What happens when a free electron and an electromagnetic wave interacts? Does it vibrate or move along the direction of the electromagnetic wave?

• Are you treating the EM wave classically here or as a "bunch of photons"? – ACuriousMind Mar 14 '15 at 17:39
• No. I consider EM wave as a combination of electric and magnetic fields. – Harish Mar 14 '15 at 17:42
• What do you know about the Lorentz force? – Kyle Kanos Mar 14 '15 at 17:43
• A charged particle experiences a force called Lorentz force when it moves in a uniform magnetic field. But it does not experience any force if the charge moves parallel or anti-parallel to the direction of the field. – Harish Mar 14 '15 at 17:47
• The Lorentz force applies to electric and magnetic fields: $\mathbf F=m\ddot{\mathbf x}=q\mathbf E+q\mathbf u\times\mathbf B$. So the answer here depends on the configuration of the total field with respect to the electron. – Kyle Kanos Mar 14 '15 at 17:57

If the frequency of the electromagnetic wave satisfies $$f< \frac{2m_e c^2}{h}$$ $$\implies \lambda > 0.002 \text{ nm, (gamma rays)}$$ Then you can study the system with classical electrodynamics, and the electron will vibrate along the amplitude of the electric field (this is exactly how antenna's work)
• I otherwise agree, but you still have to assume the electric field is not strong enough to accelerate the electron to relativistic energies. Otherwise $v_{max} \approx c \Rightarrow F_{Electric} = qE_{max} \approx qE_{max} v_{max}/c = qv_{max}B = F_{Lorentz}$, the magnetic components will also have an influence. Note that in electron's frame the EM-wave-s 4-wavevector is oscillating, it accelerates (and it seems more like a GR-problem), so this does not simplify the problem. – kristjan Mar 14 '15 at 20:01
• Correct. Also my comment was not about the electrons rest frame during oscillation which is clearly non-inertial, but the frame in which the electrons plane of motion is stationary. So as you correctly stated when the amplitude is large enough the magnetic field will become more relevant resulting in some rotational motion in that plane in addition to the vibrational motion. But again for such strong fields, the electrons radiation $\propto\gamma^6 a^2$ will be really strong... So it's interesting and nontrivial to carefully study the dynamics of the problem. – Ali Moh Mar 14 '15 at 20:27