Particle orbit

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(D. R. Nicholson ~2.1) Consider a particle of charge \(q>0\) and mass \(m\), initially at rest at \((x,y,z)=(0,0,0)\), in the presence of a static magnetic field \(\vec{B}=B_0\vec{u}_z\) and \(\vec{E}=E_0\vec{u}_y\), with \(E_0,B_0>0\).

(a) Sketch the orbit of the particle.

(b) Derive an exact expression for the orbit of the particle.

(c) Show that the orbit can be separated into an oscillatory term and a constant drift term. After averaging in time over the oscillatory motion, is there any net acceleration? If not, how are the forces balanced?

(d) In a neutral plasma, with positive and negative particles and ions of different masses, would there be any net current?

(e) Suppose the electric field were replaced by a gravitational force in the \(yy\) direction, would there be a net current?