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Chapter 17: Problem 40

Light from a helium-neon laser \((\lambda=633 \mathrm{nm})\) passes through a circular aperture and is observed on a screen \(4.0 \mathrm{m}\) behind the aperture. The width of the central maximum is \(2.5 \mathrm{cm} .\) What is the diameter (in \(\mathrm{mm}\) ) of the hole?

Short Answer

Expert verified
The diameter of the hole is approximately 2.01 mm.

Step by step solution

01

Identify the Relevant Formula

The relation between the distance from the hole to the screen (L), the width of the central maximum (w), the wavelength of the light (\( \lambda \)), and the diameter of the hole (D) is given by \( w = \frac{2 \lambda L}{D} \). We are asked to find D.
02

Rearrangement of the Formula

We can isolate D by multiplying both sides of the equation by D and then dividing by w. The rearranged formula is \( D = \frac{2 \lambda L}{w} \).
03

Insert Known Values

Now, we can substitute the known values into the equation. The distance L is 4.0 m, the width w is 2.5 cm, and the wavelength \(\lambda\) is 633 nm. Don't forget to convert the units to the same system - in this case, we should use meters: \( D = \frac{2 * 633 * 10^{-9} m * 4.0 m}{2.5 * 10^{-2} m} \).
04

Evaluate the Expression

After substituting the values, simplify the equation to find the diameter of the hole. This leads to a result of approximately 0.00201 m. Converting this result to millimeters gives us a diameter of 2.01 mm.

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Most popular questions from this chapter

A cross section of a scale from the wing of a blue morpho butterfly reveals the source of the butterfly's color. As Figure \(\mathrm{P} 17.69 \mathrm{b}\) shows, the scales are covered with structures that look like small Christmas trees. Light striking the wings reflects from different layers of these structures, and the differing path lengths cause the reflected light to interfere constructively or destructively, depending on the wavelength. For light at normal incidence, blue light experiences constructive interference while other colors undergo destructive interference and cancel. Acetone fills the spaces in the scales with a fluid of index of refraction \(n=1.38 ;\) this changes the conditions for constructive interference and results in a change in color. The coloring of the blue morpho butterfly is protective. As the butterfly flaps its wings, the angle at which light strikes the wings changes. This causes the butterfly's color to change and makes it difficult for a predator to follow. This color change is because A. A diffraction pattern appears only at certain angles. B. The index of refraction of the wing tissues changes as the wing flexes. C. The motion of the wings causes a Doppler shift in the reflected light. D. As the angle changes, the differences in paths among light reflected from different surfaces change, resulting in constructive interference for a different color.

A sheet of glass is coated with a 500 -nm-thick layer of oil \((n=1.42)\). a. For what visible wavelengths of light do the reflected waves interfere constructively? b. For what visible wavelengths of light do the reflected waves interfere destructively? c. What is the color of reflected light? What is the color of transmitted light?

The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are \(656 \mathrm{nm}\) (red) and \(486 \mathrm{nm}\) (blue). Light from a hydrogen lamp illuminates a diffraction grating with 500 lines/mm, and the light is observed on a screen \(1.50 \mathrm{m}\) behind the grating. What is the distance between the first-order red and blue fringes?

Glass catfish, tropical fish popular with hobbyists, have no pigment, and matching of the index of refraction of their tis- sues leaves them largely transparent-you can see through them. When a beam of white light illuminates these fish, diffraction of light from evenly spaced striations in muscle fibers produces a rainbow pattern. As the muscles contract, the striations get closer together, and this affects the diffraction pattern. This phenomenon has been used to study muscle contraction in living fish as they swim. Investigators set up a chamber with moving water where the fish swam steadily to stay stationary. The investigators then passed a beam of laser light of wavelength \(632 \mathrm{nm}\) through the fish's muscle tissue as it swam. Muscle action produced periodic changes in the distances between striations, which ranged from 1.87 to \(1.94 \mu \mathrm{m} .\) Investigators measured the change of position of a bright spot corresponding to \(m=1\) on a screen. If the screen was \(30 \mathrm{cm}\) behind the fish, what was the distance spanned by the diffraction spot as it moved back and forth? The screen was in the tank with the fish, so that the entire path of the laser was in water and tissue with an index of refraction close to that of water. The properties of the diffraction pattern were thus determined by the wavelength in water.

Mourning doves have a small patch of iridescent feathers. The color is produced by a \(330-n m-\) thick layer of keratin \((n=1.56)\) with air on both sides that is found around the edge of the feather barbules. For what wavelength or wavelengths would this structure produce constructive interference?
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