# How to think about the transverse force imparted to an electron by an EM wave?

This is a question that came to mind while dusting off the Feynman Lectures. I know he discusses some of this in a subsequent chapter, but I'm trying to understand it based on what has already been presented. I hope I have the directions of the forces correct. I don't do well with lots of plus and minus signs canceling each other out.

So my first question is: do I have the direction of the acceleration induced by the field correct?

The second question is, what is a good way to understand how the force on one charged particle is propagated by the EM wave and induces a force on a second particle; particularly in terms of transverse momentum?

The electric field observed at a distance $\rho$ from an accelerating electric charge is given by (Feynman Lectures, V-I, eq. 28.5)

$$\mathfrak{E}=-\frac{q}{4\pi\varepsilon_{o}c^{2}}\frac{d^{2}\hat{\rho}}{dt^{2}}.$$

Here, $\hat{\rho}$ is the retarded unit direction vector pointing from the point of measurement to the field source. So if the charged particle is an electron, accelerating in the $Z$ direction, while passing through the $X\times Y$ plane in which the stationary observer lies, the observed field will point in the positive $Z$ direction, since the charge of the electron is negative, and the formula has a minus sign in front of it.

A positive electric field is one which points away from a positive charge. An electron is attracted to a positive charge, so it will move in the opposite direction to the field vector. So the field produced by an upwardly accelerated electron $e_{1}^{-}$ will push another electron $e_{2}^{-}$ downward when it arrives at its location.

Obviously, work was done in producing the propagating field, since it carries energy. So some part of the driving force which produced the acceleration of $q_{1}$ went into producing the wave. When the wave interacts with $e_{2}^{-}$ it imparts a force, and therefore some energy and momentum (in the $-Z$ direction) to $e_{2}^{-}$.

It's almost as if the reaction force of the field of $e_{1}^{-}$ was stored in the wave and propagated to $e_{2}^{-}$.

But what happens when an identical wave interacts with a positron $e_{3}^{+}$? It will accelerate in the same direction as $e_{1}^{-}$ did.

I do recall that Feynman explains that momentum is imparted to an electron by an EM wave, in the direction of the wave. That is because the now moving electron $e_{2}^{-}$ encounters the co-propagating magnetic field, and the Lorentz force acts to push the electron forward.

I'm asking about the transverse force. Is there some way to make sense of this?