# Classical EM neglects electron recoil?

Imagine two electrons $A$ and $B$ at rest.

Electron $B$ is at a vertical distance $r$ above electron $A$.

Let us assume that the electrons are constrained to move on horizontal rails.

At time $t=0$ I give electron $A$ a horizontal acceleration $a$ for a time interval $\Delta t$.

At time $t=r/c$, due to the impulse provided by the classical Lienard-Weichert radiation field of electron $A$, electron $B$ gains horizontal momentum:

$$\Delta p = F\Delta t=-\frac{e^2a\Delta t}{4\pi\epsilon_0c^2r}$$

In terms of quantum electrodynamics a photon is exchanged between electron $A$ and electron $B$.

Therefore if electron $B$ receives a momentum $\Delta p$ from the photon does that mean that electron $A$ must recoil with momentum $-\Delta p$ as it emits the photon?

If this is true then the classical (retarded) Lienard-Weichert interaction theory describes (later) momentum transfer to the static charge $B$ due to the (earlier) accelerating charge $A$ but it neglects the fact that charge $A$ receives (earlier) recoil momentum during the process.

Maybe this recoil momentum is described by a time-reversed advanced Lienard-Weichert interaction in which the earlier momentum transfer to the accelerating charge $A$ is due to the later interaction with the static charge $B$?

Since it is proportional to $\dot{\mathbf{a}}$ in your example it will be felt by electron $A$ as a jerk at the start and end of interval $\Delta t$ and had to be compensated by whatever external force is used to provide acceleration.
This force accounts for EM waves emitted in all (allowed) directions and so, do not need subsequent interaction with electron $B$, although of course electron $B$ would emit its own (retarded) EM wave which could be additionally felt by electron $A$ later, when it arrives. Nevertheless, time-reversable interpretation of such recoil (and Abraham-Lorenz force), similar to what you are suggesting, is available within Wheeler-Feynman absorber theory