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Chapter 4: Problem 115

(a) What visible wavelength has its fourth-order maximum at an angle of \(25.0^{\circ}\) when projected on a 25,000-line per centimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Short Answer

Expert verified
a) The wavelength of the fourth-order maximum at an angle of \(25.0^{\circ}\) is 680 nm. b) The result is reasonable since it falls within the range of visible light. c) Our assumptions used in solving the problem were reasonable and consistent.

Step by step solution

01

Write down the diffraction grating formula

We start with the diffraction grating formula, which relates the wavelength of the light (λ), the order of the maximum (m), the angle of the maximum (θ), and the distance between the lines on the grating (d). \[d \sin{\theta} = m\lambda\]
02

Determine the value of d

The problem states that there are 25,000 lines per centimeter that has 1 cm = 0.01 m, so we can find the value of d as follows: \[d = \frac{0.01}{25,000} = 4 \times 10^{-7} m\]
03

Solve for the wavelength (λ)

Use the information in the problem to solve for λ using the diffraction grating formula. We are given the angle θ = 25.0° and the order of the maximum m = 4. \[4 \times 10^{-7} \sin{(25.0^{\circ})} = 4\lambda\] Now solve for λ. \[\lambda = \frac{4\times10^{-7}\sin{(25^{\circ})}}{4} = 6.8 \times 10^{-7} m\]
04

Convert the wavelength to nanometers

Convert the wavelength to nanometers (nm) for easier comparison to the visible light range. \[\lambda = 6.8 \times 10^{-7} m = 680 nm\]
05

Evaluate whether the result is reasonable

The visible light range is approximately between 400 nm and 700 nm. Our calculated wavelength of 680 nm falls within this range and is therefore reasonable.
06

Discuss the assumptions made in the problem

The main assumptions in this problem are: 1. The wavelength lies within the visible spectrum range. 2. The interference is constructive and obeys the integer value of m. Both assumptions were used to solve the problem, and the result we found falls within the visible light range. Therefore, our assumptions are reasonable and consistent. Here is a summary of our solutions: a) The wavelength of the fourth-order maximum at an angle of \(25.0^{\circ}\) is 680 nm. b) The result is reasonable since it falls within the range of visible light. c) Our assumptions used in solving the problem were reasonable and consistent.

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Most popular questions from this chapter

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

Blue light of wavelength 450 nm falls on a slit of width \(0.25 \mathrm{mm}\). A converging lens of focal length \(20 \mathrm{cm}\) is placed behind the slit and focuses the diffraction pattern on a screen. (a) How far is the screen from the lens? (b) What is the distance between the first and the third minima of the diffraction pattern?

An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form firstorder maxima at angles \(24.2^{\circ}, 25.7^{\circ}, 29.1^{\circ},\) and \(41.0^{\circ}\) when projected on a diffraction grating having 10,000 lines per centimeter?

An amateur astronomer wants to build a telescope with a diffraction limit that will allow him to see if there are people on the moons of Jupiter. (a) What diameter mirror is needed to be able to see \(1.00-\mathrm{m}\) detail on a Jovian moon at a distance of \(7.50 \times 10^{8} \mathrm{km}\) from Earth? The wavelength of light averages \(600 \mathrm{nm}\). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

X-rays of wavelength 0.103 nm reflects off a crystal and a second-order maximum is recorded at a Bragg angle of \(25.5^{\circ}\). What is the spacing between the scattering planes in this crystal?
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