Non uniform B field adiabatic invariance We now consider the case of a non uniform still time independent magnetic field We suppose the magnetic field non uniformity is slow i e small compare to the gyro radius Then motion is said to be adiabatic and there exist an invariant called the adiabatic invariant P Piot PHYS 571 Fall 2007 Non uniform B field adiabatic invariance Let s explicit but so P Piot PHYS 571 Fall 2007 Non uniform B field adiabatic invariance The previous equation implies that is an adiabatic invariant xxxx P Piot PHYS 571 Fall 2007 Magnetic mirror The previous equation implies that Using the adiabatic invariant P Piot PHYS 571 Fall 2007 Magnetic mirror Trajector top in a non uniform B field for two cases of injection angle P Piot PHYS 571 Fall 2007 Non adiabatic invariance the solenoid Consider a short magnetic solenoidal lens In cylindrical coordinate compute the component of the Lorentz force this gives the angular momentum p P Piot PHYS 571 Fall 2007 Non adiabatic invariance the solenoid Integrating over a Gauss surface Consequently the charge pick up the angular With P Piot PHYS 571 Fall 2007 Generation of angular momentum dominated beams magnetic field maps L1 L2 L3 magnetic solenoidal lenses Radio frequency gun P Piot PHYS 571 Fall 2007 e m Field tensor covariant equation of motion As we showed we expect Quadratic with e radial position Full conversion of CAM to MAM as the electrons exit the magnetic field A becomes zero P Piot PHYS 571 Fall 2007
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