Charged Particles in Electric and Magnetic Fields

1
Charged Particles in Electric and Magnetic
Fields

• Motion of charged particles
• Lorentz Force
• Examples cyclotron, mass spectrometer

2
Recall
in an electric field
in a magnetic field
in general,
Lorentz Force
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Example
A positive charge q3.2x10-19 C moves with a
velocity v(2i3j-k) m/s through a region where
both a uniform magnetic and electric field
exist.Calculate the force on the charge if
B(2i4jk) T andE(4i-j-2k) V/m.
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Magnetic Fields F qv?B
q
v

• The force is perpendicular to the velocity (and
  path), so
• no work is done by B
• kinetic energy is constant
• speed is constant

x x x x x x x x
F
B
Only the direction of the motion changes due to
the magnetic force. Lets look at
these two special cases
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1) Uniform B, v perpendicular to B
q
Magnetic force (?v ) is Fm qvB
(constant)
x x x x x x x x
r
But Fc mv2/r , so the radius of circle is
Motion is a circle.
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Example
A proton is moving in a circular orbit of radius
14cm in a uniform magnetic field of 0.35T
directed perpendicular to the velocity of the
proton. Find the orbital speed of the proton.
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2) Uniform B, v not perpendicular to B

• path is a circular helix along a field line
• the component v?? (parallel to B) is constant

On a large scale, the particles follow the field
lines.
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Frequency
rad/s
Angular frequency
is independent of speed and radius. 1/T (period)
The kinetic energy of a particle at radius r is
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The Cyclotron accelerates particles to very
high velocities
x x x x x x x x x x
B
-
dees

AC Voltage Supply
The electric field across the gap is reversed
each time the proton arrives, so that its speed
in the gap continually increases. Because the
time for each half-circle is the same for any
proton speed, the voltage supply can just be set
to a constant frequency.
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Example
A cyclotron needs to accelerate electrons to a
speed of 0.95c for experiments. Given that it
can provide a magnetic field of 0.5T a) what
is the cyclotron frequency?b) what is the radius
of the cyclotron?
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Quiz
A cyclotron can accelerate protons to a maximum
kinetic energy of 1 MeV. You want to design a
new, improved model with a higher maximum proton
energy. Which of the following changes would
help?

1. Double the magnetic field
2. Double the diameter of the dees
3. Double the proton current
4. Double the voltage of the voltage supply

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The Velocity Selector allows to select particles
that move
at the same velocity
x x x x x x x x x x
E
Source
Detector
v
B
Find Conditions so that the path is straight.
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x
Free-body diagram
Need for straight path
B
v
E
for straight path
14
x
Question
How are the charges deflected that are either
traveling too slow or too fast to travel in a
straight line?
B
v
E
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Mass Spectrometer separates ions according to
their
masscharge ratio
x x x Uniform B x x x x x x x x x
radius depends on mass
Ion Source
Exercise
Assume 12C and 14C have same charge. Find the
ratio of the diameters of the circular paths if
the ions have the same speeds