Post on 04-Sep-2018
(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.
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 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
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.
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
+
http://en.wikipedia.org/wiki/Aurora_%28astronomy%29
http://en.wikipedia.org/wiki/Aurora_%28astronomy%29
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
==
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.
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
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
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