GradB Parallel to the Magnetic Field: Magnetic Mirror

1
Grad-B Parallel to the Magnetic Field Magnetic
Mirror
Cylindrical coordinates
r
z
q

• Magnetic field directed along r-z (Bq0)
• No variations only along q (azimuthally symmetric
  field)
• A relationship between Br and Bz can be
  established from divB0

2
Magnetic Mirror (II) Particle Motion

• Approximation ?Bz/?z does not vary with r
• By solving with respect Br via integration with
  respect r it is found

• The components of the Lorentz force in
  cylindrical coordinates are

3
Magnetic Mirror (III) Particle Motion

• Axial force

• Orbit-averaging for a particle with guiding
  center on the axis. In the chosen cylindrical
  coordinates the azimuthal component of the
  velocity will be negative for a positive charge.

therefore
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Magnetic Mirror (IV) Invariance of Magnetic
Moment

• Define the magnetic moment for a gyrating
  particle

then

• The magnetic moment for a gyrating particle
  corresponds to the usual definition of magnetic
  moment for a current loop enclosing an area

5
Magnetic Mirror (V) Invariance of Magnetic Moment

• The magnetic moment is an invariant for the
  particle motion for a generic coordinate s
  parallel to the magnetic field the equation of
  motion along B is

by multiplying both members by vds/dt
(dB/dt is the variation of the field seen by
the particle)
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Magnetic Mirror (VI) Invariance of Magnetic
Moment

• Conservation of energy

or
and finally
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Magnetic Mirror (VII) Reflection

• As a consequence of the invariance of m the
  particle must change its perpendicular
  energy/velocity when the magnetic field changes
• Higher magnetic field will require larger
  perpendicular velocity
• Conservation of energy then will require smaller
  parallel energy/velocity
• Magnetic mirror the parallel velocity can go to
  zero in high magnetic field regions, then causing
  the particle reflection
• If the parallel velocity does not go to zero at
  the highest magnetic field region the particle
  exits the mirror

8
Magnetic Mirror (VIII) Loss Cone

• A particle in correspondence of the minimum B0
  has velocity v0(v0, v-0)
• The particle is being reflected in correspondence
  of the maximum field Bm the velocity at the
  reflection point will be vm(0, vm-)
• Conservation of Energy implies

• The invariance magnetic moment requires

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Magnetic Mirror (IX) Loss Cone

• By expressing everything in terms of the particle
  velocity at the minimum B it is found

v-0
v0
q
v0
Loss Cone

• Mirror Ratio