-3-
be initially traveling in the z direction with the velocity vo.
Fig. 1. The equations of motion referred to this coordinate system, (see
Fig. 1.) are:
= -b1 sin(2nft) +
b2 cos(2nft) .....(1)
= b1 sin(2nft) .....(2)
= -b2 co(2nft) .....(3)
Where the dot implies differentiation with respect to time and the abbreviation
bi = Biemc ..... (4)
has been employed to represent the cyclotron frequency (here in e.s.u)
1.2 Non-rotating case. (B1 = 0)
If we set B1 = 0 in the equations of motion of 1.1 we obtain simply a non-rotating uniform magnetic field. Clearly from Eq. (2) of 1.1 if is originally zero it will remain so, and we shall have motion in the y-z plane. If we differentiate 1.1 (3) and eliminate and we obtain
Upon employing the transformation u = sin(2nft), (1) becomes
which is readily seen to have sinusoidal solutions. If we take as boundary conditions at t = 0; = = = = 0 = v0