In the Extended Third Edition of "Physics for Engineers and Scientists" by Ohanian and Markert (pp. 722-723), it is written:
According to the naive interpretation of Coulomb’s Law, the electric forces between charges are action-at-a-distance, that is, a charge q' exerts a direct force on a charge q even though these charges are separated by a large distance and are not touching. However, such an action-at-a-distance interpretation of electric forces leads to serious difficulties in the case of moving charges. Suppose that we suddenly move the charge q' somewhat nearer to the charge q; then the electric force on q has to increase, according to the inverse-square law. But the required increase cannot occur instantaneously — the increase can be regarded as a signal from q' to q, and it is a fundamental principle of physics, based on the theory of relativity, that no signal can propagate faster than the speed of light. This suggests that, when we suddenly move the charge q', some kind of physical disturbance propagates through space from q' to q and adjusts the electric force to the new increased value (see Fig. 23.1). Thus, charges exert forces on one another by means of disturbances that they generate in the space surrounding them. These disturbances are called electric fields.
Fields are a form of matter — they are endowed with energy and momentum (as we shall find in Chapter 33), and they therefore exist in a material sense. In the context of the above example, it is easy to see why the disturbance, or field, generated by the sudden displacement of q' must carry momentum: when we suddenly move q' toward q, the increase in the force that q' exerts on q will be delayed until a signal has had time to propagate from q' carrying the information regarding the changed position of q'. But the increase in the force that q exerts on q' does not suffer a similar delay. Since we have not moved q, we have not disturbed the way in which q causes electric forces on other charges, according to Coulomb’s Law. This temporary deviation between the forces exerted by q' on q and by q on q' means that, when only the charges are considered, Newton’s Third Law on the balance of action and reaction fails. Thus an extra entity, such as the field, is needed to take up the momentum and energy missing from the particles.
Now, why does not the force that q exerts on q' also suffer a certain delay? I cannot understand this premise, and so I cannot understand the necessity of the conclusion, namely the existence of the electric field. Also, I cannot understand why such conclusion is derived only in the case of moving charges and not also when the charges are standing still.
Here's my reasoning:
If I move a charge q' towards another charge q, then the distance between q' and q decreases progressively. And the displacement of q' is transmitted to q with a certain delay, OK. But q' must also receive the signal from q with a certain delay (even if such delay must be smaller than when the charges were still, because they are getting closer to each other). So, the force exerted by q on q' must also increase progressively. So, both forces increase progressively, and Newton's Third Law is not violated. Why it is instead?
Even without a field, why can't we suppose that the charges 'know' of each other with a certain delay? And so that each charge 'adjusts' its force accordingly, keeping the balance? And why is the problem of the mutual delay only considered when the charges are moving (or at least when one of them is moving) and not also when they are standing still? Why, to suppose a field, it is necessary to consider the case of a moving charge? To avoid action-at-a-distance, can't we just suppose a field in a static system too? Please, help me understand all this. I think I must be wrong, because my reasoning seems to go against physics. But why am I wrong? (There's a voice in my head saying "Momentum!", but I cannot find a clear picture. What am I missing? A lot, it seems...!)