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Disclaimer, so that you can provide a more informative answers: I have a degree in mathematics but I am quite ignorant in physics.

I was reflecting about the ability of a charged particle (any fundamental charge, but let's restrict ourselves to an electric charge) of exerting a force without expenditure of energy.

I mean, let's imagine a charged particle in motion with respect to an observer's frame of reference. It radiates electro-magnetic waves which carry energy and momenta away from it. Once those quantities are depleted, the particle is reduced to be at rest wrt such reference frame.

This makes sense.

Now imagine a particle at rest wrt the observer's frame. It has nothing to lose (except for its mass, which is finite anyhow). No energy, no momenta (again, aprt from mass-bound energy). Still, it produces a static field which can exert a force (indefinitely) that can accelerate objects, thus making them gain energy.

At the prensent state of my knowledge, I find it strange to understand. May you provide a simple explanation?

EDIT: To rephrase my question, I find it unsettling that an entity can exert a force upon other entities without expenditure of other quantities (energy of whatever).

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  • $\begingroup$ "Once those quantities are depleted, the particle is reduced to be at rest wrt its reference frame" Probably isn't what you meant to say. There is not, in general, any predetermined rest frame for particles to be "reduced to". $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jul 1, 2018 at 18:01
  • $\begingroup$ No time to write a full blown answer, but ... you seem to be ignoring the possible presence of potential energies while at the same time assuming a conservative force field which will, in general, generate s potential contribution to the total energy of the system. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jul 1, 2018 at 18:03
  • $\begingroup$ @dmckee : May you elaborate a bit further? I thought that we could bestow a rest frame upon any material particle. $\endgroup$
    – MadHatter
    Commented Jul 1, 2018 at 18:05
  • $\begingroup$ Your intuition is reasonable. Two attracting particles do have a smaller mass when they are closer to each other. You just need to include displacement with force to get energy. The Earth does not spend any energy attracting you when you just stand, but only when you jump down. $\endgroup$
    – safesphere
    Commented Jul 1, 2018 at 18:48
  • $\begingroup$ @safesphere: "Your intuition is reasonable. Two attracting particles do have a smaller mass when they are closer to each other." Astonishing. Could you provide some links to where such matter is treated in depth? Thanks! $\endgroup$
    – MadHatter
    Commented Jul 2, 2018 at 16:23

2 Answers 2

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Electromagnetic theory obeys law of local conservation of energy. This means that we can define energy as a quantity distributed in space, and its amount in some observed region changes solely by transport through boundaries of that region.

When one particle's field acts and accelerates another (second) particle, energy is being sucked away from the space near the second particle and is being transformed into kinetic energy of that particle. So, the kinetic energy that appeared "from nothing" was actually already there in the space around the second particle, it just had a different form.

There are formulae for energy density in space due to EM interaction. In macroscopic theory (the particles have to have their charge distributed in some non-zero volume), the energy density is given by squares of electric and magnetic field: $$ \frac{dE}{dV} = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2. $$

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  • $\begingroup$ Ok, let's imagine an isolated system with a charged particle (at rest) inside its boundaries, say an electron. The sole source of (potential?) energy within such system is the electric field radiated by the electron. Note that such energy cannot be depleted: it will continue to exist indefinitely. If another particle is placed into the system, it will experience a force which never ceases to exist. Where that energy comes from? $\endgroup$
    – MadHatter
    Commented Jul 1, 2018 at 18:02
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    $\begingroup$ @Mad Several comments here. First, applying a force requires no energy so long as no relative motion occurs (see physics.stackexchange.com/q/1984). Second if you add another charged particle you are changing the potential energy landscape, but if the second particle is not charged there is not force and so no problem. Finally (and least important) a physicist would not use the word "radiated" with a static field because "radiation" means something specific. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jul 1, 2018 at 18:07
  • $\begingroup$ Thank you, this is very informative (the observation about terminology as well). Let me read that post about force vs. energy, probably the key resides there. $\endgroup$
    – MadHatter
    Commented Jul 1, 2018 at 18:12
  • $\begingroup$ Ok, that question you linked was interesting as well, but my perplexity still stands. Once we established the useful concept that a force does not require energy expenditure to be exerted (it's counterintuitive but acceptable), one has to consider what follows: by the virtue of its electrostatic field, a particle generates a potential across space and ultimately a force. That force can accelerate objects, so those objects acquire velocity, i.e. energy. It seems that energy is created from nothing: I'm still puzzled. $\endgroup$
    – MadHatter
    Commented Jul 1, 2018 at 19:15
  • $\begingroup$ @MadHatter, energy is not created, that would indeed be strange (it would violate the law of conservation of energy). When objects acquire kinetic energy, other parts of the system lose energy (possibly of different form). In your example, electromagnetic energy distributed in space is decreased by the same amount by which kinetic energy is increased. Sum of all energies is conserved. $\endgroup$
    – Ján Lalinský
    Commented Jul 1, 2018 at 19:22
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Here is a possible, although somewhat simplistic, explanation.

Let's use an electron as a charged particle to make the explanation more concrete.

The energy of the electrostatic field around an electron is generated at the time it is separated from a neutral atom and is taken away. This energy is equal to the work that had to be done by someone to move the electron against the attraction force from the positively charged atom (positive ion) left behind.

NOTE: In fact, a similar field will be created around the positive ion and those two fields could be treated as one, but, if the distance between the particles is significant, the fact that the fields are shared may not be obvious from examining the fields in their immediate neighborhoods.

This electrostatic field can perform work, say, by accelerating some charges, but any work performed by the field will reduce its energy and its capacity to do work is limited by its initial energy (or the work done to create it).

For instance, if the positive ion left behind is allowed to fly all the way toward the electron, the electrostatic field will perform exactly the same work (by accelerating the ion) someone has originally performed to separate the charges and create the field.

Once the electron and the ion reunite (with a little bang - dissipating the kinetic energy acquired by the ion), there won't be any electrostatic field or electrostatic energy left.

So, if we understand the origin of the electrostatic field, we should not be surprised by it ability to perform some limited work.

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  • $\begingroup$ Thank you, let me address your points: 1. "The energy of the electrostatic field around an electron is generated at the time it is separated from a neutral atom and is taken away." But the electron possessed an electrostatic field already back when it was bound to the atom. 2. "but any work performed by the field will reduce its energy and its capacity to do work is limited by its initial energy" I don't understand: indeed, if you take that electron's electrostatic field, you can put whatever number of particles whithin it: it will continue to accelerate them tirelessly. $\endgroup$
    – MadHatter
    Commented Jul 1, 2018 at 19:20
  • $\begingroup$ @MadHatter 1. The energy of electrostatic field in the atom is small. As you pull the electron away it increases. Think about a spring or a rubber band: the more it is stretched, the more potential energy it acquires. 2. Not really. When charges accelerate by the field, they move toward the electron and eventually will neutralize it and kill the field, as described in my example. $\endgroup$
    – V.F.
    Commented Jul 1, 2018 at 19:48
  • $\begingroup$ Understood. It's fascinating. $\endgroup$
    – MadHatter
    Commented Jul 2, 2018 at 16:27
  • $\begingroup$ @MadHatter It is fascinating to me too... as most things in physics. $\endgroup$
    – V.F.
    Commented Jul 2, 2018 at 17:45
  • $\begingroup$ I really don't know what answer to accept. Both have been informative. $\endgroup$
    – MadHatter
    Commented Jul 2, 2018 at 19:05

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