A resonant cavity
We want to look now at what our solution gives for the electric field between the plates of the capacitor as we continue to go to higher and higher frequencies. For large $\omega$, the parameter $x=\omega r/c$ also gets large, and the first few terms in the series for $J_0$ of $x$ will increase rapidly. That means that the parabola we have drawn in Fig. 23–5 curves downward more steeply at higher frequencies. In fact, it looks as though the field would fall all the way to zero at some high frequency, perhaps when $c/\omega$ is approximately one-half of $a$. Let’s see whether $J_0$ does indeed go through zero and become negative. We begin by trying $x=2$: \begin{equation*} J_0(2)=1-1+\tfrac{1}{4}-\tfrac{1}{36}=0.22. \end{equation*}  The function is still not zero, so let’s try a higher value of $x$, say, $x=2.5$. Putting in numbers, we write \begin{equation*} J_0(2.5)=1-1.56+0.61-0.11=-0.06. \end{equation*}  The function $J_0$ has already gone through zero by the time we get to $x=2.5$. Comparing the results for $x=2$ and $x=2.5$, it looks as though $J_0$ goes through zero at one-fifth of the way from $2.5$ to $2$. We would guess that the zero occurs for $x$ approximately equal to $2.4$. Let’s see what that value of $x$ gives: \begin{equation*} J_0(2.4)=1-1.44+0.52-0.08=0.00. \end{equation*}  We get zero to the accuracy of our two decimal places. If we make the calculation more accurate (or since $J_0$ is a well-known function, if we look it up in a book), we find that it goes through zero at $x=2.405$. We have worked it out by hand to show you that you too could have discovered these things rather than having to borrow them from a book.  As long as we are looking up $J_0$ in a book, it is interesting to notice how it goes for larger values of $x$; it looks like the graph in Fig. 23–6. As $x$ increases, $J_0(x)$ oscillates between positive and negative values with a decreasing amplitude of oscillation. We have gotten the following interesting result: If we go high enough in frequency, the electric field at the center of our condenser will be one way and the electric field near the edge will point in the opposite direction. For example, suppose that we take an $\omega$ high enough so that $x=\omega r/c$ at the outer edge of the capacitor is equal to $4$; then the edge of the capacitor corresponds to the abscissa $x=4$ in Fig. 23–6. This means that our capacitor is being operated at the frequency $\omega=4c/a$. At the edge of the plates, the electric field will have a rather high magnitude opposite the direction we would expect. That is the terrible thing that can happen to a capacitor at high frequencies. If we go to very high frequencies, the direction of the electric field oscillates back and forth many times as we go out from the center of the capacitor. Also there are the magnetic fields associated with these electric fields. It is not surprising that our capacitor doesn’t look like the ideal capacitance for high frequencies. We may even start to wonder whether it looks more like a capacitor or an inductance. We should emphasize that there are even more complicated effects that we have neglected which happen at the edges of the capacitor. For instance, there will be a radiation of waves out past the edges, so the fields are even more complicated than the ones we have computed, but we will not worry about those effects now. We could try to figure out an equivalent circuit for the capacitor, but perhaps it is better if we just admit that the capacitor we have designed for low-frequency fields is just no longer satisfactory when the frequency is too high. If we want to treat the operation of such an object at high frequencies, we should abandon the approximations to Maxwell’s equations that we have made for treating circuits and return to the complete set of equations which describe completely the fields in space. Instead of dealing with idealized circuit elements, we have to deal with the real conductors as they are, taking into account all the fields in the spaces in between. For instance, if we want a resonant circuit at high frequencies we will not try to design one using a coil and a parallel-plate capacitor.  We have already mentioned that the parallel-plate capacitor we have been analyzing has some of the aspects of both a capacitor and an inductance. With the electric field there are charges on the surfaces of the plates, and with the magnetic fields there are back emf’s. Is it possible that we already have a resonant circuit? We do indeed. Suppose we pick a frequency for which the electric field pattern falls to zero at some radius inside the edge of the disc; that is, we choose $\omega a/c$ greater than $2.405$. Everywhere on a circle coaxial with the plates the electric field will be zero. Now suppose we take a thin metal sheet and cut a strip just wide enough to fit between the plates of the capacitor. Then we bend it into a cylinder that will go around at the radius where the electric field is zero. Since there are no electric fields there, when we put this conducting cylinder in place, no currents will flow in it; and there will be no changes in the electric and magnetic fields. We have been able to put a direct short circuit across the capacitor without changing anything. And look what we have; we have a complete cylindrical can with electrical and magnetic fields inside and no connection at all to the outside world. The fields inside won’t change even if we throw away the edges of the plates outside our can, and also the capacitor leads. All we have left is a closed can with electric and magnetic fields inside, as shown in Fig. 23–7(a). The electric fields are oscillating back and forth at the frequency $\omega$—which, don’t