Charged Particles in Electric
and
Magnetic Fields
• Motion of charged particles
• Lorentz Force
• Examples: cyclotron, mass spectrometer
Recall:


F  qE
in an electric field

 
F  qv  B
in a magnetic field




F  qv  B  qE
in general,
“Lorentz Force”
Example:
A positive charge q=3.2x10-19 C moves with a velocity
v=(2i+3j-k) m/s through a region where both a uniform
magnetic and electric field exist.
Calculate the force on the charge if B=(2i+4j+k) T and
E=(4i-j-2k) V/m.
Magnetic Fields: F = qvB
q
The force is perpendicular to x
the velocity (and path), so:
• no work is done by B
• kinetic energy is constant
• speed is constant
x
v
+
x
x
F
x
x
Only the direction of the motion changes due to
the magnetic force.
Let’s look at these two special cases:
x
Bx
1) Uniform B, v perpendicular to B
Magnetic force (v ) is
x
Fm = qvB (constant)

v
x
x
+
q
x
F
x
But Fc = mv2/r , so the radius of
circle is :
mv
r
qB
x
Motion is a circle.
x
r
x
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.
2) Uniform B, v not perpendicular to B
x
z
v
+
y
-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.
Frequency
Angular frequency
v qBr qB
 

r mr
m
[rad/s]

= "cyclotron frequency f " is independent of speed
2
and radius. = 1/T (period)
1
1  qBr  q B r
KE  mv  m 
 
2
2  m 
2m
qBr
sin ce v 
m
2
The kinetic energy of a
particle at radius r is
2
2
2
2
The Cyclotron : accelerates particles to very high
velocities
x
“dees”
x
Bx
x
x
x
x
x
+
+ x
x
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.
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?
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?
A)
B)
C)
D)
Double
Double
Double
Double
the magnetic field
the diameter of the dees
the proton current
the voltage of the voltage supply
The Velocity Selector: allows to select particles that
move at the same velocity
E
x
Source
x
+
x
x
v
x
x
x
x
x
Detector
B
x
  
E, B, v all 
Find: Conditions so that the path is straight.
Free-body diagram:
FM  qvB
x
B
+
E
FE  qE
v

Need F  0
for straight path
 qvB  qE

E
for straight path
v
B
Question:
FM  qvB
x
B
+
E
FE  qE
v
How are the charges deflected
that are either traveling too
slow or too fast to travel in a
straight line?
Mass Spectrometer : separates ions according to
their mass:charge ratio
Ion
Source
Exercise:
x
x
x
x
x
x
x
x
x
x
x
x
Uniform B
radius depends on
mass
Assume 12C and 14C have same charge.
Find the ratio of the diameters of the circular
paths if the ions have the same speeds