Let a steady current I flow up the y-axis and let the initial position and velocity of a particle with mass m and charge q be r₀ = (x_0, 0, 0) and v₀ = (0, ν₀, 0).(a) Show that the motion of the particle is confined to the x-y plane.(b) Prove that ν_y = v₀ + β ln(x/x₀) (where β is a constant) and use the fact that the Lorentz force does no work to prove that the particle never leaves the interval x₀ ≤ x ≤ x₀ exp(−2ν₀/β).(c) Use (but do not solve) the equation of motion for ν_x and the results obtained so far to sketch a typical particle trajectory. Justify your sketch.(d) Prove that the particle trajectory is given by

Transcribed Image Text:

dy dx Vo + B ln(x/xo) -ß ln(x/xo)[2vo + ß ln(x/xo)]