Transcript Applications of the Motion of Charged Particles in a

Applications of the Motion of
Charged Particles in a Magnetic
Field
AP Physics C
Montwood High School
R. Casao
Lorentz Force
• In many devices that involve the motion of
charged particles in uniform magnetic fields,
the charge under consideration will be moving
with velocity v in the presence of both an
electric field E and a magnetic field B.
• The charge will experience both an electric
force q·E and a magnetic force q·(v × B).
• The total force, called the Lorentz force, on the
charge is:


F  q  E  q  (v x B)
Velocity Selector
• Experiments involving the motion of charged
particles often require particles that move in a straight
line with the same velocity.
• A combination of an electric field and a magnetic
field can produce this stream of particles.
• A uniform electric field E is provided by a pair of
charged parallel plates.
• A uniform magnetic field B is applied perpendicular
to E such that Fmag is equal to and opposite to Felectric.


q E  q vB

E
v 
B
The two fields, called crossed fields can be manipulated to
produce velocities in other directions.
Mass Spectrometer
• Separates atomic and molecular ions based on their
mass-to-charge ratio.
• A beam of ions first passes through a velocity selector
and then enters a uniform magnetic field Bo, where the
ions move in a semicircle of radius r before striking a
photographic plate at P.
• Mass-to-charge ratio:

m r  Bo
 
q
v
• If the magnetic field in the velocity selector is
 
B, then:
m r  Bo  B


q
E
• Charge-to-mass ratio for electrons:
– Electrons are accelerated from the cathode to the
anode, passing through slits in the anodes, and
allowed to drift into a region of perpendicular
electric and magnetic fields.
– The crossed fields are first adjusted to produce an
undeflected beam.
• The magnetic field B is turned off and the electric field
E produces a measureable beam deflection on the
screen.
• From the size of the deflection and the values for E and
B, the charge-to-mass ratio can be determined.
Cyclotron
• Cyclotron accelerates charged particles to very
high velocities using both electric and magnetic
fields.
• The high energy particles that emerge from the
cyclotron are used to bombard other nuclei to
produce nuclear reactions for researchers to
study.
• Hospitals use cyclotrons to produce radioactive
substances used in diagnosis and treatment.
• Motion of the charges occurs in two semicircular
containers D1 and D2 (called dees).
• The dees are evacuated to prevent energy losses
in collisions with the ions and air molecules.
• A high frequency alternating voltage is applied
to the dees and an electromagnet provides a
uniform magnetic field directed perpendicular to
the dees.
• Positive ions released at P near the center of the
electromagnet move in a semicircle and arrive
back at the gap in a time T/2.
• The frequency of the applied voltage is adjusted
so that the polarity of the dees is reversed in the
same time it takes for the ions to complete one
half of a revolution.
• If the phase of the applied
voltage is adjusted so that
D2 is a a lower potential
than D1 by an amount V, the
ion will accelerate across
the gap to D2 and its kinetic
energy will increase by an
amount q·V.
• Ion continues to move in D2
in a semicircular path of
larger radius due to the
velocity increase.
• After a time T/2 it arrives at
the gap.
• The voltage across the gap is reversed so that D1 is now
negative and the ion is again accelerated across the gap.
• The ion is accelerated at each half revolution, gaining a
kinetic energy equal to q·V each time.
• When the radius of the orbit is nearly that of the dees,
the ion leave the system through an exit slit.
• The maximum kinetic energy of the ion can be
obtained upon the exit of the ion from the cyclotron:

 q Br
v
m
2 2
2 q  B  r
K  0.5  m  v 
2m
• Frequncy of oscillation of the cyclotron:
2
ω
f
2π
• Period:
2π r 2π m v 2π m
T


v
q Bv
q B
• Angular frequency (rad/s):
• Cyclotron frequency (Hz):
q B
f 
2πm
ω cyclotron
q B

m
• When the cyclotron principle is used to accelerated
electrons, it has been historically called a betatron. The
cyclotron principle as applied to electrons is illustrated
below.