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Chapter 34: Problem 18

A red laser pointer is shined on a diffraction grating, producing a diffraction pattern on a screen behind the diffraction grating. If the red laser pointer is replaced with a green laser pointer, will the green bright spots on the screen be closer together or farther apart than the red bright spots were?

Short Answer

Expert verified
Answer: The green bright spots in a diffraction pattern are closer together than the red bright spots. This is due to the shorter wavelength of green light, which results in a more compact diffraction pattern as the bright spots are closer to the central maximum.

Step by step solution

01

Understand the concept of diffraction grating

A diffraction grating is a device that diffracts light into its constituent wavelengths. It usually consists of a surface with multiple parallel slits, which allow light waves to interfere with each other, creating a diffraction pattern on a screen behind the grating. The distance between the bright spots in the diffraction pattern depends on the wavelength of the light and the properties of the grating.
02

Determine the formula for calculating the angle of diffraction

To analyze the diffraction patterns created by the red and green laser pointers, we can use the formula for calculating the angle of diffraction (θ) in a diffraction grating: \(theta = sin^{–1}\left(\dfrac{m\lambda}{d}\right)\) where \(m\) is an integer representing the diffraction order, \(\lambda\) is the wavelength of the light, and \(d\) is the distance between adjacent slits on the grating. This formula is derived from the condition for constructive interference, which states that the bright spots in a diffraction pattern are formed when the path difference between adjacent slits is an integer multiple of the wavelength.
03

Compare red and green light wavelengths

Red light has a longer wavelength than green light. Typically, red light has a wavelength of about 650 nm, while green light has a wavelength of about 532 nm. To compare the diffraction patterns formed by the two colors of light, we can plug their respective wavelengths into the formula for the angle of diffraction: θ_red = sin^(-1)((m * 650 nm) / d) θ_green = sin^(-1)((m * 532 nm) / d)
04

Compare the angles of diffraction for red and green light

From the previous step, we can see that the angle of diffraction for green light is smaller than the angle of diffraction for red light for the same diffraction order (m) and distance between slits (d). This means that the green bright spots in the diffraction pattern formed by the green laser pointer will be closer to the central maximum (which corresponds to m = 0) than the red bright spots formed by the red laser pointer.
05

Conclude whether green bright spots are closer together or farther apart

Since the angles of diffraction for green light are smaller than the angles of diffraction for red light, the green bright spots in the diffraction pattern will be closer together than the red bright spots. This is because green light has a shorter wavelength, which results in a more compact diffraction pattern as the bright spots are closer to the central maximum.

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Most popular questions from this chapter

Calculate and compare the angular resolutions of the Hubble Space Telescope (aperture diameter \(2.40 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\); illustrated in the text), the Keck Telescope (aperture diameter \(10.0 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\) ), and the Arecibo radio telescope (aperture diameter \(305 \mathrm{~m}\), wavelength \(0.210 \mathrm{~m}\) ). Assume that the resolution of each instrument is diffraction limited.

In a single-slit diffraction pattern, there is a bright central maximum surrounded by successively dimmer higher-order maxima. Farther out from the central maximum, eventually no more maxima are observed. Is this because the remaining maxima are too dim? Or is there an upper limit to the number of maxima that can be observed, no matter how good the observer's eyes, for a given slit and light source?

What minimum path difference is needed to cause a phase shift by \(\pi / 4\) in light of wavelength \(700 . \mathrm{nm} \)

Think of the pupil of your eye as a circular aperture \(5.00 \mathrm{~mm}\) in diameter. Assume you are viewing light of wavelength \(550 \mathrm{nm}\), to which your eyes are maximally sensitive. a) What is the minimum angular separation at which you can distinguish two stars? b) What is the maximum distance at which you can distinguish the two headlights of a car mounted \(1.50 \mathrm{~m}\) apart?

Suppose the thickness of a thin soap film \((n=1.32)\) surrounded by air is nonuniform and gradually tapers. Monochromatic light of wavelength \(550 \mathrm{nm}\) illuminates the film. At the thinnest end, a dark band is observed. How thick is the film at the next two dark bands closest to the first dark band?
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