forget, determined the diameter of the can. The amplitude of the oscillating $E$ field varies with the distance from the axis of the can, as shown in the graph of Fig. 23–7(b). This curve is just the first arch of the Bessel function of zero order. There is also a magnetic field which goes in circles around the axis and oscillates in time $90^\circ$ out of phase with the electric field. We can also write out a series for the magnetic field and plot it, as shown in the graph of Fig. 23–7(c). How is it that we can have an electric and magnetic field inside a can with no external connections? It is because the electric and magnetic fields maintain themselves, the changing $\FLPE$ makes a $\FLPB$ and the changing $\FLPB$ makes an $\FLPE$—all according to the equations of Maxwell. The magnetic field has an inductive aspect, and the electric field a capacitive aspect; together they make something like a resonant circuit. Notice that the conditions we have described would only happen if the radius of the can is exactly $2.405 c/\omega$. For a can of a given radius, the oscillating electric and magnetic fields will maintain themselves—in the way we have described—only at that particular frequency. So a cylindrical can of radius $r$ is resonant at the frequency \begin{equation} \label{Eq:II:23:18} \omega_0=2.405\,\frac{c}{r}. \end{equation}  We have said that the fields continue to oscillate in the same way after the can is completely closed. That is not exactly right. It would be possible if the walls of the can were perfect conductors. For a real can, however, the oscillating currents which exist on the inside walls of the can lose energy because of the resistance of the material. The oscillations of the fields will gradually die away. We can see from Fig. 23–7 that there must be strong currents associated with electric and magnetic fields inside the cavity. Because the vertical electrical field stops suddenly at the top and bottom plates of the can, it has a large divergence there; so there must be positive and negative electric charges on the inner surfaces of the can, as shown in Fig. 23–7(a). When the electric field reverses, the charges must reverse also, so there must be an alternating current between the top and bottom plates of the can. These charges will flow in the sides of the can, as shown in the figure. We can also see that there must be currents in the sides of the can by considering what happens to the magnetic field. The graph of Fig. 23–7(c) tells us that the magnetic field suddenly drops to zero at the edge of the can. Such a sudden change in the magnetic field can happen only if there is a current in the wall. This current is what gives the alternating electric charges on the top and bottom plates of the can. You may be wondering about our discovery of currents in the vertical sides of the can. What about our earlier statement that nothing would be changed when we introduced these vertical sides in a region where the electric field was zero? Remember, however, that when we first put in the sides of the can, the top and bottom plates extended out beyond them, so that there were also magnetic fields on the outside of our can. It was only when we threw away the parts of the capacitor plates beyond the edges of the can that net currents had to appear on the insides of the vertical walls. Although the electric and magnetic fields in the completely enclosed can will gradually die away because of the energy losses, we can stop this from happening if we make a little hole in the can and put in a little bit of electrical energy to make up the losses. We take a small wire, poke it through the hole in the side of the can, and fasten it to the inside wall so that it makes a small loop, as shown in Fig. 23–8. If we now connect this wire to a source of high-frequency alternating current, this current will couple energy into the electric and magnetic fields of the cavity and keep the oscillations going. This will happen, of course, only if the frequency of the driving source is at the resonant frequency of the can. If the source is at the wrong frequency, the electric and magnetic fields will not resonate, and the fields in the can will be very weak. The resonant behavior can easily be seen by making another small hole in the can and hooking in another coupling loop, as we have also drawn in Fig. 23–8. The changing magnetic field through this loop will generate an induced electromotive force in the loop. If this loop is now connected to some external measuring circuit, the currents will be proportional to the strength of the fields in the cavity. Suppose we now connect the input loop of our cavity to an RF signal generator, as shown in Fig. 23–9. The signal generator contains a source of alternating current whose frequency can be varied by varying the knob on the front of the generator. Then we connect the output loop of the cavity to a “detector,” which is an instrument that measures the current from the output loop. It gives a meter reading proportional to this current. If we now measure the output current as a function of the frequency of the signal generator, we find a curve like that shown in Fig. 23–10. The output current is small for all frequencies except those very near the frequency $\omega_0$, which is the resonant frequency of the cavity. The resonance curve is very much like those we described in Chapter 23 of Vol. I. The width of the resonance is however, much narrower than we usually find for resonant circuits made of inductances and capacitors; that is, the $Q$ of the cavity is very high. It is not unusual to find $Q$’s as high as $100{,}000$ or more if the inside walls of the cavity are made of some material with a very good conductivity, such as silver.