27.5 Single slit diffraction  (Page 3/4)

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Conceptual questions

As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change?

Problems&Exercises

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μm}$ ? (b) Will there be a second minimum?

(a) $\text{33}\text{.}4º$

(b) No

(a) Calculate the angle at which a $2\text{.}\text{00}\text{-μm}$ -wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

(a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of $\text{28}\text{.}0º$ ? (b) At what angle will the second minimum be?

(a) $1\text{.}\text{35}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}$

(b) $\text{69}\text{.}9º$

(a) What is the width of a single slit that produces its first minimum at $\text{60}\text{.}0º$ for 600-nm light? (b) Find the wavelength of light that has its first minimum at $\text{62}\text{.}0º$ .

Find the wavelength of light that has its third minimum at an angle of $\text{48}\text{.}6º$ when it falls on a single slit of width $3\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μm}$ .

750 nm

Calculate the wavelength of light that produces its first minimum at an angle of $\text{36}\text{.}9º$ when falling on a single slit of width $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μm}$ .

(a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width $7\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\text{μm}$ . At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

(a) $9\text{.}\text{04º}$

(b) 12

(a) Find the angle of the third diffraction minimum for 633-nm light falling on a slit of width $\text{20}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{μm}$ . (b) What slit width would place this minimum at $\text{85}\text{.}0º$ ? Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics

(a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width $2\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μm}$ . (b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit? (c) Discuss the ease or difficulty of measuring such a distance.

(a) $0\text{.}\text{0150º}$

(b) 0.262 mm

(c) This distance is not easily measured by human eye, but under a microscope or magnifying glass it is quite easily measurable.

(a) What is the minimum width of a single slit (in multiples of $\lambda$ ) that will produce a first minimum for a wavelength $\lambda$ ? (b) What is its minimum width if it produces 50 minima? (c) 1000 minima?

(a) If a single slit produces a first minimum at $\text{14}\text{.}5º$ , at what angle is the second-order minimum? (b) What is the angle of the third-order minimum? (c) Is there a fourth-order minimum? (d) Use your answers to illustrate how the angular width of the central maximum is about twice the angular width of the next maximum (which is the angle between the first and second minima).

(a) $\text{30}\text{.}1º$

(b) $\text{48}\text{.}7º$

(c) No

(d) ${2\theta }_{1}=\left(2\right)\left(\text{14.5º}\right)=\text{29º},\phantom{\rule{0.25em}{0ex}}{\theta }_{2}-{\theta }_{1}=\text{30}\text{.}\text{05º}-\text{14}\text{.}5º\text{=}\text{15}\text{.}\text{56º}$ . Thus, $\text{29º}\approx \left(2\right)\left(\text{15}\text{.}\text{56º}\right)=\text{31}\text{.}1º$ .

A double slit produces a diffraction pattern that is a combination of single and double slit interference. Find the ratio of the width of the slits to the separation between them, if the first minimum of the single slit pattern falls on the fifth maximum of the double slit pattern. (This will greatly reduce the intensity of the fifth maximum.)

Integrated Concepts

A water break at the entrance to a harbor consists of a rock barrier with a 50.0-m-wide opening. Ocean waves of 20.0-m wavelength approach the opening straight on. At what angle to the incident direction are the boats inside the harbor most protected against wave action?

$\text{23}\text{.}6º$ and $\text{53}\text{.}1º$

Integrated Concepts

An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a 600-Hz sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is 0.800 m wide and the speed of sound is 340 m/s?

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