### Additional Problems

For 600-nm wavelength light and a slit separation of 0.12 mm, what are the angular positions of the first and third maxima in the double slit interference pattern?

If the light source in the preceding problem is changed, the angular position of the third maximum is found to be $0.57\text{\xb0}$. What is the wavelength of light being used now?

Red light ($\lambda =710.\phantom{\rule{0.2em}{0ex}}\text{nm}$) illuminates double slits separated by a distance $d=0.150\phantom{\rule{0.2em}{0ex}}\text{mm}.$ The screen and the slits are 3.00 m apart. (a) Find the distance on the screen between the central maximum and the third maximum. (b) What is the distance between the second and the fourth maxima?

Two sources as in phase and emit waves with $\lambda =0.42\phantom{\rule{0.2em}{0ex}}\text{m}$. Determine whether constructive or destructive interference occurs at points whose distances from the two sources are (a) 0.84 and 0.42 m, (b) 0.21 and 0.42 m, (c) 1.26 and 0.42 m, (d) 1.87 and 1.45 m, (e) 0.63 and 0.84 m and (f) 1.47 and 1.26 m.

Two slits $4.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{m}$ apart are illuminated by light of wavelength 600 nm. What is the highest order fringe in the interference pattern?

Suppose that the highest order fringe that can be observed is the eighth in a double-slit experiment where 550-nm wavelength light is used. What is the minimum separation of the slits?

The interference pattern of a He-Ne laser light $(\lambda =632.9\phantom{\rule{0.2em}{0ex}}\text{nm})$ passing through two slits 0.031 mm apart is projected on a screen 10.0 m away. Determine the distance between the adjacent bright fringes.

Young’s double-slit experiment is performed immersed in water ($n=1.333$). The light source is a He-Ne laser, $\lambda =632.9\phantom{\rule{0.2em}{0ex}}\text{nm}$ in vacuum. (a) What is the wavelength of this light in water? (b) What is the angle for the third order maximum for two slits separated by 0.100 mm.

A double-slit experiment is to be set up so that the bright fringes appear 1.27 cm apart on a screen 2.13 m away from the two slits. The light source was wavelength 500 nm. What should be the separation between the two slits?

An effect analogous to two-slit interference can occur with sound waves, instead of light. In an open field, two speakers placed 1.30 m apart are powered by a single-function generator producing sine waves at 1200-Hz frequency. A student walks along a line 12.5 m away and parallel to the line between the speakers. She hears an alternating pattern of loud and quiet, due to constructive and destructive interference. What is (a) the wavelength of this sound and (b) the distance between the central maximum and the first maximum (loud) position along this line?

A hydrogen gas discharge lamp emits visible light at four wavelengths, $\lambda =$ 410, 434, 486, and 656 nm. (a) If light from this lamp falls on a *N* slits separated by 0.025 mm, how far from the central maximum are the third maxima when viewed on a screen 2.0 m from the slits? (b) By what distance are the second and third maxima separated for $l=486\phantom{\rule{0.2em}{0ex}}\text{nm}$?

Monochromatic light of frequency $5.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{14}\phantom{\rule{0.2em}{0ex}}\text{Hz}$ falls on 10 slits separated by 0.020 mm. What is the separation between the first and third maxima on a screen that is 2.0 m from the slits?

Eight slits equally separated by 0.149 mm is uniformly illuminated by a monochromatic light at $\lambda =523\phantom{\rule{0.2em}{0ex}}\text{nm}$. What is the width of the central principal maximum on a screen 2.35 m away?

Eight slits equally separated by 0.149 mm is uniformly illuminated by a monochromatic light at $\lambda =523\phantom{\rule{0.2em}{0ex}}\text{nm}$. What is the intensity of a secondary maxima compared to that of the principal maxima?

A transparent film of thickness 250 nm and index of refraction of 1.40 is surrounded by air. What wavelength in a beam of white light at near-normal incidence to the film undergoes destructive interference when reflected?

An intensity minimum is found for 450 nm light transmitted through a transparent film $(n=1.20)$ in air. (a) What is minimum thickness of the film? (b) If this wavelength is the longest for which the intensity minimum occurs, what are the next three lower values of $\lambda $ for which this happens?

A thin film with $n=1.32$ is surrounded by air. What is the minimum thickness of this film such that the reflection of normally incident light with $\lambda =500\phantom{\rule{0.2em}{0ex}}\text{nm}$ is minimized?

Repeat your calculation of the previous problem with the thin film placed on a flat glass ($n=1.50$) surface.

After a minor oil spill, a think film of oil ($n=1.40$) of thickness 450 nm floats on the water surface in a bay. (a) What predominant color is seen by a bird flying overhead? (b) What predominant color is seen by a seal swimming underwater?

