Cavities and resonant circuits
Although the resonant cavity we have been describing seems to be quite different from the ordinary resonant circuit consisting of an inductance and a capacitor, the two resonant systems are, of course, closely related. They are both members of the same family; they are just two extreme cases of electromagnetic resonators—and there are many intermediate cases between these two extremes. Suppose we start by considering the resonant circuit of a capacitor in parallel with an inductance, as shown in Fig. 23–16(a). This circuit will resonate at the frequency $\omega_0=1/\sqrt{LC}$. If we want to raise the resonant frequency of this circuit, we can do so by lowering the inductance $L$. One way is to decrease the number of turns in the coil. We can, however, go only so far in this direction. Eventually we will get down to the last turn, and we will have just a piece of wire joining the top and bottom plates of the condenser. We could raise the resonant frequency still further by making the capacitance smaller; however, we can also continue to decrease the inductance by putting several inductances in parallel. Two one-turn inductances in parallel will have only half the inductance of each turn. So when our inductance has been reduced to a single turn, we can continue to raise the resonant frequency by adding other single loops from the top plate to the bottom plate of the condenser. For instance, Fig. 23–16(b) shows the condenser plates connected by six such “single-turn inductances.” If we continue to add many such pieces of wire, we can make the transition to the completely enclosed resonant system shown in part (c) of the figure, which is a drawing of the cross section of a cylindrically symmetrical object. Our inductance is now a cylindrical hollow can attached to the edges of the condenser plates. The electric and magnetic fields will be as shown in the figure. Such an object is, of course, a resonant cavity. It is called a “loaded” cavity. But we can still think of it as an $L$-$C$ circuit in which the capacity section is the region where we find most of the electric field and the inductance section is that region where we find most of the magnetic field. If we want to make the frequency of the resonator in Fig. 23–16(c) still higher, we can do so by continuing to decrease the inductance $L$. To do that, we must decrease the geometric dimensions of the inductance section, for example by decreasing the dimension $h$ in the drawing. As $h$ is decreased, the resonant frequency will be increased. Eventually, of course, we will get to the situation in which the height $h$ is just equal to the separation between the condenser plates. We then have just a cylindrical can; our resonant circuit has become the cavity resonator of Fig. 23–7. You will notice that in the original $L$-$C$ resonant circuit of Fig. 23–16 the electric and magnetic fields are quite separate. As we have gradually modified the resonant system to make higher and higher frequencies, the magnetic field has been brought closer and closer to the electric field until in the cavity resonator the two are quite intermixed. Although the cavity resonators we have talked about in this chapter have been cylindrical cans, there is nothing magic about the cylindrical shape. A can of any shape will have resonant frequencies corresponding to various possible modes of oscillations of the electric and magnetic fields. For example, the “cavity” shown in Fig. 23–17 will have its own particular set of resonant frequencies—although they would be rather difficult to calculate.