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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.

(My 'strong'.)

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...!)

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  • $\begingroup$ Ultimately, this explanation fails because it tries to avoid (special) relativity (presumably deemed to hard for engineers). Ultimately, the reference frame used (origin of velocity) should not matter for the emerging physics, but this explanation violates that. Also it presents a picture of the fields to live in absolute space, which is fundamentally wrong. $\endgroup$
    – Walter
    Nov 7, 2017 at 12:19
  • $\begingroup$ But relativity is explicitly mentioned in the argument. Are you saying that nonetheless the authors are oversimplifying? And that this might be a reason why I am confused? $\endgroup$
    – gpascal
    Nov 7, 2017 at 12:55

3 Answers 3

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I think the key to the argument lies in "we have not disturbed the way in which q causes electric forces on other charges." What other charges?

Let's imagine a third charged particle q'' that is a copy of q' (same charge and same mass) and is located right next to the final position of q' after it moves.

To illustrate:

Before q' moves:

q'                                  q
            q''

After q' moves:

            q'                      q
            q''

If there were a delay in the force on q' changing, then q' and q'' would feel different forces despite being located at the same place with respect to q. That is, you would have identical charges in the same location feeling different forces, despite the origin of the both forces being q. Imagine you started watching the particles q' and q'' just after q' stopped moving but before the delayed change in force. You would immediately see that q' and q'' accelerate at different rates despite being identical in every respect. This indeterminism would be disturbing to a physicist, since identical starting conditions should produce identical results (since we're still in the realm of classical physics).

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  • $\begingroup$ Thank you, Mark! Your point is very interesting and helpful. I am trying to incorporate it in the picture, and beginning to see it more clearly. $\endgroup$
    – gpascal
    Nov 7, 2017 at 12:09
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In the frame of reference of q (the frame in which q is stationary), when q' moves towards q, it is moving through the static electric field due to q. q' must therefore experience a change (due to the $r^{-2}$ law) in local field strength as soon as it starts to move. But q must wait to experience the changed field strength due to q' starting to move. O&M justify this using the relativistic idea that forces and changes in forces can't propagate instantaneously. Indeed this is really their justification for the notion of a field rather than action at a distance.

Reverse q and q' in the last paragraph to summarise what happens in the q' frame of reference. What you mustn't do is to try and compare what happens "at the same time" in the two frames! The concept doesn't make sense. You would need to use the relativistic view of space-time.

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  • $\begingroup$ Thank you Philip. I am trying to assume the relativistic point of view. So, the field of q "touches" (so to say) q'. And, as soon as q' starts moving towards q, it feels an increase in this field. Instead, even if the field of q' also "touches" q, q cannot feel anything until the increase in the field of q' is transmitted. Is it correct? But, precisely because there is a field, is not such transmission instantaneous? I am thinking about 3 billiard balls in a line: A touches B and B touches C. Suppose B "is" the field. If I push A, then C feels the push immediately, or not? $\endgroup$
    – gpascal
    Nov 7, 2017 at 12:50
  • $\begingroup$ Hello gpascal. Yes, I'd say that the first three lines of your comment $are$ correct. But, moving on, fields don't imply instantaneous transmission – quite the reverse. Interestingly, your billiard balls example makes the point: the impact force is transmitted at finite speed as a mechanical wave passes through the balls. Maxwell first arrived at his equations for e-m fields and their finite propagation speed by modelling them as disturbances in a medium with finite inertia and elastic modulus. $\endgroup$ Nov 7, 2017 at 15:43
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The problem is that the charge can not change the place at the moment, and when the charge is moved, a magnetic field arises and becomes an electromagnetic field. The picture is different then.

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  • $\begingroup$ It's the justification of the field itself, as given in that textbook, that I was finding problematic. But my ideas are slowly becoming clearer. Thanks! $\endgroup$
    – gpascal
    Nov 7, 2017 at 16:35
  • $\begingroup$ @gpascal The process of thinking is the information processing process. If you include the wrong information, you come to the wrong conclusion. Since it is not possible to move the charge right now, any answer to your question is wrong. Maybe we can try something similar: Proton is placed in point A, a neutron at point B, and the electric field is measured at point C. Now the electron hits the proton and neutralizes it, the resulting photon hits the neutron and causes beta decay and the charge in point B occurs. It's not the same, but it's kind of similar. $\endgroup$
    – FPosta
    Nov 7, 2017 at 19:53

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