91Ó°ÊÓ

Figure 36–35 shows the bright fringes that lie within the central diffraction envelope in two double-slit diffraction experiments using the same wavelength of light. Is (a) the slit width $a$, (b) the slit separation $d$, and (c) the ratio $d/a$in experiment B greater than, less than, or the same as those quantities in experiment A?

In three arrangements, you view two closely spaced small objects that are the same large distance from you. The angles that the objects occupy in your field of view and their distances from you are the following: (1) $2\mathrm{Ï•}$and ; (2) $2\mathrm{Ï•}$and $2R$; (3) $\mathrm{Ï•}/2$and $R/2$. (a) Rank the arrangements according to the separation between the objects, with the greatest separation first. If you can just barely resolve the two objects in arrangement 2, can you resolve them in (b) arrangement 1 and (c) arrangement 3?

In the single-slit diffraction experiment of $\mathit{F}\mathit{i}\mathit{g}\mathbf{.}\mathbf{36}\mathbf{-}\mathbf{4}$,let the wavelength of the light be $\mathbf{500}\mathbf{}\mathbf{nm}$, the slit width be localid="1664272054434" $\mathbf{6}\mathbf{μ³¾}$, and the viewing screen be at distance localid="1664272062951" $\mathbf{D}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{m}$. Let y axis extend upward along the viewing screen, with its origin at the center of the diffraction pattern. Also let ${\mathit{I}}_{\mathbf{p}}$represent the intensity of the diffracted light at point P at $\mathbf{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}$. (a) What is the ratio of ${\mathit{I}}_{\mathbf{p}}$to the intensity ${\mathit{I}}_{\mathbf{m}}$ at the center of the pattern? (b) Determine where point P is in the diffraction pattern by giving the maximum and minimum between which it lies, or the two minima between which it lies.

The full width at half-maximum (FWHM) of a central diffraction maximum is defined as the angle between the two points in the pattern where the intensity is one-half that at the center of the pattern. (See $\mathit{F}\mathit{i}\mathit{g}\mathbf{.}\mathbf{}\mathbf{36}\mathbf{-}\mathbf{8}\mathit{b}\mathbf{.}$) (a) Show that the intensity drops to one-half the maximum value when ${\mathbf{sin}}^{\mathbf{2}}\mathbf{Î\pm }\mathbf{=}\frac{{\mathbf{Î\pm }}^{\mathbf{2}}}{\mathbf{2}}$. (b) Verify that $\mathbf{Î\pm }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{39}\mathbf{}\mathbf{rad}$. (about $\mathbf{80}\mathbf{°}$) is a solution to the transcendental equation of (a). (c) Show that the FWHM is $\mathbf{Δθ}\mathbf{=}\mathbf{2}{\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{0}\mathbf{.}\mathbf{443}\mathbf{λ}\mathbf{/}\mathbf{a}\right)$where a is the slit width. Calculate the FWHM of the central maximum for slit width (d) $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{λ}$ ,(e) $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{λ}$,and (f) $\mathbf{10}\mathbf{.}\mathbf{00}\mathbf{λ}$.

(a) Show that the values of a at which intensity maxima for single-slit diffraction occur can be found exactly by differentiating Eq. 36-5 with respect to a and equating the result to zero, obtaining the condition $\mathit{t}\mathit{a}\mathit{n}\mathit{Î\pm }\mathbf{=}\mathbf{Î\pm }$. To find values of a satisfying this relation, plot the curve $\mathbf{y}\mathbf{=}\mathbf{³Ù²¹²ÔÎ\pm }$ and the straight line $\mathbf{y}\mathbf{=}\mathbf{Î\pm }$ and then find their intersections, or use $\mathbf{Î\pm }$calculator to find an appropriate value of a by trial and error. Next, from $\mathbf{Î\pm }\mathbf{=}\mathbf{(}\mathbf{m}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{)}\mathbf{Ï€}$, determine the values of $\mathit{m}$ associated with the maxima in the singleslit pattern. (These m values are not integers because secondary maxima do not lie exactly halfway between minima.) What are the (b) smallest and (c) associated , (d) the second smallest $\mathbf{Î\pm }$(e) and associated , (f) and the third smallest (g) and associated ?

The wall of a large room is covered with acoustic tile in which small holes are drilled $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{mm}$from centre to centre. How far can a person be from such a tile and still distinguish the individual holes, assuming ideal conditions, the pupil diameter of the observer’s eye to be $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{mm}$, and the wavelength of the room light to be $\mathbf{550}\mathbf{nm}$?

How far from grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{mm}$, the grains are spherical with radius$\mathbf{50}\mathbf{}\mathit{μ}\mathit{m}$ , and the light from the grains has wavelength $\mathbf{650}\mathbf{}\mathit{n}\mathit{m}$? (a) If the grains were blue and the light from them had wavelength $\mathbf{400}\mathbf{}\mathit{n}\mathit{m}$, would the answer to (b) be larger or smaller?

The distance between the first and fifth minima of a single slit diffraction pattern is $\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{}\mathbf{mm}$with the screen $\mathbf{40}\mathbf{}\mathbf{cm}$away from the slit, when light of wavelength role="math" localid="1663070418419" $\mathbf{550}\mathbf{}\mathbf{nm}$is used. (a) Find the slit width. (b) Calculate the angle role="math" localid="1663070538179" $\mathit{θ}$ of the first diffraction minimum.

You are conducting a single slit diffraction experiment with a light of wavelength $\mathit{λ}$. What appears , on a distant viewing screen, at a point at which the top and bottom rays through the slit have a path length difference equal to (a) $\mathbf{5}\mathit{λ}$and (b) $\mathbf{4}\mathbf{.}\mathbf{5}\mathit{λ}$?

The radar system of a navy cruiser transmits at a wavelength of 1.6 cm, from a circular antenna with a diameter of 2.3 m. At a range of 6.2 km, what is the smallest distance that two speedboats can be from each other and still be resolved as two separate objects by the radar system?