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Chapter 4: Problem 113

Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600 -nm light?

Short Answer

Expert verified
The angle of the first-order maximum for 600-nm light reflecting from a grating with 8000 lines per centimeter is approximately 28.50°.

Step by step solution

01

Convert lines per centimeter to line spacing

The given information is the number of lines on the grating per centimeter (8000 lines/cm). We must convert this into the spacing between lines, which is the reciprocal of the number of lines per unit length. Therefore, we have: Line spacing (d) = \(\frac{1}{8000 \text{ lines/cm}}\) Don't forget to convert the line spacing to meters, since the wavelength is given in nanometers. 1 cm = 0.01 m d = \(\frac{1}{8000 \times 0.01 \text{ m}}\) = \(1.25 \times 10^{-6}\) m
02

Convert the wavelength of light from nanometers to meters

Next, we should convert the wavelength of the light from nanometers to meters: 1 nm = \(1 \times 10^{-9}\) m So, 600 nm = \(600 \times 10^{-9}\) m = \(6 \times 10^{-7}\) m
03

Use the grating equation to find the angle of the first-order maximum

The grating equation is given by: \(d \sin{\theta} = m \lambda\) where θ is the angle of the first-order maximum, m is the order of the maximum (m=1 for the first-order maximum), λ is the wavelength of the light and d is the line spacing. Solving for θ in our given problem: \(1.25 \times 10^{-6} \sin{\theta} = 1(6 \times 10^{-7})\) \(\sin{\theta} = \frac{6 \times 10^{-7}}{1.25 \times 10^{-6}}\) Now, we need to find the inverse sine (sin^(-1)) to get the angle in degrees: \(\theta = \sin^{-1} \left( \frac{6 \times 10^{-7}}{1.25 \times 10^{-6}} \right)\) Using a calculator: \(\theta \approx 28.50°\)
04

State the final answer

The angle of the first-order maximum for 600-nm light reflecting from a grating with 8000 lines per centimeter is approximately 28.50°.

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Most popular questions from this chapter

(a) What is the width of a single slit that produces its first minimum at \(60.0^{\circ}\) for 600 -nm light? (b) Find the wavelength of light that has its first minimum at \(62.0^{\circ}\)

Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally \(5.00 \mathrm{cm}\) in diameter with an average wavelength of \(600 \mathrm{nm}\)

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?

(a) What is the minimum angular spread of a 633 -nm wavelength He-Ne laser beam that is originally \(1.00 \mathrm{mm}\) in diameter? (b) If this laser is aimed at a mountain cliff 15.0 km away, how big will the illuminated spot be? (c) How big a spot would be illuminated on the moon, neglecting atmospheric effects? (This might be done to hit a corner reflector to measure the round- trip time and, hence, distance.)

A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the moon. (a) If this is done with the Mount Wilson telescope, producing a 2.54 -m-diameter beam of 633 -nm light, what is the minimum angular spread of the beam? (b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the moon, assuming a lunar distance of \(3.84 \times 10^{8} \mathrm{m} ?\)
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