# Optics and Diffraction Problem Set

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1) The drawing shows a crystalline slab (n = 1.402) with a rectangular cross section. A ray of light strikes the slab at an incident angle of theta_1 = 44 degree, enters the slab, and travels to point P. This slab is surrounded by a fluid with a refractive index n. What is the maximum value of n such that total internal reflection occurs at point P?

2) Red light (n = 1.520) and violet light (n = 1.538) traveling in air are incident on a slab of crown glass. Both colors enter the glass at the same angle of refraction. The red light has an angle of incidence of 35.96 degrees. What is the angle of incidence of the violet light? Give your answer to four significant figures.

3) Horizontal rays of red light (lambda = 660 nm, in vacuum) and violet light (lambda = 410 nm, in vacuum) are incident on the prism shown in the drawing. The indices of refraction for the red and violet light are 1.651 and 1.698, respectively. What is the angle of refraction for (a) red light and (b) violet light as they emerge from the prism?

4) The drawing shows a horizontal beam of light that is incident on a prism. The base of the prism is also horizontal. The prism (n = 1.40) is surrounded by a liquid whose index of refraction is 1.6-. Determine the angle theta that the exiting light makes with the normal to the right face of the prism.

5) The distance between an object and its image formed by a diverging lens is 9.5 cm. The focal length of the lens is -4.4 cm. Find (a) the image distance and (b) the object distance.

6) For a distance of 55m, a photographer uses a telephoto lens (f = 390.0 mm) to take a picture of a charging rhinoceros. How far from the rhinoceros would the photographer have to be to record an image of the same size using a lens whose focal length is 50.0 mm?

7) A converging lens (f = 37.3 cm) is used to project an image of an object onto a screen. The object and the screen are 150 cm apart, and between them the lens can be placed at either of two locations. Find the two object distances, the smaller being the answer to part (a).

8) A nearsighted person cannot read a sign that is more than 5.0 m from his eyes. To deal with this problem, he wears contact lenses that do not correct his vision completely, but do allow him to read signs located up to distances of 12.2 m from his eyes. What is the focal length of the contacts?

9) The maximum angular magnification of a magnifying glass is 17.3 when a person uses it who has a near point that is 25.0 cm from his eyes. The same person finds that a microscope, using this magnifying glass as the eyepiece, has an angular magnification of -637. The separation between the eyepiece and the objective of the microscope is 17.9 c,. Obtain the focal length of the objective.

10) A telescope has an objective whose focal length is 20.8 m. Its eyepiece has a focal length of 7.45 cm. (a) What is the angular magnification of the telescope? (b) If the telescope is used to look at a lunar crater (diameter = 1960 m), what is the size (assume positive) of the first image, assuming the surface of the moon is 3.77 x 10^8 m from the surface of the earth? (c) How close does the crater appear to be when seen through the telescope?

11) A man in a boat is looking straight down at a fish in the water directly beneath him. The fish is looking straight up at the man. They are equidistant from the air/water interface. To the man, the fish appears to be 1.9 m beneath his eyes. To the fish, how far above its eyes does the man appear to be?

12) In Young's experiment a mixture of orange light (611 nm) and blue light (471 nm) shines on the double slit. The centers of the first-order bright blue fringes lie at the outer edges of a screen that is located 0.698 m away from the slits. However, the first-order bright orange fringes fall off the screen. By how much (toward or away from the slits) should the screen be moved, so that the centers of the first-order bright orange fringes just appear on the screen? It may be assumed that is small, so that sin(theta) is approximately equal to tan(theta).

13) A mixture of red light (lambda_vacuum = 605 nm) and green light (lambda_vacuum = 521 nm) shines perpendicularly on a soap film (n = 1.333) that has air on either side. What is the minimum nonzero thickness of the film, so that destructive interference removes the latter wavelength from the reflected light?

Diffraction Problems

1) How many dark fringes will be produced on either side of the central maximum if light (lambda = 668 nm) is incident on a single slit that is 4.17 x 10^-6 m wide?

2) The width of a slit is 2.0 x 10^-5 m. Light with a wavelength of 540 nm passes through this slit and falls on a screen that is located 0.41 m away. In the diffraction pattern, find the width of the bright fringe that is next to the central bright fringe.

3) In a single-slit diffraction pattern, the central fringe is 360 times wider than the slit. The screen is 15,000 times farther from the slit than the slit is wide. What is the ratio lambda/W, where lambda is the wavelength of the light shining through the slit and W is the width of the slit? Assume that the angle that locates a dark fringe on the screen is small, so that sin(theta) is approximately equal to tan(theta).

4) You are looking down at the earth from inside a jetliner flying at an altitude of 8500 m. The pupil of your eye has a diameter of 2.00 mm. Determine how far apart two cars must be on the ground if you are to have any hope of distinguishing between them in (a) red light (wavelength = 665 nm in vacuum) and (b) violet light (wavelength = 405 nm in vacuum).

5) Two stars are 4.1 x 10^11 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.66 m and just detects these stars as separate objects. Assume that light of wavelength 600 nm is being observed. Also, assume that diffraction effects, rather than atmospheric turbulence, limit the resolving power of the telescope. Find the maximum distance that these stars could be from the earth.

6) Astronomers have discovered a planetary system orbiting a star, which is at a distance of 5.8 x 10^20 m from the earth. One planet is believed to be located at a distance of 1.7 x 10^11 m from the star. Using visible light with a vacuum wavelength of 550 nm, what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

7) The pupil of an eagle's eye has a diameter of 6.0 mm. Two field mice are separated by 0.012 m. From a distance of 184 m, the eagle sees them as one unresolved object and dives toward them at a speed of 19 m/s. Assume that the eagle's eye detects light that has a wavelength of 550 nm in a vacuum. How much time passes until the eagle sees the mice as separate objects?

8) Two concentric circles of light emit light whose wavelength is 581 nm. The larger circle has a radius of 3.2 cm, while the smaller circle has a radius of 1.0 cm. When taking a picture of these lighted circles, a camera admits light through an aperture whose diameter is 12.7 mm. What is the maximum distance at which the camera can (a) distinguish one circle from the other and (b) reveal that the inner circle is a circle of light rather than a solid disk of light?

9) For a wavelength of 480 nm, a diffraction grating produces a bright fringe at an angle of 21 degrees. For an unknown wavelength, the same grating produces a bright fringe at an angle of 32 degrees. In both cases the bright fringes are of the same order m. What is the unknown wavelength?

10) The wavelength of the laser beam used in a compact disc player is 634 nm. Suppose that a diffraction grating produces first-order tracking beams that are 0.93 mm apart at a distance of 2.8 mm from the grating. Estimate the spacing between the slits of the grating.

11) Three, and only three, bright fringes can be seen on either side of the central maximum when a grating is illuminated with light (lambda = 462 nm). What is the maximum number of lines/cm for the grating?

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This complete set of solutions includes formula, calculations, answers and explanation for each problem. Please refer to the attached Word documents.

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Optics

Problem 1: We need the angle for the refracted ray which is dependent on n of the medium as

sin(theta_1)/sin(theta_2) = (n_2)/(n_1) where

n1 is the medium. Also the triangle made with the ...

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