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Unformatted text preview: ξ ( t ) = i η ω exp(-iωt + iθ ) we see that the position and velocity have phases that diFer by a ±actor o± i. Thus we’ve recovered the ±act that velocity is perpendicular to position ±or uni±orm circular motion. [N.B. this is not true when the ±requenct ω is a ±unction o± time. In ±act it is straight-±orward to fnd the answer in this case as long as you remember that dω dt is no longer 0.] ²inally absorbing some constants in r, θ we have the equations o± motion x ( t ) = r cos(-ωt + θ ) y ( t ) = r sin(-ωt + θ ) z ( t ) = z + v z + 1 2 a z t 2 This simply describes a helix with pitch that increases as a ±unction o± time. ± 1.0 ± 0.5 0.0 0.5 1.0 ± 1.0 ± 0.5 0.0 0.5 1.0 1 2 3 4 ²igure 1: Charged Particle in Constant aligned E and B felds 2...
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