   Next: Planetary Motion Up: Multi-Dimensional Motion Previous: Charged Particle Motion in

# Exercises

1. An electron of mass and charge moves in a uniform -directed electric field of magnitude , and a uniform -directed magnetic field of magnitude . The electron is situated at the origin at with an initial -directed velocity of magnitude . Show that the electron traces out a cycloid of the general form         Find the values of , , , and , and sketch the electron's trajectory in the - plane when , , and .

2. A particle of mass and charge moves in the - plane under the influence of a constant amplitude rotating electric field which is such that and . The particle starts at rest from the origin. Determine its subsequent motion. What shape is the particle's trajectory?

3. A particle of mass slides on a frictionless surface whose height is a function of only: i.e., . The function is specified by the parametric equations      where is the parameter. Show that the total energy of the particle can be written where . Deduce that the particle undergoes periodic motion whose frequency is amplitude independent (even when the amplitude is large). Demonstrate that the frequency of the motion is given by .   Next: Planetary Motion Up: Multi-Dimensional Motion Previous: Charged Particle Motion in
Richard Fitzpatrick 2011-03-31