Problems & Exercises

Find the range of visible wavelengths of light in crown glass.

Analysis of an interference effect in a clear solid shows that the wavelength of light in the solid is 329 nm. Knowing this light comes from a He-Ne laser and has a wavelength of 633 nm in air, is the substance zircon or diamond?

Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.

Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of $\text{45}\text{.}\mathrm{0º}$.

What is the wavelength of light falling on double slits separated by $2\text{.}\text{00}\text{μm}$ if the third-order maximum is at an angle of $\text{60}\text{.}\mathrm{0º}$?

What is the highest-order maximum for 400-nm light falling on double slits separated by $\text{25}\text{.}0\text{μm}$?

What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?

(a) If the first-order maximum for pure-wavelength light falling on a double slit is at an angle of $\text{10}\text{.}\mathrm{0º}$, at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?

Using the result of the problem above, calculate the distance between fringes for 633-nm light falling on double slits separated by 0.0800 mm, located 3.00 m from a screen as in Figure 27.56.

Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a diffraction grating having 1500 lines per centimeter.

What is the distance between lines on a diffraction grating that produces a second-order maximum for 760-nm red light at an angle of $\text{60}\text{.}\mathrm{0º}$?

An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles of $\text{24}\text{.}\mathrm{2º}$, $\text{25}\text{.}\mathrm{7º}$, $\text{29}\text{.}\mathrm{1º}$, and $\text{41}\text{.}\mathrm{0º}$ when projected on a diffraction grating having 10,000 lines per centimeter? Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics

What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light?

What is the spacing between structures in a feather that acts as a reflection grating, given that they produce a first-order maximum for 525-nm light at a $\text{30}\text{.}\mathrm{0º}$ angle?

An opal such as that shown in Figure 27.17 acts like a reflection grating with rows separated by about $8\text{μm}$. If the opal is illuminated normally, (a) at what angle will red light be seen and (b) at what angle will blue light be seen?

Show that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than $\text{30}\text{.}\mathrm{0º}$.

(a) Find the maximum number of lines per centimeter a diffraction grating can have and produce a maximum for the smallest wavelength of visible light. (b) Would such a grating be useful for ultraviolet spectra? (c) For infrared spectra?

A He–Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD?

Unreasonable Results

Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Construct Your Own Problem

Consider a spectrometer based on a diffraction grating. Construct a problem in which you calculate the distance between two wavelengths of electromagnetic radiation in your spectrometer. Among the things to be considered are the wavelengths you wish to be able to distinguish, the number of lines per meter on the diffraction grating, and the distance from the grating to the screen or detector. Discuss the practicality of the device in terms of being able to discern between wavelengths of interest.

(a) Calculate the angle at which a $2\text{.}\text{00}\text{-μm}$-wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

(a) What is the width of a single slit that produces its first minimum at $\text{60}\text{.}\mathrm{0º}$ for 600-nm light? (b) Find the wavelength of light that has its first minimum at $\text{62}\text{.}\mathrm{0º}$.

Calculate the wavelength of light that produces its first minimum at an angle of $\text{36}\text{.}\mathrm{9º}$ when falling on a single slit of width $1\text{.}\text{00}\text{μm}$.

(a) Find the angle of the third diffraction minimum for 633-nm light falling on a slit of width $\text{20}\text{.}0\text{μm}$. (b) What slit width would place this minimum at $\text{85}\text{.}\mathrm{0º}$? Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics

(a) What is the minimum width of a single slit (in multiples of $\lambda$) that will produce a first minimum for a wavelength $\lambda$? (b) What is its minimum width if it produces 50 minima? (c) 1000 minima?

A double slit produces a diffraction pattern that is a combination of single and double slit interference. Find the ratio of the width of the slits to the separation between them, if the first minimum of the single slit pattern falls on the fifth maximum of the double slit pattern. (This will greatly reduce the intensity of the fifth maximum.)

Integrated Concepts

An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a 600-Hz sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is 0.800 m wide and the speed of sound is 340 m/s?

Assuming the angular resolution found for the Hubble Telescope in Example 27.5, what is the smallest detail that could be observed on the Moon?

(a) What is the minimum angular spread of a 633-nm wavelength He-Ne laser beam that is originally 1.00 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.) Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics.

The limit to the eye’s acuity is actually related to diffraction by the pupil.

