Consider an electron acted upon by a constant and uniform magnetic field B = B₀ẑ, and a uniform but time-varying electric field E = ŷE_y0 sin(ωt). Assume that the initial conditions are such that the motion takes place in the (x, y) plane and that at t = 0 the electron is at rest (v₀ = 0) at the origin.

(a) Show that the orbit of the electron is given by

(b) In the low-frequency limit, w ≪ Ω_c, show that the electron orbits at the angular frequency ω around an ellipse that has its major axis perpendicular to the electric field. Determine the ratio of the minor to the major axis of the ellipse.

(c) In the high-frequency limit, w ≫ Ω_c, show that the electron moves in a circle at the cyclotron frequency Ω_c.

Transcribed Image Text:

x(t) = e Eyo m(w² - 1²) Nc y(t) = = √ -(cos Net - 1) - Ως (cos wt - 1)] e Eyo m(w² - N²) (sin Net - sin wt) -