The motion of a charged particle in an electromagnetic field can be obtained from the lorentz equation. if the electric field vector is e and the magnetic field is b, the force on a particle of mass m that carries a charge q and has a velocity v where we assume u < < c(speed of light) (a) if there is no electric field and if the particle enters the magnetic field in a direction perpendicular to the lines of magnetic flux, show that the trajectory is a circle with radius mu where wqb/m is the cyclotron frequency the yz-plane. thus show that the z component of the motion is given by (b) choose the z-axis to lie in the direction of b and let the plane containing e and b be 2m where (c) continue the calculation and obtain expressions for č(t) and y(t). show that the time averages of these velocity components are (show that the motion is periodic and then average over one complete period.) (d) integrate the velocity equations found in (c) and show (with the initial conditions cos wet + t, y(t)- sin wct. these are the parametric equations of a trochoid. sketch the projections of the tra- jectory on the xy-plane for the cases (i) a 〉 ey/bi, (ii) a < ie,/bi, and (iii) a = ie, /bi. (the last case yields a cycloid.)