# How can a plane wave induce perpendicular motion in particles, when the plane wave only carries longitudinal momentum? [closed]

An electromagnetic plane wave moving in the $$z$$-direction

$$E(x,t) = E_x \cos(k z - \omega t), \hspace{1cm} B(x,t) = B_y \sin(kz-\omega t)$$

has field momentum in $$z$$-direction. But according to this paper, it induces motion in $$x$$ and $$y$$ direction in a charged particle. How is this possible? Does conservation of momentum not apply? If not, are field and particle momentum completely separated concepts?

Remark: My question is less focused on the precise movement described in the paper, but generally how it is possible for a wave with momentum in $$z$$ direction to induce motion in $$x$$ and $$y$$ direction.

• When the particle acquires x momentum, the field acquires negative x momentum. (Don’t forget that charged particles make their own fields.) Nov 2, 2022 at 20:48
• Can this be true? The particles momentum is dependent on the mass, but the charged particles field is indipendent of the mass. So how can they cancel out reliably? Or did I missunderstand you? Nov 2, 2022 at 22:07
• The field the charged particle makes depends on how fast it's moving, which in turn depends on its mass, because $F = ma$. Nov 2, 2022 at 22:10
• Nov 3, 2022 at 6:01
• The paper you link in your question is too involved for me to check, In the linked answer to a similar question it is seen that the oscilations happen in the direction of the propagation of the wave, Nov 3, 2022 at 6:43

It is easiest to think of this in terms of scattering. A photon, with momentum in $$z$$ collides elastically with a particle at rest. After the collision the particle has momentum along $$x$$ so the photon is scattered and has equal momentum along $$-x$$ in addition to its momentum along $$z$$
In the simple classical picture, the electromagnetic field does no work on the charged particle (because the E-field and particle velocity are $$\pi/2$$ out of phase) in a time-averaged sense. Similarly, there is no time-averaged momentum transferred to the particle because it oscillates up and down along the polarisation vector of the E-field. In a bit more detail we would include the Lorentz force due to the magnetic part of the field and find that there was some momentum transferred to the particle along the z-axis.