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Grad-B Parallel to the Magnetic Field: Magnetic Mirror. Cylindrical coordinates. r. z. q. Magnetic field directed along r-z ( B q =0 ) No variations only along q (azimuthally symmetric field) A relationship between B r and B z can be established from div B=0 :.

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grad b parallel to the magnetic field magnetic mirror
Grad-B Parallel to the Magnetic Field: Magnetic Mirror

Cylindrical coordinates

r

z

q

  • Magnetic field directed along r-z (Bq=0)
  • No variations only along q (azimuthally symmetric field)
  • A relationship between Brand Bz can be established from divB=0:
magnetic mirror ii particle motion

Azimuthal force => radial drift

Magnetic Mirror (II): Particle Motion
  • Approximation: ∂Bz/∂z does not vary with r
  • By solving with respect Brvia integration with respect r it is found
  • The components of the Lorentz force in cylindrical coordinates are
magnetic mirror iii particle motion
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

magnetic mirror iv invariance of magnetic moment
Magnetic Mirror (IV): Invariance of Magnetic Moment
  • Define the magnetic moment for a gyrating particle:

then

  • The magnetic momentfor a gyrating particle corresponds to the usual definition of magnetic moment for a current loop enclosing an area
magnetic mirror v invariance of magnetic moment
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 v||=ds/dt

(dB/dt is the variation of the field “seen” by the particle)

magnetic mirror vi invariance of magnetic moment
Magnetic Mirror (VI): Invariance of Magnetic Moment
  • Conservation of energy:

or

and finally

magnetic mirror vii reflection
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
magnetic mirror viii loss cone
Magnetic Mirror (VIII): Loss Cone
  • A particle in correspondence of the minimumB0 has velocity v0=(v║0, 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
magnetic mirror ix loss cone
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

v║0

Loss Cone

  • Mirror Ratio: