P. Piot, PHYS 571 – Fall 2007Non-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 2007Non-uniform B-field: adiabatic invariance• Let’s explicit• but soP. Piot, PHYS 571 – Fall 2007Non-uniform B-field: adiabatic invariance• The previous equation implies thatis an adiabatic invariant• xxxxP. Piot, PHYS 571 – Fall 2007Magnetic mirror• The previous equation implies that• Using the adiabatic invariantP. Piot, PHYS 571 – Fall 2007Magnetic mirrorTrajector (top) in a non-uniform B-field for two cases of injection angleP. Piot, PHYS 571 – Fall 2007Non-adiabatic invariance: the solenoid• Consider a short magnetic solenoidal lens• In cylindrical coordinate, compute the θ-component of the Lorentzforce (this gives the angular momentum pθ)P. Piot, PHYS 571 – Fall 2007Non-adiabatic invariance: the solenoid• Integrating over a Gauss-surface• Consequently the charge pick-up the angular:•WithP. Piot, PHYS 571 – Fall 2007Generation of angular-momentum dominated beamsmagnetic field mapsL1, L2, L3: magnetic solenoidal lensesRadio-frequency gunP. Piot, PHYS 571 – Fall 2007e.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 magneticfield (A becomes
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