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Chapter 17: Problem 43

Infrared light of wavelength \(2.5 \mu \mathrm{m}\) illuminates a \(0.20-\mathrm{mm}-\) diameter hole. What is the angle of the first dark fringe in radians? In degrees?

Short Answer

Expert verified
The angle of the first dark fringe in radians is 0.0125 and in degrees is approximately 0.7162.

Step by step solution

01

Understand the problem and identify the required formula

In such type of diffraction problem, we commonly use the formula \(\theta = \lambda/d\), where \(\lambda\) is the wavelength of the light, \(d\) is the diameter of the hole, and \(\theta\) is the angle of diffraction. We need to find the value of \(\theta\) in radians, and subsequently convert it to degrees.
02

Convert the diameter

The diameter is given in millimeters but we need it in the same unit as the wavelength for the formula to work properly. So, we convert it to micrometers(m). \(1 \mathrm{mm} = 1000 \mu \mathrm{m}\), hence \(0.20 \mathrm{mm} = 0.20 * 1000 \mu \mathrm{m} = 200 \mu \mathrm{m}\)
03

Solve for the angle in diffraction formula

Substitute \( \lambda = 2.5 \mu \mathrm{m} \), and \(d = 200 \mu \mathrm{m}\) into the formula \(\theta = \lambda/d\). So, \(\theta = 2.5/200 = 0.0125\) radians
04

Conversion of radians to degrees

Now, convert the radian measure to degree measure by using the conversion factor \(1 \mathrm{rad} = 57.296 \mathrm{degrees}\). So, \( \theta = 0.0125 * 57.296 = 0.7162 \mathrm{degrees}\)

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Most popular questions from this chapter

Quality control systems have been developed to remotely measure the diameter of wires using diffraction. A wire with a stated diameter of \(170 \mu \mathrm{m}\) blocks the beam of a \(633 \mathrm{nm}\) laser, producing a diffraction pattern on a screen \(50.0 \mathrm{cm}\) distant. The width of the central maximum is measured to be \(3.77 \mathrm{mm}\). The wire should have a diameter within \(1 \%\) of the stated value. Does this wire pass the test?

A helium-neon laser beam has a wavelength in air of \(633 \mathrm{nm}\). It takes 1.38 ns for the light to travel through \(30.0 \mathrm{cm}\) of an unknown liquid. What is the wavelength of the laser beam in the liquid?

Solar cells are given antireflection coatings to maximize their efficiency. Consider a silicon solar cell \((n=3.50)\) coated with a layer of silicon dioxide \((n=1.45) .\) What is the minimum coating thickness that will minimize the reflection at the wavelength of \(700 \mathrm{nm},\) where solar cells are most efficient?

The blue-ringed octopus reveals the bright blue rings that give it its name as a warning display. The rings have a stack of reflectin (a protein used for structural color in many cephalopods) plates with index of refraction \(n=1.59 \quad\) separated \(\quad\) by cells with index \(n=1.37 .\) The plates have thickness \(62 \mathrm{nm} .\) What is the longest wavelength, in air, of light that will give constructive interference from opposite sides of the reflecting plates?

\(A\) diffraction grating with 600 lines/mm is illuminated with light of wavelength 500 nm. A very wide viewing screen is \(2.0 \mathrm{m}\) behind the grating. a. What is the distance between the two \(m=1\) fringes? b. How many bright fringes can be seen on the screen?
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