Suppose that I have two charged particles in the configuration below.
Let us assume the following:
- We apply a constant force $f$ to the the bottom particle so that it has a constant acceleration $a(t)=f/m$.
- The velocity of the bottom particle is negligible to simplify the calculation of the electric field.
- The top particle is initially stationary with a large mass $M$.
- The distance $r$ is large enough so that the Coulomb repulsion between the particles, which is inversely proportional to $r^2$, is negligible.
Under these conditions the Lienard-Wiechert retarded electric field due to the bottom particle, accelerating at time $t=0$, produces a force $F$ on the top particle, at a later time $t=r/c$, given by:
Let us say that in a time interval $\Delta t$ the top particle gains a momentum $F\Delta t$ towards the left.
My question is the following: How is this momentum change balanced?
The conventional answer is to say that the EM field gains an opposite momentum to the right.
But the only way I can see the EM field changing is if the top particle accelerates to the left, under the action of $F$, producing its own counter electric field towards the right. However this won't work as the mass $M$ of the top particle is assumed to be large so that its acceleration, and thus its induced electric field, is negligible.
In summary I can see how the electric field in the vicinity of the top particle can transfer momentum to it but I can't see any mechanism whereby the top particle transfers momentum back to the field if we are free to make the assumption that it is so heavy that it can absorb the momentum without changing its motion appreciably.
P.S. My hypothesis is that a balancing momentum $F\Delta t$ to the right is transmitted backwards in time from the top particle at time $t=r/c$ to the bottom particle at time $t=0$ using an advanced electromagnetic interaction. This momentum then has the effect of reducing the effective mass of the bottom particle (less external force has to be supplied to produce a given acceleration $a$).