Examples of energy flow
Our formula for the energy flow vector $\FLPS$ is something quite new. We want now to see how it works in some special cases and also to see whether it checks out with anything that we knew before. The first example we will take is light. In a light wave we have an $\FLPE$ vector and a $\FLPB$ vector at right angles to each other and to the direction of the wave propagation. (See Fig. 27–2.) In an electromagnetic wave, the magnitude of $\FLPB$ is equal to $1/c$ times the magnitude of $\FLPE$, and since they are at right angles, \begin{equation*} \abs{\FLPE\times\FLPB}=\frac{E^2}{c}. \end{equation*}  Therefore, for light, the flow of energy per unit area per second is \begin{equation} \label{Eq:II:27:16} S=\epsO cE^2. \end{equation}  For a light wave in which $E=E_0\cos\omega(t-x/c)$, the average rate of energy flow per unit area, $\av{S}$—which is called the “intensity” of the light—is the mean value of the square of the electric field times $\epsO c$: \begin{equation} \label{Eq:II:27:17} \text{Intensity} = \av{S} = \epsO c\av{E^2}. \end{equation}  Believe it or not, we have already derived this result in Section 31–5 of Vol. I, when we were studying light. We can believe that it is right because it also checks against something else. When we have a light beam, there is an energy density in space given by Eq. (27.14). Using $cB=E$ for a light wave, we get that \begin{equation*} u=\frac{\epsO}{2}\,E^2+\frac{\epsO c^2}{2}\biggl( \frac{E^2}{c^2}\biggr)=\epsO E^2. \end{equation*}  But $\FLPE$ varies in space, so the average energy density is \begin{equation} \label{Eq:II:27:18} \av{u} = \epsO\av{E^2}. \end{equation}  Now the wave travels at the speed $c$, so we should think that the energy that goes through a square meter in a second is $c$ times the amount of energy in one cubic meter. So we would say that \begin{equation*} \av{S} = \epsO c\av{E^2}. \end{equation*}  And it’s right; it is the same as Eq. (27.17). Now we take another example. Here is a rather curious one. We look at the energy flow in a capacitor that we are charging slowly. (We don’t want frequencies so high that the capacitor is beginning to look like a resonant cavity, but we don’t want dc either.) Suppose we use a circular parallel plate capacitor of our usual kind, as shown in Fig. 27–3. There is a nearly uniform electric field inside which is changing with time. At any instant the total electromagnetic energy inside is $u$ times the volume. If the plates have a radius $a$ and a separation $h$, the total energy between the plates is \begin{equation} \label{Eq:II:27:19} U=\biggl(\frac{\epsO}{2}\,E^2\biggr)(\pi a^2h). \end{equation}  This energy changes when $E$ changes. When the capacitor is being charged, the volume between the plates is receiving energy at the rate \begin{equation} \label{Eq:II:27:20} \ddt{U}{t}=\epsO\pi a^2hE\dot{E}. \end{equation}  So there must be a flow of energy into that volume from somewhere. Of course you know that it must come in on the charging wires—not at all!  It can’t enter the space between the plates from that direction, because $\FLPE$ is perpendicular to the plates; $\FLPE\times\FLPB$ must be parallel to the plates. You remember, of course, that there is a magnetic field that circles around the axis when the capacitor is charging. We discussed that in Chapter 23. Using the last of Maxwell’s equations, we found that the magnetic field at the edge of the capacitor is given by \begin{equation*} 2\pi ac^2B=\dot{E}\cdot\pi a^2, \end{equation*}  or \begin{equation*} B=\frac{a}{2c^2}\,\dot{E}. \end{equation*}  Its direction is shown in Fig. 27–3. So there is an energy flow proportional to $\FLPE\times\FLPB$ that comes in all around the edges, as shown in the figure. The energy isn’t actually coming down the wires, but from the space surrounding the capacitor.  Let’s check whether or not the total amount of flow through the whole surface between the edges of the plates checks with the rate of change of the energy inside—it had better; we went through all that work proving Eq. (27.15) to make sure, but let’s see. The area of the surface is $2\pi ah$, and $\FLPS=\epsO c^2\FLPE\times\FLPB$ is in magnitude \begin{equation*} \epsO c^2E\biggl(\frac{a}{2c^2}\,\dot{E}\biggr), \end{equation*}  so the total flux of energy is \begin{equation*} \pi a^2h\epsO E\dot{E}. \end{equation*}  It does check with Eq. (27.20). But it tells us a peculiar thing: that when we are charging a capacitor, the energy is not coming down the wires; it is coming in through the edges of the gap. That’s what this theory says! How can that be? That’s not an easy question, but here is one way of thinking about it. Suppose that we had some charges above and below the capacitor and far away. When the charges are far away, there is a weak but enormously spread-out field that surrounds the capacitor. (See Fig. 27–4.) Then, as the charges come together, the field gets stronger nearer to the capacitor. So the field energy which is way out moves toward the capacitor and eventually ends up between the plates. As another example, we ask what happens in a piece of resistance wire when it is carrying a current. Since the wire has resistance, there is an electric field along it, driving the current. Because there is a potential drop along the wire, there is also an electric field just outside the wire, parallel to the surface. (See Fig. 27–5.) There is, in addition, a magnetic field which goes around the wire because of the current. The $\FLPE$ and $\FLPB$ are at right angles; therefore there is a Poynting vector directed radially inward, as shown in the figure. There is a flow of energy into the wire all around. It is, of course, equal to the energy being lost in the wire in the form of heat. So our “crazy” theory says that the electrons are getting their energy to generate heat because of the energy flowing into the wire from the field outside. Intuition would seem to tell us that the electrons get their energy from being pushed along the wire, so the energy should be flowing down (or up) along the wire. But the theory says that the electrons are really being pushed by an electric field, which has come from some charges very far away, and that the electrons get their energy for generating heat from these fields. The energy somehow flows from the distant charges into a wide area of space and then inward to the wire. Finally, in order to really convince you that this theory is obviously nuts, we will take one more example—an example in which an electric charge and a magnet are at rest near each other—both sitting quite still. Suppose we take the example of a point charge sitting near the center of a bar magnet, as shown in Fig. 27–6. Everything is at rest, so the energy is not changing with time. Also, $\FLPE$ and $\FLPB$ are quite static. But the Poynting vector says that there is a flow of energy, because there is an $\FLPE\times\FLPB$ that is not zero. If you look at the energy flow, you find that it just circulates around and around. There isn’t any change in the energy anywhere—everything which flows into one volume flows out again. It is like incompressible water flowing around. So there is a circulation of energy in this so-called static condition. How absurd it gets! Perhaps it isn’t so terribly puzzling, though, when you remember that what we called a “static” magnet is really a circulating permanent current. In a permanent magnet the electrons are spinning permanently inside. So maybe a circulation of the energy outside isn’t so queer after all. You no doubt begin to get the impression that the Poynting theory at least partially violates your intuition as to where energy is located in an electromagnetic field. You might believe that you must revamp all your intuitions, and, therefore have a lot of things to study here. But it seems really not necessary. You don’t need to feel that you will be in great trouble if you forget once in a while that the energy in a wire is flowing into the wire from the outside, rather than along the wire. It seems to be only rarely of value, when using the idea of energy conservation, to notice in detail what path the energy is taking. The circulation of energy around a magnet and a charge seems, in most circumstances, to be quite unimportant. It is not a vital detail, but it is clear that our ordinary intuitions are quite wrong.