Consider two particles with charges $q_1 = q$ and $q_2 = -q$ which are kept stationary a (large) distance $d$ from each other. The two charges attract each other via the Coulomb force, which we can write as for example $F_2 = -q E_1$, where $E_1 = \frac{q}{4\pi\epsilon_0 d^2}$ is the electrostatic field of charge 1 evaluated at the position of charge 2 (see figure below).
From Newton's third law, we know that $F_1 = - F_2$, and this is also clear from the Lorentz force law.
At time $t=0$ the particles are suddenly released, so that they can accelerate toward each other. Now, here is where my confusion comes in. Assume that one particle is much heavier than the other, e.g. $m_1 \gg m_2$. The lighter particle will then accelerate much faster than the heavier one. After a short time interval $\Delta t$, the situation will look like figure 2, where the light particle (charge 2) has moved a further distance than the heavy one:
Now if $\Delta t \ll \frac{d}{c}$, then the information that charge 1 has started moving has not yet reached charge 2, because this information cannot propagate faster than the speed of light. Therefore, I think that charge 2 still feels initial electrostatic field of charge 1. But because it has moved closer to the initial position of charge 1, the field it experiences now is stronger than the initial field. On the other hand, charge 1 has barely moved, and therefore still experiences almost the same force as in the beginning.
But this means that $F_1 \neq - F_2$, and so Newton's third law is broken! Moreover, total force on the combined system is $F_1 + F_2 < 0$, which indicates that the system as a whole is accelerating to the left, breaking the conservation of momentum! Something seems to be very wrong! Can anyone help find the error in my analysis?
