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1. Magnetic field B (Tesla)
2. Force acting on a moving charge
in B field: Fq
3. Cyclotron motion
Ω
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Historical milestones
• 5-th century B.C. Greeks knew about rocks in Magnesia
(Western Turkey) that attract each other. They named
rocks “magnetite”, chemical formula
Fe O
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• ~110 A.D. Chinese use compass to navigate
• Late 12-th century. Europeans “invent” loadstone=leading
stone
• ~1600 William Gilbert. The Earth is a giant magnet
• ~1800 Hans Christian Oerstead. Compass is sensitive to
electric current
• >1800 Ampere, Faraday, Maxwell. Universal theory of
magnetism
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Magnetic field
• We recognize the fact that the magnetic
field is established around us by placing a
compass needle (test magnet) that would
point to magnetic SOUTH
• Postulate: Needle’s North pole shows the
direction of magnetic field (towards South
Pole)
• Compass cannot say anything
about the magnetic field magnitude
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Bar magnet (Permanent magnet)
North pole is marked by RED color
South pole is marked by BLUE color
Magnetic field flux lines can be visualized with iron filings acting as little magnets
Magnetic field flux lines are continuous loops, they emerge from N pole, arrive to S
pole and “continue” inside the magnet
Magnetic flux lines do not touch and cross each other
One cannot get a magnetic monopole – say S Pole; N and S poles always coexist
Magnetic flux lines show the direction of magnetic field at every spot
Magnetic flux lines act as rubber bends – try to squeeze themselves
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Earth magnetic field – Permanent
magnet
• North Magnetic Pole is
located on Canadian
territory
• North Magnetic Pole is
off the geographic
Pole; the separation is
~17 deg
• North Magnetic Pole
should be actually
called South Magnetic
Pole, for doing physics
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North (South)
magnetic pole
This figure shows the path of
the North Magnetic Pole since
its discovery in 1831 to the
last observed position in 2001.
During the last century the
Pole has moved a remarkable
1100 km. What is more, since
about 1970 the NMP has
accelerated and is now
moving at more than 40 km
per year. If the NMP maintains
its present speed and
direction it will reach Siberia in
about 50 years. Such an
extrapolation is, however,
tenuous. It is quite possible
that the Pole will veer from its
present course, and it is also
possible that the pole will slow
down sometime in the next
half century.
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Properties
• Like poles repel
each other
• Unlike poles attract
each other
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Why North magnetic Pole should be
South magnetic Pole?
Because unlike poles
attract each other, the
needle’s North end points
towards the South Pole
in the Earth magnetic
field, but we call this
“South Pole” the North
Pole, nevertheless
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Force on a moving charge in B field

 

Fq = q[V × B]; | Fq |= qVB sin θ
(Vector product)
θ
This equation can be used
as a definition of B field
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Direction of the vector product
V
=
[V1 × V2 ]
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Use ALWAYS right hand (right hand rule)
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Direction of the vector product
V
=
[V1 × V2 ]
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Use ALWAYS right hand (right hand rule)
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Force on a moving charge in B field

 

Fq = q[V × B]; | Fq |= qVB sin θ
Examples:
F=0
F=qVB
F=qVBsin()
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Important points regarding magnetic
force Fq
1) POLARITY of magnetic force depends on charge;
the force acts in opposite directions for electrons
(“-”charge) and protons (“+” charge)
2) Magnetic force is zero for
non-moving charge
for motion along the B field direction
3) Notation for directions:
away,
towards you,
down to the page
out of page
B
B
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Units of B field
Unit for B field magnitude is Tesla (T), after
Nikola Tesla, Croatian-born US engineer
B=F/ (q v sin )
[1 T ] =N/C/(m/s)
or since 1 C/s= 1Amper
[1 T ]= N/(Am)
Non-SI unit for B is Gauss (G):
I G=10-4 T
Nikola Tesla
Inventor
(1856 – 1943)
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Work done by B field
Magnetic field does
not do work over a
particle and does not
change its kinetic
energy as the force is
always perpendicular
to the velocity (scalar
product of force and
displacement vectors
or F and velocity is
zero)
 