(a) What is the angle between two just-resolvable points of light for a 3.00-mm-diameter pupil, assuming an average wavelength of 550 nm?

(b) Take your result to be the practical limit for the eye. What is the greatest possible distance a car can be from you if you can resolve its two headlights, given they are 1.30 m apart?

(c) What is the distance between two just-resolvable points held at an arm’s length (0.800 m) from your eye?

(d) How does your answer to (c) compare to details you normally observe in everyday circumstances?

You are told not to shoot until you see the whites of their eyes. If the eyes are separated by 6.5 cm and the diameter of your pupil is 5.0 mm, at what distance can you resolve the two eyes using light of wavelength 555 nm?

The headlights of a car are 1.3 m apart. What is the maximum distance at which the eye can resolve these two headlights? Take the pupil diameter to be 0.40 cm.

When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh’s criterion. Take the pupil of the eye to be 3.0 mm and the distance from the paper to the eye of 35 cm; find the minimum separation of two dots such that they cannot be resolved. How many dots per inch (dpi) does this correspond to?

Unreasonable Results

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 m detail on a Jovian Moon at a distance of $7\text{.}\text{50}\times {\text{10}}^8\text{km}$ from Earth? The wavelength of light averages 600 nm.

(b) What is unreasonable about this result?

(c) Which assumptions are unreasonable or inconsistent?

Construct Your Own Problem

Consider diffraction limits for an electromagnetic wave interacting with a circular object. Construct a problem in which you calculate the limit of angular resolution with a device, using this circular object (such as a lens, mirror, or antenna) to make observations. Also calculate the limit to spatial resolution (such as the size of features observable on the Moon) for observations at a specific distance from the device. Among the things to be considered are the wavelength of electromagnetic radiation used, the size of the circular object, and the distance to the system or phenomenon being observed.

An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?

Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be 680 nm, and assume the same index of refraction as water.

What are the three smallest non-zero thicknesses of soapy water ($n=1\text{.}\text{33}$) on Plexiglas if it appears green (constructively reflecting 520-nm light) when illuminated perpendicularly by white light? Explicitly show how you follow the steps in Problem Solving Strategies for Wave Optics.

(a) As a soap bubble thins it becomes dark, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the bubble can be and appear dark at all visible wavelengths? Assume the same index of refraction as water. (b) Discuss the fragility of the film considering the thickness found.

Figure 27.34 shows two glass slides illuminated by pure-wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a 0.100-mm-diameter hair at the other end, forming a wedge of air. (a) How far apart are the dark bands, if the slides are 7.50 cm long and 589-nm light is used? (b) Is there any difference if the slides are made from crown or flint glass? Explain.

Repeat Exercise 27.70, but take the light to be incident at a $\text{45º}$ angle.

Unreasonable Results

To save money on making military aircraft invisible to radar, an inventor decides to coat them with a non-reflective material having an index of refraction of 1.20, which is between that of air and the surface of the plane. This, he reasons, should be much cheaper than designing Stealth bombers. (a) What thickness should the coating be to inhibit the reflection of 4.00-cm wavelength radar? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

The angle between the axes of two polarizing filters is $\text{45}\text{.}\mathrm{0º}$. By how much does the second filter reduce the intensity of the light coming through the first?

What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity $1\text{.}\text{00}{\text{kW/m}}^2$ to reduce the intensity to $10\text{.}0{\text{W/m}}^2$?

Show that if you have three polarizing filters, with the second at an angle of $\text{45º}$ to the first and the third at an angle of $\text{90}\text{.}\mathrm{0º}$ to the first, the intensity of light passed by the first will be reduced to $\text{25}\text{.}\mathrm{0\%}$ of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)

At what angle will light reflected from diamond be completely polarized?

A scuba diver sees light reflected from the water’s surface. At what angle will this light be completely polarized?

Light reflected at $\text{55}\text{.}\mathrm{6º}$ from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made?

If ${\theta }_{\text{b}}$ is Brewster’s angle for light reflected from the top of an interface between two substances, and ${\theta \prime }_{\text{b}}$ is Brewster’s angle for light reflected from below, prove that ${\theta }_{\text{b}}+{\theta \prime }_{\text{b}}=\text{90}\text{.}\mathrm{0º.}$

Integrated Concepts

Suppose you put on two pairs of Polaroid sunglasses with their axes at an angle of $\text{15}\text{.}\mathrm{0º}$. How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics.