Phased-Array Radar. In one common type of radar installation, a rotating antenna sweeps a radio beam around the sky. But in a phased-array radar system, the antennas remain stationary and the beam is swept electronically. To see how this is done, consider an array of $N$ antennas that are arranged along the horizontal $x$ -axis at $x=0, \pm d, \pm 2 d, \ldots, \pm(N-1) d / 2 .$ (The

number $N$ is odd.) Each antenna emits radiation uniformly in all directions in the horizontal $x y$ -plane. The antennas all emit radiation coherently, with the same amplitude $E_{0}$ and the save-

length $\lambda$ . The relative phase $\delta$ of the emission from adjacent antennas

can be varied, however. If the antenna at $x=0$ emits a signal that is given by $E_{0} \cos \omega t,$ as measured at a point next to the antenna, the antenna at $x=d$ emits a signal given by $E_{0} \cos (\omega t+\delta),$ as measured at a point next to that antenna. The corresponding quantity for the antenna at $x=-d$ is $E_{0} \cos (\omega t-\delta) ;$ for the antennas at $x=\pm 2 d,$ it is $E_{0} \cos (\omega t \pm 2 \delta) ;$ and so on. (a) If $\delta=0,$ the interference pattern at a distance from the antennas is large compared to $d$ and has a principal maximum at $\theta=0$ (that is, in the $+y$ -direction, perpendicular to the line of the antennas. Show that if $d<\lambda,$ this is the only principal interference maximum in the angular range $-90^{\circ}<\theta<90^{\circ}$ . Hence this principal maximum describes a beam emitted in the direction $\theta=0 .$ As described in

Section 36.4 , if $N$ is large, the beam will have a large intensity and be quite narrow. (b) If $\delta \neq 0,$ show that the principal intensity maximum described in part (a) is located at

$$\theta=\arcsin \left(\frac{\delta \lambda}{2 \pi d}\right)$$ where $\delta$ is measured in radians. Thus, by varying $\delta$ from positive to negative values and back again, which can easily be done electronically, the beam can be made to sweep back and forth around $\theta=0 .(\mathrm{c})$ A weather radar unit to be installed on an airplane emits radio waves at 8800 $\mathrm{MHz}$ . The unit uses 15 antennas in an array 28.0 $\mathrm{cm}$ long (from the antenna at one end of the array to the antenna at the other end. What must the maximum and minimum values of $\delta$ be (that is, the most positive and most negative values) if the radar beam is to sweep $45^{\circ}$ to the left or right of the airplane's direction of flight? Give your answer in radians.