stationary charge, but the amount of energy that is going past, in watts per square meter,is proportional to the square of the electric field. We shall derive the constant of proportionality in the next chapter If we look at the array from the w side both oscillators contribute equally and in phase, so the electric field is twice as strong as it would be from a single oscillator. Therefore the intensity is four times as strong as it would be if there were only one oscillator. (The numbers in Fi 29-5 represent how strong the intensity would be in this case, compared with what it would be if there were only a single oscillator of unit strength. )Now, in either the nor S direction along the line of the oscillators, since they are half a wavelength apart, the effect of one oscillator turns out to be out of phase by exactly half an oscillation from the other, and therefore the fields add to zero. At a certain par- ticular intermediate angle(in fact, at 30%)the intensity is 2, and it falls off, 4, 2, and so forth. We have to learn how to find these numbers at other angles. It is a question of adding two oscillations with different phases Let us quickly look at some other cases of interest. Suppose the oscillators are again one-half a wavelength apart, but the phase a of one is set half a period behind the other in its oscillation(Fig. 29-5b). In the w direction the intensity is now zero, because one oscillator is"pushing"when the other one is"pulling "But in the n direction the signal from the near one comes at a certain time and that of the other comes half a period later. But the latter was originally half a period behind in timing, and therefore it is now exactly in time with the first one, and so the intensity in this direction is 4 units. The intensity in the direction at 30%is still 2, as we can prove later. Let us remark that one of the reasons that phase relations of oscillators are in teresting is for beaming radio transmitters. For instance, if we build an antenna system and want to send a radio signal, say, to Hawaii, we set the antennas up as Fig 29-5(a) and we broa as in phase, because is to the west of us. Then we decide that tomorrow we are going to broadcast toward Alberta, Canada. Since that is north, not west, all we have to do is to phase of So we can build antenna systems with various arrangements. Ours is one of the simplest possible ones; we can make them much more complicated, and by chang ing the phases in the various antennas we can send the beams in various directions and send most of the power in the direction in which we wish to transmit, without ever moving the antenna! In both of the preceding cases, however, while we are A1-2 broadcasting toward alberta we are wasting a lot of power on Easter Island, and it would be interesting to ask whether it is possible to send it in only one direction At first sight we might think that with a pair of antennas of this natur is always going to be symmetrical. So let us consider a case that comes out un- symmetrical, to show the I variety Fig. 29-6. A pair of dipole antennas If the antennas are separated by one- quarter wavelength, and if the n one giving maximum power in one direction is one-fourth period behind the S one in time, then what happens(Fig. 29-6)? In the w direction we get 2, as we will see later. In the s direction we get zero, because the signal from S comes at a certain time; that from N comes 90% later in time, but it is already 90 behind in its built- in phase, therefore it arrives, altogether, 180 out of phase, and there is no effect. On the other hand. in the n direction the n signal arrives earlier than the S signal by 90 in time, because it is a quarter wavelength closer. But its phase is set sa that it is oscillating 90 behind in time which just compensates the delay difference, and therefore the two signals appear together in phase, ma field strength twice as larg nergy Thus, by using some cleverness in spacing and phasing our antennas, we can send the power all in one direction. But still it is distributed over a great angles. Can we arrange it so that it is focused still more sharply in a particular direction? Let us consider the case of Hawaii again, where we are sending the beam east and west but it is spread over quite an angle, because even at 30 we are still getting half the intensity-we are wasting power. Can we do better th that? Let us take a situation in which the separation is ten wavelengths (Fi