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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

A Young's interference experiment is performed with monochromatic green light \((\lambda=540 \mathrm{nm}) .\) The separation between the slits is \(0.100 \mathrm{~mm},\) and the interference pattern on a screen shows the first side maximum \(5.40 \mathrm{~mm}\) from the center of the pattern. How far away from the slits is the screen?

White light is shone on a very thin layer of mica \((n=1.57),\) and above the mica layer, interference maxima for two wavelengths (and no other in between) are seen: one blue wavelength of \(480 \mathrm{nm},\) and one yellow wavelength of \(560 \mathrm{nm} .\) What is the thickness of the mica layer?

Many times, radio antennas occur in pairs. The effect is that they will then produce constructive interference in one direction while producing destructive interference in another direction - a directional antenna-so that their emissions don't overlap with nearby stations. How far apart at a minimum should a local radio station, operating at \(88.1 \mathrm{MHz},\) place its pair of antennae operating in phase such that no emission occurs along a line \(45.0^{\circ}\) from the line joining the antennae?

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