Light waves are one half kinetic and one half potential energy

[Philip Wood]

Going back - with apologies for being boring - to my post (hash20), we find that the energy in a charged capacitor with uniform field is proportional to the volume occupied by the field. This strongly suggests associating the energy with the field itself.

This is supported by considering other geometries of capacitor where the field is non uniform. We get the same value for energy stored using U=\frac{1}{2}C V^2 as by using \int\frac{1}{2}\epsilon_{0} E^2 d(Vol) in which the integral is taken over the volume of the field.

So we can, it seems, associate a given amount of stored energy with each volume element of the region occupied by the field. That's as far as I'd want to go. I wouldn't really want to say that this is where you would find the energy, or this is where the energy hangs out. Though I suppose you could say this if you wanted to; it would certainly have more going for it than saying that the energy lives on the plates with the charges!

 

[Ken G]

A way to argue that this is indeed "where the energy hangs out" is to look at time varying fields and Maxwell's equations. Via the Poynting flux concept, Maxwell's equations can be cast in the form of conservation equations for currents of electromagnetic energy. So if you have a charge, with a static field, and you shake the charge, you set up a blip in the field that corresponds to a local increase in the field energy density. That local energy blip then propagates along the field, where it can deposit energy somewhere else if it is absorbed by another charge. At any time, if you want to know where to put another charge to absorb some excess field energy, the answer is, "in the region where the excess field resides." This makes it very straightforward to associate that energy with a kind of "location" within the field.

Indeed, those who view fields as just placekeepers for what charges are doing may benefit from switching hats, and viewing the charges as placekeepers for what the fields are doing. I don't think either concept stands alone very well-- they work off each other.

 

[fizzle]

... So we can, it seems, associate a given amount of stored energy with each volume element of the region occupied by the field. That's as far as I'd want to go. I wouldn't really want to say that this is where you would find the energy, or this is where the energy hangs out. Though I suppose you could say this if you wanted to; it would certainly have more going for it than saying that the energy lives on the plates with the charges!

Sounds like good advice to me. Heaviside, who came up with the Poynting Vector, warned against giving electromagnetic energy a "personal identity" and following it from place to place. That's especially important when trying to break it down into electric and magnetic energy since they can "vanish" at any given time when em waves intersect. For example, if you send equal step waves down each end of a transmission line, when they overlap in the middle the magnetic energy in the resulting field will go to zero while the electric energy doubles.

 

[fizzle]

Indeed, those who view fields as just placekeepers for what charges are doing may benefit from switching hats, and viewing the charges as placekeepers for what the fields are doing. I don't think either concept stands alone very well-- they work off each other.

What's interesting about that is the complete flip-flop in the viewpoint required. The photon viewpoint is extreme in that it appears to be purely charge-to-charge transfer of state. To see this, start with the old Bohr model of hydrogen (electron orbiting a proton at various stable radii/energy level) and do a semiclassical calculation of the fields and energies involved. Have the electron drop from a high energy state to the lowest state (ignore the various forbidden transition issues). It emits a photon of a certain energy which corresponds to a certain frequency. That frequency is somewhere between the starting orbit frequency and the ending orbit frequency, asymptotically approaching 50% as the photon energy increases. You can calculate the radial electric field for the frequency during this transition. The obvious physical picture from the Bohr model is that the electron, for whatever reason, doesn't emit an em wave in the stable states but does during the transition and some sort of mean frequency is assigned to the photon. Regardless of what actually happens, we now assume a photon of the given energy goes flying off into space.

Now for the extreme charge-to-charge state transfer part. There's an electron sitting out in space somewhere, minding its own business (stationary). Along comes the photon emitted by the hydrogen atom. It's incident on the electron and absorbed temporarily. Let's say the electron was along the axis of the original hydrogen atom so that the photon's em wave is circularly-polarized. Do a semiclassical analysis of this CP em wave interacting with the free electron - with the constraint that the electron's total energy is increased by the incident photon's energy. What do you get for the required electric field in the CP em wave to make the electron behave properly? Well, you get, essentially, the electric field strength of the original electron in the emitting hydrogen atom. The free electron mimics the motion of the original electron in the hydrogen atom. It behaves as if orbiting a central positive charge at a given radius and associated frequency; i.e. the state of the hydrogen electron has been transferred completely to the free electron, at least temporarily. The two most troubling things from a semiclassical viewpoint are:

1) The required electric field strength in the CP em wave is *very* high
2) The CP em wave's electric field strength is independent of the distance (instead of 1/r)

Either one of these would be a showstopper from a semiclassical viewpoint. This is why any attempt to reconcile electromagnetic theory with QM is mostly doomed to failure. I tried a handwaving explanation in a previous post a while back but it was deleted by the mods here, so I won't do it again and be banned. I just think that going through all the calculations is a fascinating exercise in semiclassical em theory that others might enjoy.