A microscope slide 10 cm long is separated from a glass plate at one end by a sheet of paper. As shown below, the other end of the slide is in contact with the plate. The slide is illuminated from above by light from a sodium lamp ($\lambda =589\phantom{\rule{0.2em}{0ex}}\text{nm}$), and 14 fringes per centimeter are seen along the slide. What is the thickness of the piece of paper?

Suppose that the setup of the preceding problem is immersed in an unknown liquid. If 18 fringes per centimeter are now seen along the slide, what is the index of refraction of the liquid?

A thin wedge filled with air is produced when two flat glass plates are placed on top of one another and a slip of paper is inserted between them at one edge. Interference fringes are observed when monochromatic light falling vertically on the plates are seen in reflection. Is the first fringe near the edge where the plates are in contact a bright fringe or a dark fringe? Explain.

Two identical pieces of rectangular plate glass are used to measure the thickness of a hair. The glass plates are in direct contact at one edge and a single hair is placed between them hear the opposite edge. When illuminated with a sodium lamp ($\lambda =589\phantom{\rule{0.2em}{0ex}}\text{nm}$), the hair is seen between the 180th and 181st dark fringes. What are the lower and upper limits on the hair’s diameter?

Two microscope slides made of glass are illuminated by monochromatic ($\lambda =589\phantom{\rule{0.2em}{0ex}}\text{nm}$) light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a thin copper wire at the other end, forming a wedge of air. The diameter of the copper wire is $29.45\phantom{\rule{0.2em}{0ex}}\mu \text{m}$. How many bright fringes are seen across these slides?

A good quality camera “lens” is actually a system of lenses, rather than a single lens, but a side effect is that a reflection from the surface of one lens can bounce around many times within the system, creating artifacts in the photograph. To counteract this problem, one of the lenses in such a system is coated with a thin layer of material ($n=1.28$) on one side. The index of refraction of the lens glass is 1.68. What is the smallest thickness of the coating that reduces the reflection at 640 nm by destructive interference? (In other words, the coating’s effect is to be optimized for $\lambda =640\phantom{\rule{0.2em}{0ex}}\text{nm}$.)

Constructive interference is observed from directly above an oil slick for wavelengths (in air) 440 nm and 616 nm. The index of refraction of this oil is $n=1.54$. What is the film’s minimum possible thickness?

A soap bubble is blown outdoors. What colors (indicate by wavelengths) of the reflected sunlight are seen enhanced? The soap bubble has index of refraction 1.36 and thickness 380 nm.

A Michelson interferometer with a He-Ne laser light source ($\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$) projects its interference pattern on a screen. If the movable mirror is caused to move by $8.54\phantom{\rule{0.2em}{0ex}}\mu \text{m}$, how many fringes will be observed shifting through a reference point on a screen?

An experimenter detects 251 fringes when the movable mirror in a Michelson interferometer is displaced. The light source used is a sodium lamp, wavelength 589 nm. By what distance did the movable mirror move?

A Michelson interferometer is used to measure the wavelength of light put through it. When the movable mirror is moved by exactly 0.100 mm, the number of fringes observed moving through is 316. What is the wavelength of the light?

A 5.08-cm-long rectangular glass chamber is inserted into one arm of a Michelson interferometer using a 633-nm light source. This chamber is initially filled with air $(n=1.000293)$ at standard atmospheric pressure but the air is gradually pumped out using a vacuum pump until a near perfect vacuum is achieved. How many fringes are observed moving by during the transition?

Into one arm of a Michelson interferometer, a plastic sheet of thickness $75\phantom{\rule{0.2em}{0ex}}\mu \text{m}$ is inserted, which causes a shift in the interference pattern by 86 fringes. The light source has wavelength of 610 nm in air. What is the index of refraction of this plastic?

The thickness of an aluminum foil is measured using a Michelson interferometer that has its movable mirror mounted on a micrometer. There is a difference of 27 fringes in the observed interference pattern when the micrometer clamps down on the foil compared to when the micrometer is empty. Calculate the thickness of the foil?

The movable mirror of a Michelson interferometer is attached to one end of a thin metal rod of length 23.3 mm. The other end of the rod is anchored so it does not move. As the temperature of the rod changes from $15\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ to $25\phantom{\rule{0.2em}{0ex}}\text{C}$, a change of 14 fringes is observed. The light source is a He Ne laser, $\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$. What is the change in length of the metal bar, and what is its thermal expansion coefficient?

In a thermally stabilized lab, a Michelson interferometer is used to monitor the temperature to ensure it stays constant. The movable mirror is mounted on the end of a 1.00-m-long aluminum rod, held fixed at the other end. The light source is a He Ne laser, $\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$. The resolution of this apparatus corresponds to the temperature difference when a change of just one fringe is observed. What is this temperature difference?

A 65-fringe shift results in a Michelson interferometer when a $42.0\text{-}\mu \text{m}$ film made of an unknown material is placed in one arm. The light source has wavelength 632.9 nm. Identify the material using the indices of refraction found in Table 1.1.