Chapter 28: Magnetic Fields
PHY2049: Chapter 28
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Magnetic Fields
ÎMagnetic
field (units, field lines)
‹ Magnetic
ÎEffects
field of the earth and other astronomical objects
of magnetic fields on charges and currents
‹ Force
on a moving charge
‹ Force on a current
‹ Torque on a current loop
‹ Path followed by particle in magnetic field
ÎInstruments
‹ Mass
spectrometers
‹ Cyclotrons and synchrotrons
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Reading Quiz
ÎWhen
I cut a magnet into two pieces I get:
‹ An
isolated north and south magnetic pole
‹ Two smaller magnets
‹ The two pieces are no longer magnets
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Bar Magnets
ÎTwo
poles: “north” and “south”
ÎLike
poles repel
ÎUnlike
poles attract
ÎMagnetic
poles cannot be isolated
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Magnetic Monopoles?
ÎCan
any isolated magnetic charge exist?
‹ We
would call this a “magnetic monopole”
‹ It would have a + or – magnetic charge
ÎHow
can we isolate this magnetic charge?
‹ Cut
a bar magnet in half? NO!
What you get
is a bunch of
little magnets!
No one has ever found magnetic monopoles in nature
Listen to Magnetic Monopoles Song:
http://www.haverford.edu/physics-astro/songs/monopoles.htm
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Bar Magnets (2)
S
N
Similar to dipole field from electrostatics
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Earth is a big magnet!!
The North pole of a small magnet
(compass) points towards
geographic North because Earth’s
magnetic South pole is presently up
there!!
Earth’s magnetic poles have reversed very frequently on a geological time scale:
http://science.nasa.gov/headlines/y2003/29dec_magneticfield.htm
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What Causes Magnetism?
ÎWhat
is the origin of magnetic fields?
‹ Electric
charge in motion!
‹ For example, a current in a wire loop produces a field very similar
to that of a bar magnet (as we shall see in Chapter 29).
ÎUnderstanding
the source of bar magnet field lies in
understanding currents at the atomic level within matter
(More on this in Chapter 32)
Orbits of electrons about nuclei
Intrinsic “spin” of
electrons (more
important effect)
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Law of Magnetism
ÎAnalogy
with Coulomb’s Law does not work
‹ So
far no one has found magnetic monopoles (=magnetic
charges)
‹ Force between two small bar magnets is complicated; turns out to
be not fundamental (should be deduced from a law that governs
more fundamental phenomena)
ÎWhat
does this last statement mean?
‹ Magnetic
field produced by bar magnet is not fundamental
‹ Magnetic force on bar magnet is not fundamental either
ÎTwo
phenomena turn out to be fundamental
‹ Electric
current produces magnetic field
‹ Magnetic field exerts force on moving charge
ÎThe
law consists of two parts, two equations
‹ Magnetic
field produced by electric current (Chapter 29)
‹ Force due to magnetic field on moving charge (Chapter 28)
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Magnetic Field Units
ÎFrom
the expression for force on a current-carrying wire:
‹B
= Fmax / I L
‹ Units:
newtons/A⋅m ≡ tesla (SI unit)
‹ Another unit: gauss = 10-4 tesla
ÎSome
sample magnetic field strengths:
‹ Earth:
B = 0.5 gauss = 0.5 x 10-4 T
‹ Galaxy: B ∼ 10-6 gauss = 10-10 T
‹ Bar magnet: B ∼ 100 gauss = 10-2 T
‹ Strong electromagnet: B = 2 T (35 T in Tallahassee)
‹ Superconducting magnet: B = 5 – 10 T (20 T in Tallahassee,
also coming soon to UF)
‹ Pulsed magnet: B ∼ 100 T
‹ Neutron star: B ∼ 108 – 109 T
‹ Magnetar: B ∼ 1011 T
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Pulsars
Rapidly Rotating Neutron Stars
Enormous Magnetic Fields
Lighthouse effect
Beam off
Beam on
Crab Pulsar
R = 10 km
M = 1.4 solar mass
B ≈ 108 T
Period = 1/30 sec
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Magnetic fields on this planet — large and small
45 tesla magnet during assembly,
National High Magnetic Field
Laboratory, Tallahassee, FL
Requires 30 MW (300 thousand light
bulbs!)
