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I am having difficulty in understanding whether fields store momentum and energy or particles store them or both fields and particles store them?

System of potential energy

In the above source it mentions that energy is stored in system but in this source

Energy stored in fields

they mention energy is stored in field. So which one is correct?

Please also give an explanation why?

I am trying to ask here if any of the choices I mentioned is correct, why is it correct? Why are other choices wrong? But I am not trying to ask if momentum and energy can exist together. So I feel it's not a duplicate of the question mentioned.

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  • $\begingroup$ In what context are you asking this question? The answer is similar in both classical electromagnetism and quantum electrodynamics, but there are some important caveats in the latter case. $\endgroup$ – probably_someone Jun 17 at 8:43
  • $\begingroup$ Actually I am trying to understand where in general energy is stored or momentum like one case is electrostatics other could be gravitation( energy stored in Mass or field of gravity? Why so?) $\endgroup$ – Trilok Girish Kamagond Jun 17 at 8:50
  • $\begingroup$ @probably_someone classical electrodynamics $\endgroup$ – Trilok Girish Kamagond Jun 17 at 9:14
  • $\begingroup$ @John Rennie I am trying to ask the one edited. $\endgroup$ – Trilok Girish Kamagond Jun 17 at 15:45
  • $\begingroup$ @TrilokGirishKamagond OK, I have reopened it. I think the title of the question is misleading and you might want to consider editing it in case anyone else closes the question. $\endgroup$ – John Rennie Jun 17 at 15:56
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They are both correct, because they talk about a different concept of energy. There are dozens of different concepts of energy in physics. In this case, one is related to another in a subordinate matter; energy of EM field is a generalization, a broader concept than electric potential energy.

Electric potential energy of system of particles is a pre-relativistic concept. Let there be some charged particles in some definite region of space, let us call these particles "a system". Potential energy of this system is sum of contributions that depend solely on positions of the particles in space, or mathematically, on coordinates of these particles in some reference frame; it does not depend on their velocities or other variable quantities. It is a property of the system, but it is not necessarily "distributed" across all points of the region. It is just a number assigned to the system, there is no spatial localization of it implied. In non-relativistic theory, the law of conservation of energy states that sum of kinetic energies of all particles and total potential energy is constant:

$$ E_{kinetic} + E_{potential} = const., $$ but where in space exactly this energy "is" is immaterial, all that matters is numerical value of this energy and how it depends on positions and velocities of particles.

This concept of potential energy as function of particle positions works (in the sense of allowing formulation of conservation of energy) in many scenarios that are studied in electrostatics.

It however becomes insufficient whenever induced electric fields or other electromotive forces need to be considered. We know that outside the phenomena of electrostatics, the law of conservation of energy needs to be modified to include other contributions to energy, so it will still hold.

In a relativistic theory such as classical electromagnetic theory, whenever the charged particles move, the potential energy assigned in the usual manner, based on particle positions, doesn't give all electromagnetic energy. In other words, in relativistic EM theory one cannot, in general case where particles move, formulate law of conservation of energy using just kinetic energy of particles and potential energy based on particle positions.

However, Maxwell's equations allow us to formulate the law of local conservation of energy, where all energies, including electromagnetic energy, are assigned to a region of space via a distribution, which gives "density" of that energy type at every point of the region. Total energy of this region can be calculated based on values of these densities there. Thus EM energy is thought of as localized to space, with some definite density. Sometimes it is called EM field energy because in macroscopic EM theory, the distribution at any point of space is a function of total electromagnetic field at that point.

However, it should be noted that the formula for this electromagnetic energy is still not unique - there is an infinite number of formulae that give valid distribution of EM energy density and all are equivalent, at least in special relativistic setting, where gravity can be ignored.

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  • $\begingroup$ Thank you for such a wonderful explanation which I never read about. One doubt I have is when they say capacitor stores energy in the field, it is the energy stored in system right? Not the electromagnetic energy? $\endgroup$ – Trilok Girish Kamagond Jun 18 at 8:47
  • $\begingroup$ Can also provide a reference for detailed understanding of the answer you mentioned? Thank you $\endgroup$ – Trilok Girish Kamagond Jun 18 at 8:51
  • $\begingroup$ The capacitor stores energy as electromagnetic energy. One can view this electromagnetic energy either as potential energy of the positive and negative charges on the capacitor plates (in case of plate capacitor), or as field energy of strong electric field between those plates. Both views are valid, but the second one is more easily generalized to more complicated situations, such as energy stored in an inductor. $\endgroup$ – Ján Lalinský Jun 18 at 13:24
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    $\begingroup$ EM energy is more general concept than potential energy. It includes electric and magnetic energy contributions that are not part of electrostatic potential energy. In special case when electric field is static, electrostatic potential energy of a system of all charges is the same as whole EM energy of such system. $\endgroup$ – Ján Lalinský Jun 19 at 10:06
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    $\begingroup$ Energy stored in field can often be regarded as potential energy, for example in capacitor or even in inductor. In the inductor case, the energy is potential in the sense it can be stored and then released and turned back into other kinds of energy, but it is not the potential energy in the classical sense - "function of particles positions", because it is a function particle velocities (electric current). In case of EM wave energy, one usually does not talk about it as potential energy at all, so one says "electromagnetic energy" or "EM field energy". $\endgroup$ – Ján Lalinský Jun 19 at 14:20

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