Page:Design and Calibration of a New Apparatus to Measure the Specific Electronic Charge.pdf/7

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be initially traveling in the z direction with the velocity v_o.

Fig. 1. The equations of motion referred to this coordinate system, (see Fig. 1.) are:

${\overset {H}{z}}$ = -b₁${\overset {I}{x}}$ sin(2nft) +

b₂${\overset {I}{y}}$ cos(2nft) .....(1)

${\overset {H}{x}}$ = b₁${\overset {I}{z}}$ sin(2nft) .....(2)

${\overset {H}{y}}$ = -b₂${\overset {I}{z}}$ co(2nft) .....(3)

Where the dot implies differentiation with respect to time and the abbreviation

b_i = B_ie/mc ..... (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 ${\overset {I}{x}}$ 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 ${\overset {I}{z}}$ and ${\overset {II}{z}}$ 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; ${\overset {I}{x}}$ = ${\overset {II}{x}}$ = ${\overset {I}{y}}$ = ${\overset {II}{y}}$ = 0 ${\overset {.}{z}}$ = v₀