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Chapter 34: Problem 51

Light from an argon laser strikes a diffraction grating that has 7020 grooves per centimeter. The central and firstorder principal maxima are separated by \(0.332 \mathrm{~m}\) on a wall \(1.00 \mathrm{~m}\) from the grating. Determine the wavelength of the laser light.

Short Answer

Expert verified
Answer: The wavelength of the laser light is approximately \(454 \mathrm{~nm}\).

Step by step solution

01

Find the Grating Separation (d)

The grating separation (d) can be determined by the formula: \(d = \frac{1}{N}\) Where N is the number of grooves per centimeter. In this case, N = 7020 grooves per centimeter. \(d = \frac{1}{7020}\) \(d = 1.42 \times 10^{-4} \mathrm{~cm}\)
02

Convert Grating Separation to Meters

To work with meters in the rest of the problem, convert the grating separation (d) to meters: \(d = 1.42\times10^{-4}\ \text{cm}\times\frac{1\text{ m}}{100\text{ cm}} = 1.42\times10^{-6}\ \text{m}\)
03

Calculate the Angle Between Central and First-Order Maxima (θ)

We are given the distance between the central and first-order maxima (y = 0.332 m) and the distance from the grating to the wall (L = 1.00 m). We can use the right triangle relationship to find the angle θ: \(\tan \theta = \frac{y}{L}\) \(\theta = \arctan{\left(\frac{0.332 \ \text{m}}{1.00 \ \text{m}}\right)}\) \(\theta \approx 18.56^{\circ}\)
04

Calculate the Wavelength Using the Grating Equation

Now, we can use the grating equation to find the wavelength (λ): \(m \lambda = d \sin \theta\) Where m is the order of the principal maximum (m = 1 for the first-order maximum). \(\lambda = \frac{d \sin \theta}{m}\) \(\lambda = \frac{1.42 \times 10^{-6} \ \text{m} \cdot \sin(18.56^{\circ})}{1}\) \(\lambda \approx 4.54 \times 10^{-7} \text{ m}\)
05

Convert Wavelength to Nanometers

To express the wavelength in nanometers, convert the result to nanometers: \(\lambda = 4.54\times10^{-7} \text{ m}\times\frac{10^9\text{ nm}}{1\text{ m}} \approx 454\ \text{nm}\) The wavelength of the laser light is approximately \(454 \mathrm{~nm}\).

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Most popular questions from this chapter

A red laser pointer with a wavelength of \(635 \mathrm{nm}\) is shined on a double slit producing a diffraction pattern on a screen that is \(1.60 \mathrm{~m}\) behind the double slit. The central maximum of the diffraction pattern has a width of \(4.20 \mathrm{~cm}\) and the fourth bright spot is missing on both sides. What is the size of the individual slits, and what is the separation between them?

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?

How many lines per centimeter must a grating have if there is to be no second- order spectrum for any visible wavelength \((400-750 \mathrm{nm})\) ?

In a double-slit arrangement the slits are \(1.00 \cdot 10^{-5} \mathrm{~m}\) apart. If light with wavelength \(500 .\) nm passes through the slits, what will be the distance between the \(m=1\) and \(m=3\) maxima on a screen \(1.00 \mathrm{~m}\) away?

White light shines on a sheet of mica that has a uniform thickness of \(1.30 \mu \mathrm{m} .\) When the reflected light is viewed using a spectrometer, it is noted that light with wavelengths of \(433.3 \mathrm{nm}, 487.5 \mathrm{nm}, 557.1 \mathrm{nm}, 650.0 \mathrm{nm}\), and \(780.0 \mathrm{nm}\) is not present in the reflected light. What is the index of refraction of the mica?
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