Work / second ≡ F ⋅ V
For mag field force


| V | ⋅ | B | ⋅ cos(π / 2) =
0
V

 
=
F q[V × B]
B
Is q positive here?
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Comparing effects of E
field and B field
• Positive charge is
shifted along E field
towards negatively
charged plate. The
trajectory is a parabola
• Positive charge is
shifted upwards. The
trajectory is a circle
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Velocity separator and making a
straight trajectory for a charged particle
Particle
detector/
collector
q ⋅ E = q ⋅ V ⋅ B ⇒ V = E/B
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Circular motion
in B field
1) Centripetal
acceleration, centripetal
force
2) Radius of the orbit
3) Radial frequency
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Note on centripetal (“center
acceleration
Even though
seeking”)


| V (t0 ) |=| V (t ) |
(circular motion with constant speed), the velocity changes its direction so that



V (t ) − V (t0 ) =
∆V ≠ 0
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Centripetal
acceleration
∆V V∆t
=
V
r
2
∆V V
=
= ac
r
∆t
Direction of ac: Toward the center
For circular motion, the resultant force
should always be directed toward the
center



V (t ) − V (t0 ) =
∆V
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Derivation of expressions for the radius of
gyration and angular (cyclotron) frequency
V2
mV
mac = Fq ; ⇒ m
= qVB; ⇒ r =
r
qB
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1
f ( frequency ) =
[ ]
T ( period ) s
so many turns per sec
radian
2π
]
= Ω [
T ( period )
sec
ang . freq. = so many radians per sec
(there are 2π radian s in one circle)
2π f = (angular frequency ) =
r ⋅ Ω =V
So much of a distance
covered per sec
V
V
qB
Ω= =
=
r (mV / qB) m
qB
Ω=
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m
Radius of the orbit
mV
r=
qB
Cyclotron frequency of gyration
qB
Ω=
m
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Aurora Borealis
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Aurora borealis is a global
phenomenon
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Plasma
(electrons
& protons)
is ejected
from the
Sun
toward the
Earth
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Particles from the Sun are diverted by the
Earth magnetic field, but some of them
precipitate into the upper atmosphere,
causing aurora borealis
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Example of a problem: aurora
Aurora borealis occurs because energetic
electrons and protons bombard the atmospheric
gases at the height of ~ 100-150 km. Assume
that the precipitating protons and electrons have
energy of 5 eV and move almost perpendicular
the Earth’s magnetic field (0.5 G). (a) Find the
radius of proton (electron) gyration and the
radial frequency of their motion. (b) If you are
looking along the Earth’s magnetic field, what is
the direction of the proton (electron) gyration,
clockwise or counterclockwise?
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Solution: Electron motion
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Solution: Proton motion
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Applications:
The mass spectrometer
• Ions are accelerated
in electric field
• Ions travel in B field
along trajectories of
different radius,
depending on mass
• Detector selects
particles of proper
mass, certainly you
have to calibrate the
instrument
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Of two particles accelerated to the same velocity, m & m1, which one is lighter?
Example of a problem:
A proton is released from rest
at point A. The proton is then
accelerated toward the
negative plate of a capacitor. It
leaves the capacitor at B
through a small hole in the
plate. The electric potential of
the positive plate with respect
to the negative plate is 2.1 kV.
Outside the capacitor, there is
constant magnetic field of 0.1
T, directed perpendicular to the
plates. Find: (a) the speed of
the proton at point B and (b)
the largest separation of the
proton from B.
Point B
A
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Solution: Mass spectrometer
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Comments of particle motion in electric field

E

Force Electric Felectr
=
charge
q
+
E
1) Electric field is directed from “+” charge towards ” – “ charge:
2) Positive charge (free of other forces) moves in the direction of E field
3) Negative charge (free of other forces) moves against the direction of E field
E
E
+
Positive q
Negative q
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Conceptual analogy between E & B fields
Electric field
Magnetic field
(positive)
(shown is force on a negative particle)
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In both cases, the field characterizes force acting on a particle
Summary of equations

 

Fq = q[V × B]; | Fq |= qVB sin θ
2
V
ac =
r
qB
Ω=
m
E
Vel = (Crossed E & B )
B
mV
r=
qB
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