Helical Orbits of Charged Particles in
Magnetic Fields
• 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 (the adjective is helical)
• The angular frequency w is also called the
cyclotron frequency.
• 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.
v parallel
p

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.
Helical Orbits of Charged Particles in
Magnetic Fields