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Chapter 4: Problem 41

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520 -nmwavelength green light?

Short Answer

Expert verified
The angle at which the first-order maximum occurs for 520-nm wavelength green light in this diffraction grating is approximately \(6.0°\).

Step by step solution

01

Convert given values to the appropriate units

First, let's convert the given values to the appropriate units for easier calculations. The number of lines per centimeter (2000 lines/cm) needs to be converted to the number of lines per meter (lines/m), and the wavelength of the light (520 nm) should be converted to meters (m). 1. Conversion of lines per centimeter to lines per meter: \(2000 \, lines/cm * 100 cm/m = 200,000 \, lines/m\) 2. Conversion of the wavelength: \(520 \, nm = 520 * 10^{-9} m\) Now we have: Number of lines per meter (lines/m) = 200,000 lines/m Wavelength of the light (m) = \(520 * 10^{-9} m\)
02

Calculate the grating spacing (d)

Next, we need to find the grating spacing (d). The grating spacing is the distance between each line in the grating, and it can be found using the following formula: \(d = \frac{1}{N}\) Where: d = grating spacing (m) N = number of lines per meter (lines/m) Now, let's plug in the values: \(d = \frac{1}{200,000 lines/m} = 5*10^{-6} m\)
03

Apply the formula for constructive interference

To find the angle of the first-order maximum (\(\theta\)), we need to use the formula for constructive interference in a diffraction grating: \(d * sin(\theta) = m * \lambda\) Where: d = grating spacing (m) \(\theta\) = angle of the maximum order m = order (1 for first-order maximum) \(\lambda\) = wavelength of the light (m) Now, plug in the values and solve for theta: \(5*10^{-6} m * sin(\theta) = 1 * (520 * 10^{-9} m)\)
04

Solve for the angle (\(\theta\))

Now let's solve for \(\theta\): \(sin(\theta) = \frac{1 * (520 * 10^{-9} m)}{5*10^{-6} m}\) \(sin(\theta) = 0.104\) Now, find the angle by taking the inverse sine (arcsin) of the result: \(\theta = arcsin(0.104)\) \(\theta \approx 6.0°\) Therefore, the angle at which the first-order maximum occurs for 520-nm wavelength green light in this diffraction grating is approximately 6.0°.

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Most popular questions from this chapter

An X-ray scattering experiment is performed on a crystal whose atoms form planes separated by \(0.440 \mathrm{nm}\). Using an X-ray source of wavelength \(0.548 \mathrm{nm}\), what is the angle (with respect to the planes in question) at which the experimenter needs to illuminate the crystal in order to observe a first-order maximum?

(a) How wide is a single slit that produces its first minimum for 633 -nm light at an angle of \(28.0^{\circ}\) ? (b) At what angle will the second minimum be?

Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally \(5.00 \mathrm{cm}\) in diameter with an average wavelength of \(600 \mathrm{nm}\)

If the separation between the first and the second minima of a single-slit diffraction pattem is \(6.0 \mathrm{mm}\), what is the distance between the screen and the slit? The light wavelength is \(500 \mathrm{nm}\) and the slit width is \(0.16 \mathrm{mm}\).

How far apart must two objects be on the moon to be resolvable by the 8.1-m-diameter Gemini North telescope at Mauna Kea, Hawaii, if only the diffraction effects of the telescope aperture limit the resolution? Assume \(550 \mathrm{nm}\) for the wavelength of light and \(400,000 \mathrm{km}\) for the distance to the moon.
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