A particle with charge q and mass m interacts with a circularly polarized plane wave in vacuum. The electric field of the wave is E(z, t) = Re {( + i ŷ)E₀ exp[i(kz − ωt)]}.

(a)Let v_± = ν_x ± iv_y and Ω = 2q E₀/mc. Show that the equations of motion for the components of the particle’s velocity υ can be written

(b) Let ± = ν ± e ^{± i(kz−ωt)} ± ic Ω ω and show that

(c) Let K be the constant of the motion defined by the two ʋ̇ ˙v_z equations above. Differentiate the equations in part (a) and establish that

Use the initial conditions ν(0) = 0 and v'_z(0) = 0 to evaluate K and solve for v_z(t). Describe the nature of the particle acceleration in the z-direction.

Transcribed Image Text:

dvz dt dv+ dt ₂+i(kz-wt) =-1/2/22 = -√2² {1+ e +i(k2-aut). +v_e_i(kz_wr)} = 2(c-v₂)eFi(kz-wt)