# The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation for the force

## Question:

(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

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