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Chapter 34: Problem 55

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

Short Answer

Expert verified
Answer: The grating must have at least 66666 lines per centimeter.

Step by step solution

01

Write down the grating equation

The grating equation is given by: \(d \cdot \sin{\theta} = m \cdot \lambda\) where \(d\) is the distance between the grating lines (in meters), \(\theta\) is the angle of diffraction, \(m\) is the order of the spectrum, and \(\lambda\) is the wavelength of light (in meters).
02

Set the longest visible wavelength and the condition to avoid the 2nd-order spectrum

The longest visible wavelength \(\lambda\) is 750 nm, or \(750 \times 10^{-9}\) m. To avoid the second-order spectrum, the angle of diffraction \(\theta\) should be equal to or greater than 90° when \(m = 2\): \(d \cdot \sin{90^\circ} \geq 2 \cdot (750 \times 10^{-9}\mathrm{m})\)
03

Find the distance between the grating lines and the number of lines per centimeter

We can now solve for \(d\): \(d \geq 2 \cdot (750 \times 10^{-9}\mathrm{m})\) The number of lines per centimeter, \(N\), is the reciprocal of the distance between lines, \(d\), converted to centimeters: \(N = \frac{1}{d (in\:cm)}\) Substitute the value we found for \(d\) in the previous step: \(N = \frac{1}{2 \cdot (750 \times 10^{-7}\mathrm{cm})}\)
04

Calculate the number of lines per centimeter

Now we just need to calculate the value of \(N\): \(N = \frac{1}{1.5 \times 10^{-5}\mathrm{cm}}\) \(N = 66666.67\) Since there can't be a fraction of a line, we round down to the nearest whole number, which gives us 66666 lines per centimeter. So, the grating must have at least 66666 lines per centimeter to avoid a second-order spectrum for any visible wavelength.

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Most popular questions from this chapter

Plane light waves are incident on a single slit of width \(2.00 \mathrm{~cm} .\) The second dark fringe is observed at \(43.0^{\circ}\) from the central axis. What is the wavelength of the light?

A diffraction grating with exactly 1000 lines per centimeter is illuminated by a He-Ne laser of wavelength \(633 \mathrm{nm}\). a) What is the highest order of diffraction that could be observed with this grating? b) What would be the highest order if there were exactly 10,000 lines per centimeter?

What is the wavelength of the X-rays if the first-order Bragg diffraction is observed at \(23.0^{\circ}\) related to the crystal surface, with inter atomic distance of \(0.256 \mathrm{nm} ?\)

A \(5.000-\mathrm{cm}\) -wide diffraction grating with 200 grooves is used to resolve two closely spaced lines (a doublet) in a spectrum. The doublet consists of two wavelengths, \(\lambda_{\mathrm{a}}=\) \(629.8 \mathrm{nm}\) and \(\lambda_{\mathrm{b}}=630.2 \mathrm{nm} .\) The light illuminates the entire grating at normal incidence. Calculate to four significant digits the angles \(\theta_{1 \mathrm{a}}\) and \(\theta_{1 \mathrm{~b}}\) with respect to the normal at which the first-order diffracted beams for the two wavelengths, \(\lambda_{\mathrm{a}}\) and \(\lambda_{\mathrm{b}}\), respectively, will be reflected from the grating. Note that this is not \(0^{\circ}\) What order of diffraction is required to resolve these two lines using this grating?

An airplane is made invisible to radar by coating it with a 5.00 -mm-thick layer of an antireflective polymer with the index of refraction \(n=1.50 .\) What is the wavelength of radar waves for which the plane is made invisible?
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