The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation for the force on a particle in such a field, if the electric field vector is E and the magnetic field vector is B, the force on a particle of mass m that carries a charge q and has a velocity v is given by F = qE + qv x B where we assume that v (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 r = mv/qB = v/wc where wc = qB/m is the cyclotron frequency.
(b) Choose the z-axis to lie in the direction of B and let the plane containing E and B be the yz-plane. Thus B = Bk, E = E, j + Ezk Show that the z component of the motion is given by z(t) = z0 + z0t qEz/2mt2 where z(0) = z0 and z(0) = z0
(c) Continue the calculation and obtain expressions for x(t) and y(t). Show that the time averages of these velocity components are (x) Ey/B€™ (y) = 0 (Show that the motion is periodic and then average over one complete period).
(d) Integrate the velocity equations found in © and show (with the initial condition x(0) = €“ A A/w, x(0) = Ey/B, y(0) = 0, y(0) €“ A) that

These are the parametric equations of a trochoid. Sketch the projection of the trajectory on the xy €“ plane for the cases (i) A > |Ey/B|, (ii) A

Transcribed Image Text:

E, x(t) -A cos wt + 4, y(1) sin w i