An electron undergoes an oscillatory circular motion in the xy plane, dictated by a uniform constant...

An electron undergoes an oscillatory circular motion in the xy plane, dictated by a uniform constant magnetic field B applied in the z direction, and by a circularly polarized electromagnetic wave that propagates in the z direction. The electric field of the wave is given by E = E_o[cos(kz - wt)x - sin(kz - wt)y]. Note that at z=0, where the electron moves, this electric field is E = E_o[cos(wt)x + sin(wt)y].

(a) This part of the problem is classical mechanics, not electrodynamics. Write down the equation of motion for the electron and solve it. In the solution, express the radius r_o of the oscillatory circular orbit in terms of w and the cyclotron resonance frequency w_c = qB/m, where q and m are the charge and the mass of the electron. Assume q>0.

(b) Find the power P radiated by the electron by using the Larmor formula generalized by Lienard, instead of starting from scratch.

(c) Write the time average of the Poynting vector <S> of the electromagnetic wave in terms of E_o. Multiply <S> by the area pi*r_o² of the circle defined by the motion of the electron to estimate the power P_o available to the electron, and compare the P you found in part (b) with P_o.

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genius_generous answered 1 year ago

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