Full question is attached.
1) In this question we will consider the motion of a charged particle in uniform electric and magnetic fields that are perpendicular. You may ignore the effects of gravity throughout this question.
i) Consider a charged particle of mass m and charge q which is moving in a uniform electric field E = Eey and a perpendicular uniform magnetic field B = Bez. Show that the equations of motion for the particle are,
ii) If the particle is at rest at time t = 0, verify that
iii) Hence determine the position of the particle at time t, assuming that it was located at the origin at t = 0
b) An infinite metal plate occupies the xz-plane (y=0). The plate is kept at zero potential, V = 0. (see figure 1 below) Photoelectrons are liberated from the plate at y = 0 by ultraviolet radiation. The initial velocity of the photoelectrons is negligible. A uniform magnetic field B is maintained parallel to the plate in the positive z-direction and a uniform electric field E is maintained perpendicular to the plate in the negative y-direction. The electric field is produced by a second infinite plate parallel to the first plate, maintained at a constant positive voltage V0 with respect to the first plate. The separation of the plated is d.
The solution analyzes the motion of a particle in uniform electric and magnetic fields in detail.