Motion in a constant uniform magnetic field

1. Motion in a constant uniform magnetic field Section 21

2. To obtain the relativistic trajectory… • Obtain equation of motion for relativistic momentum

3. Use equations of relativistic dynamics to obtain differential equation for velocity.

6. Integrate to obtain velocity vs. time.

8. Integrate again to obtain coordinates vs. time.

11. If H magnitude varies slowly in space… Magnetic bottle

15. What is the relativistically correct trajectory of an electron in a constant uniform magnetic field? • A cycloid • A catenary • A helix

16. Trajectory is helix • Difference between exact and classical results is the expression for angular frequency

17. Changing uniform magnetic field • Field need only be uniform on the scale of the particle’s orbit. • Uniform magnetic field may change in magnitude and direction. • If the changes is sufficiently slow, it is “Adiabatic”. • That means, the orbit changes only slightly during one period. • Use method of Adiabatic Invariants from Mechanics (v.1, sec. 49). • Transverse component of momentum pt varies as Sqrt[H] when H changes.

18. Magnetic bottle When H has slow spatial variations, as a particle moves through the changing field, H appears to be changing in time, but H remains uniform to the particle. • pt varies as Sqrt[H]

19. More on the magnetic bottle • Energy (and p2) remain constant, since H does no work. • If pt2 increases, the longitudinal component pl2 must decrease. • Penetration of the particle into regions of sufficiently high magnetic field is impossible.

20. Magnetic bottle continued • The radius of the helix decreases as H increases. • The longitudinal step per cycle decreases as H increases. • Eventually, the particle is reflected.

21. Magnetic bottle, cont. • Longitudinal inhomogeneity of H causes a drift of the guiding center. • The drift is transverse to H.