TTTM

Ferromagnetic materials

Non-linear response (χ depends on H and past history) – hysteresis.

Susceptibility is very large and positive: χ >> 0 → µ >> µ0.

Paramagnetic above Curie temperature (TC). Spontaneous moment (at H = 0) below TC.

Moments aligned within magnetic domains, but domains can be disordered. External field causes domains to align.

HBHM 0)1( µχχ +==

Strongly attracted to a magnetic field www.wikipedia.org

Hysteresis in ferromagnets

Remanence

Coercitivity

Saturation

Good permanent magnets need both high remanence and high coercitivity (loop area determines the ability to do magnetic work).

Hard ferromagnet Soft ferromagnet

Iron (Fe) is magnetically soft (small Hc). NdFeB is “hard” (large Hc).

non-magnetic diamagnetic paramagnetic ferromagnetic

0>χ0<χ0=χ

0µµ = 0µµ < 0µµ >

hysteresis

hysteresis

0>>χ

0µµ >>

requires atomic moments at H = 0

Curie’s law

weakly attracted to magnetic fields

no atomic moments needed at H = 0

weaker at high T

weakly repelled by magnetic fields

No real material (except the vacuum)

is non-magnetic

permanent (H = 0) magnetic domains

below TC

strongly attracted to magnetic fields

hard or soft