Grad-B Parallel to the Magnetic Field: Magnetic Mirror

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# Grad-B Parallel to the Magnetic Field: Magnetic Mirror - PowerPoint PPT Presentation

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

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
• 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
• 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
• 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
• 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
• Conservation of energy:

or

and finally

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
• 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
• 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: