Chapter 28 Subjects
ÎBar
magnets
‹Poles,
ÎEffects
magnetic field lines, etc.
of magnetic fields on charges and currents
‹Force
on a moving charge
‹Path followed by charged particle in magnetic field
‹Force on a current
‹Torque on a current loop
ÎApplications
‹Mass
spectrometers
‹Cyclotrons and synchrotrons
‹Hall effect
‹Electric motor
PHY2049: Chapter 28
1
Bar Magnets
ÎTwo
poles: “north” and “south”
ÎLike
poles repel
ÎUnlike
ÎSo
poles attract
far, the poles are like electric charges
PHY2049: Chapter 28
2
Magnetic Monopoles?
ÎCan any isolated magnetic charge
‹ We would call this a “magnetic monopole”
‹ It would have a + or – magnetic charge
exist?
ÎHow can we isolate this magnetic
‹ Cut a bar magnet in half? NO!
charge?
What you get
is a bunch of
little magnets!
No one has ever found magnetic monopoles in nature.
PHY2049: Chapter 28
3
PHY2049: Chapter 28
4
Bar Magnets (2)
S
Î Similar
Î Sign
N
to electric field produced by electric dipole
convention:
‹ Field
lines from North pole to South pole (for outside
only)
‹ When used as a compass, North pole points north
PHY2049: Chapter 28
5
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
PHY2049: Chapter 28
6
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 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 magnet is not fundamental
‹ Magnetic force on 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)
PHY2049: Chapter 28
7
Reading Quiz
ÎThe
is:
magnetic force on a moving charged particle
‹(a)
Perpendicular to the velocity
‹(b) Parallel to the velocity
‹(c) Parallel to the B field
‹(d) Independent of the velocity
‹(e) None of the above
PHY2049: Chapter 28
8
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
PHY2049: Chapter 28
9
Right Hand Rule
ÎFirst
point fingers in direction of
velocity
‹Curl
fingers toward B field
‹⇒ Thumb points toward force
‹This is for positive charge q
F
v
B
PHY2049: Chapter 28
10
Right Hand Rule
Î
Consider +q moving relative to a B field as
shown
‹
‹
‹
‹
(a) Force is parallel to v
(b) Force is parallel to B
(c) Force is into the page
(d) Force is out of the page
B
+q
PHY2049: Chapter 28
11
Right Hand Rule
Î
A charged particle moves in a straight line
through some region of space. Can you
conclude that B = 0 there?
‹
‹
(a) Yes
(b) No
B field can exist since if v || B
there is no magnetic force
B
v
q
PHY2049: Chapter 28
12
Magnetic Force
ÎA
particle with negative charge enters a magnetic
field region. What path will it follow?
‹
‹
‹
‹
‹
A
B
C
D
E
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
A
B
C
x x x x x x x x x x x x
D
x x x x x x x x x x x x
E
(1) RHR says it bends down (− charge)
(2) But force cannot instantaneously change v
(3) So the answer is D, not E
PHY2049: Chapter 28
13
Magnetic Field Units
ÎFrom
the expression for force:
‹B
= Fmax / q v
‹ Units:
newton/(coulomb⋅meter/s) = N/(A⋅m) ≡ tesla (SI unit)
‹ Another unit: gauss = 10-4 tesla
ÎExamples:
‹ Earth:
B = 0.5 gauss = 5 x 10-5 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
PHY2049: Chapter 28
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Magnetic fields — large and small
45 T magnet during assembly,
National High Magnetic Field
Laboratory, Tallahassee
~Million times the earth field.
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
PHY2049: Chapter 28
15
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
PHY2049: Chapter 28
16
Example
with m = 1.5 g, q = −2 µC moves with
velocity 2 km/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!
PHY2049: Chapter 28
17
Magnetic Field and Work
ÎMagnetic
force is always perpendicular to velocity
‹Therefore
B field does no work!
r r r r
‹Why? Because ∆K = F ⋅ ∆x = F ⋅ ( v ∆t ) = 0
ÎConsequences
‹Kinetic
energy does not change
‹Speed does not change
‹Only direction changes
r
‹Particle moves in a circle (if v ⊥
PHY2049: Chapter 28
r
B)
18
Trajectory in a Constant Magnetic Field
ÎParticle
with charge q enters B field with velocity v
perpendicular to B. What path will the particle follow?
is always ⊥ velocity and ⊥ B
‹ Path will be a circle. F is the centripetal force needed to keep the
charge in its circular orbit. Let’s calculate radius R
‹ Force
x x 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
v
x x x x x x x x x x x x x x
F
v
v
F
q
R
PHY2049: Chapter 28
19
(continued)
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 x x x x x x x x
v
B
F
q
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 x x x x x x x x
2
mv
= qvB
R
mv
R=
qB
PHY2049: Chapter 28
20
Magnetic Force
ÎTwo
particles of the same charge enter a
magnetic field with the same speed. Which one
has the bigger mass?
‹(a)
A
x
‹(b) B
x
‹(c) Both masses are equal
x
‹(d) Cannot tell without more info x
mv
R=
qB
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 x x x x x x x x
Bigger mass means larger
inertia, less acceleration,
thus bigger radius
PHY2049: Chapter 28
A
B
21
Work and Energy
ÎA
charged particle enters a uniform magnetic
field. What happens to the kinetic energy of the
particle?
‹(a)
it increases
‹(b) it decreases
‹(c) it stays the same
‹(d) it depends on the direction of the velocity
‹(e) it depends on the direction of the magnetic field
Magnetic field does no work, so kinetic energy does not
change.
PHY2049: Chapter 28
22