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Basic Plasma Physics 39 E = , (3.2-7) where the negative sign comes from the convention that the electric field always points in the direction of ion motion. 3.3 Single Particle Motions The equation of motion for a charged particle with a velocity v in a magnetic field B is given by the Lorentz force equation: F=mdv =q(E+v B). (3.3-1) dt Particle motion in a magnetic field in the zˆ direction for the case of negligible electric field is found by evaluating Eq. (3.3-1): m vx =qBvy t m vy = –qBvx t m vz =0. t (3.3-2) Taking the time derivative of Eq. (3.3-2) and solving for the velocity in each direction gives 2vx qB vy qB 2 t 2 = m t = m v x 2vy qB vx qB 2 t 2 = m t = m v y . (3.3-3) These equations describe a simple harmonic oscillator at the cyclotron frequency: c = q B . (3.3-4) m For electrons, this is called the electron cyclotron frequency. The size of the particle orbit for finite particle energies can be found from the solution to the particle motion equations in the axial magnetic field. In this case, the solution to Eq. (3.3-3) isPDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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