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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 ( 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

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  1. 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:

  2. 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

  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

  4. 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

  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 v||=ds/dt (dB/dt is the variation of the field “seen” by the particle)

  6. Magnetic Mirror (VI): Invariance of Magnetic Moment • Conservation of energy: or and finally

  7. 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 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

  9. 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:

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