# 4.1 Single-slit diffraction  (Page 3/5)

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Check Your Understanding Suppose the slit width in [link] is increased to $1.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}.$ What are the new angular positions for the first, second, and third minima? Would a fourth minimum exist?

$17.8\text{°}$ , $37.7\text{°}$ , $66.4\text{°}$ ; no

## Summary

• Diffraction can send a wave around the edges of an opening or other obstacle.
• A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.

## Conceptual questions

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

The diffraction pattern becomes wider.

Compare interference and diffraction.

If you and a friend are on opposite sides of a hill, you can communicate with walkie-talkies but not with flashlights. Explain.

Walkie-talkies use radio waves whose wavelengths are comparable to the size of the hill and are thus able to diffract around the hill. Visible wavelengths of the flashlight travel as rays at this size scale.

What happens to the diffraction pattern of a single slit when the entire optical apparatus is immersed in water?

In our study of diffraction by a single slit, we assume that the length of the slit is much larger than the width. What happens to the diffraction pattern if these two dimensions were comparable?

The diffraction pattern becomes two-dimensional, with main fringes, which are now spots, running in perpendicular directions and fainter spots in intermediate directions.

A rectangular slit is twice as wide as it is high. Is the central diffraction peak wider in the vertical direction or in the horizontal direction?

## Problems

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width $1.00\mu \text{m}$ ? (b) Will there be a second minimum?

a. $33.4\text{°}$ ; b. no

(a) Calculate the angle at which a $2.00\text{-}\mu \text{m}$ -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 $28.0\text{°}$ ? (b) At what angle will the second minimum be?

a. $1.35\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}$ ; b. $69.9\text{°}$

(a) What is the width of a single slit that produces its first minimum at $60.0\text{°}$ for 600-nm light? (b) Find the wavelength of light that has its first minimum at $62.0\text{°}$ .

Find the wavelength of light that has its third minimum at an angle of $48.6\text{°}$ when it falls on a single slit of width $3.00\mu \text{m}$ .

750 nm

(a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width $7.50\mu \text{m}$ . At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

Consider a single-slit diffraction pattern for $\lambda =589\phantom{\rule{0.2em}{0ex}}\text{nm}$ , projected on a screen that is 1.00 m from a slit of width 0.25 mm. How far from the center of the pattern are the centers of the first and second dark fringes?

2.4 mm, 4.7 mm

(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 $2.00\mu \text{m}$ . (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) 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. $1.00\lambda ;$ b. $50.0\lambda ;$ c. $1000\lambda$

(a) If a single slit produces a first minimum at $14.5\text{°},$ 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).

If the separation between the first and the second minima of a single-slit diffraction pattern is 6.0 mm, what is the distance between the screen and the slit? The light wavelength is 500 nm and the slit width is 0.16 mm.

1.92 m

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 angles to the incident direction are the boats inside the harbor most protected against wave action?

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?

$45.1\text{°}$

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