Download Motion of a Charged Particle in a Magnetic Field

Motion of a Charged Particle
in a Magnetic Field
AP Physics C
Montwood High School
R. Casao
• The magnetic force acting on a charged
particle moving in a magnetic field is
always perpendicular to the velocity of the
particle.
• From this property, the work done by the
magnetic force is 0 J since the
displacement of the charge is always
perpendicular to the magnetic force.
W = F·d·cos q
• Therefore, a static magnetic field changes
the direction of the velocity but does not
change the speed or the kinetic energy of
the charged particle.
• Consider a positively
charged particle moving
in a uniform external
magnetic field with its
initial velocity vector
perpendicular to the
magnetic field.
• The magnetic field B is
into the page (as
indicated by the x’s).
• The figure shows that
the charged particle
moves in a circle whose
plane is perpendicular to
the magnetic field.
• The circular path results
because the magnetic force
Fmag is at right angles to the
velocity v and the magnetic
field B and has a constant
magnitude equal to q·v·B.
• The force deflects the particle
and the directions of v and B
change continuously.
• The force Fmag is a centripetal
force, which changes only the
direction of the velocity while
the speed remains constant.
• The direction of the rotation
is given using the right
hand rule – right hand for
positive charges and left
hand for negative charges
(Casao’s rule).
• Since Fmag= Fcentripetal:

2

mv

qvB 
r
• This reduces to:

 mv
qB 
r
• Solving for the radius
of curvature:
• The radius of curvature
is proportional to the
momentum of the
particle and inversely
proportional to the
magnetic field B.
• The angular frequency
w of the rotating
charged particle is:

v qB
ω 
r
m

mv

r
q B
• The period T of the circular motion (time
for one revolution) is equal to the
circumference of the circle divided by the
speed of the particle:
d 2π r 2π 2π m

T 


v
v
ω
q B
• The angular frequency and the period of
the circular motion do not depend on the
speed of the particle or the radius of the
orbit.
• The angular frequency w is also called the
cyclotron frequency since charged particles
circulate at this frequency in a particle
accelerator called a cyclotron.
• If a charged particle moves in a uniform
magnetic field B with its velocity at some angle q
to B, its path is a helix.
• For the field B in the
x-direction, there is no
component of force in
the x direction, therefore
ax = 0 m/s2 and the
x component of v is
constant.
• The magnetic force q·(v x B) causes the
components of vx and vy to change in time
and the resulting motion is a helix having
its axis parallel to the magnetic field B.
• The projection of the path onto the yz
plane (viewed along the x axis) is a circle.
• The distance between
successive rotations
in the helical path is
called the pitch, p.
• The pitch is parallel to the
magnetic field B.
• The perpendicular velocity
influences how much time
it takes to complete the
circular path.
• The parallel velocity
determines the pitch.
p
v parallel 
t
p  v parallel  t
2πm

p  v parallel 
qB
• The motion of a charged particle in a
nonuniform magnetic field is complex.
• If a magnetic field is strong at the ends
and weak in the middle, the particles
oscillate back and forth between the end
points.
• Such a field can be produced by two current loops
at the ends of the “bottle” to produce a strong
magnetic field to pinch off the ends.
• A charged particle starting at one end will spiral
along the field lines until it reaches the other end,
where it reverses directions and spirals back. This
configuration is known as a “magnetic bottle”
because charged particles can be trapped in it.
– This concept has been used to confine plasmas (hot
gases consisting of electrons and protons).
– The magnetic bottle may pay a role in achieving a
controlled nuclear fusion process.
– The problem is that if a large number of particles are
trapped in the magnetic bottle, collisions between the
particles cause them to “leak” from the system.
• The Van Allen radiation belts
consist of charged particles (e& p+) surrounding the earth.
• The charged particles are
trapped by the earth’s
nonuniform magnetic field and
spiral around the earth’s field
lines from pole to pole.
• Most of the charged particles
come from the sun.
• When the charged particles
are in the atmosphere over the
poles, they can collide with
other atoms, causing them to
emit visible light, the Aurora
Borealis and Aurora Australis.