Magnetoencephalography (MEG)
detects magnetic fields produced by
brain activity (electric currents):
~10-12 T
Sensors require a 4 K temperature
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Reading Quiz
ÎThe
magnetic force on a moving charged particle is:
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
Perpendicular to the velocity
Parallel to the velocity
Parallel to the B field
Independent of the velocity
None of the above
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Magnetic Field B
ÎPostponing
until Chapter 29 the questions of how magnetic field is
produced and how its strength is varied, large number of experiments
show
r
r r
v
Fm ∝ qv × B
‹
Choose the unit (tesla) such that
r
r r
F = qv × B
F = qvB sin φ
ÎForce
v is parallel to B
‹ v is perpendicular to B
‹ v is at angle 45° to B
‹
ÎFor
+q
F (into page)
magnitude depends on direction of v relative to B
‹
ÎForce
B
⇒ sinφ = 0
⇒ sinφ = 1
⇒ sinφ = 0.71
F =0
F = qvB
F = qvB sin 45
direction is perpendicular to both B and v
Right hand rule (next slide)
given direction of v, force magnitude is proportional to v and B
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Right Hand Rule
ÎFirst
point fingers in direction of velocity
‹ Curl
fingers toward B field
‹ ⇒ Thumb points toward force
F
v
B
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Example
with m = 1.5 g, q = −2µC moves with velocity
2,000 m/s through a magnetic field of 2.5 T at an angle of
30° to the field.
ÎParticle
‹ Magnitude
of force
(
)
F = qBv sin φ = 2 × 10−6 ( 2.5)( 2000 )( 0.5) = 0.005 N
‹ Direction
of force?
Up out of the page, from RHR
v
B
−q
F (out of page)
Note the negative charge!
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Reading Quiz
Î
Consider +q moving relative to a B field as shown
Force
‹ Force
‹ Force
‹ Force
‹
is
is
is
is
parallel to v
parallel to B
into the page
out of the page
B
+q
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A charged particle moves in a straight line through some
region of space. Can you conclude that B = 0 here?
1.
2.
Yes
No
A B field can exist since if v || B
there is no magnetic force
B
v
q
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Magnetic Force
ÎA
particle with a negative charge enters a magnetic field
region. What path will it follow?
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
A
B
C
D
E
x x x x x x x x x x x x
A
x x x x x x x x x x x x
B
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
C
D
E
(1) RHR says it bends down (− charge)
(2) But force cannot instantaneously change v
(3) So the answer is D, not E
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Magnetic Force on Current-Carrying Wire
ÎMagnitude
of force F = iBL sin φ
‹ Easy
to derive from charge, number density & drift velocity of
individual charge carriers
ÎDirection
of force: RHR
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Example
ÎA
4 m long wire carries current of 500A in NE direction
‹ Magnitude
of force (B = 0.5 gauss = 5 × 10-5 T, pointing N)
(
)
F = iBL sin φ = ( 500 ) 5 ×10−5 ( 4 )( 0.71) = 0.071N
‹ Direction
of force?
Upwards, from RHR
ÎCan
adjust current in wire to balance against gravity
iBL sin φ = mg
‹ Calculate
mass from density, length and cross-sectional area
m = ρ LA
‹ Good
exam problem!
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Magnetic Force
ÎA
vertical wire carries a current in a vertical magnetic
field. What is the direction of the force on the wire?
‹ (a)
left
‹ (b) right
‹ (c) zero
‹ (d) into the page
‹ (e) out of the page
B
I is parallel to B, so
no magnetic force
I
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