Physics III

Homework IX CJ

Chapter 21; 21, 24, 66 23; 37, 73, 74

21.21. Model: The interference of two waves depends on the difference between the phases of the two waves.

Solve: (a) Because the speakers are in phase, . Let d represent the path-length difference. Using m  0 for the smallest d and the condition for destructive interference, we get

m 0, 1, 2, 3 …

(b) When the speakers are out of phase, . Using m 1 for the smallest d andthe condition for constructive interference, we get

m 0, 1, 2, 3, …

21.24. Model: Reflection is maximized if the two reflected waves interfere constructively.

Solve: The film thickness that causes constructive interference at wavelength  is given by Equation 21.32:

where we have used m  1 to calculate the thinnest film.

Assess: The film thickness is much less than the wavelength of visible light. The above formula is applicable because nairnfilmnglass.

21.66. Model: A light wave that reflects from a boundary at which the index of refraction increases has a phase shift of  rad.

Solve: (a) Because nfilmnair, the wave reflected from the outer surface of the film (called 1) is inverted due to the phase shift of  rad. The second reflected wave does not go through any phase shift of  rad because the index of refraction decreases at the boundary where this wave is reflected, which is on the inside of the soap film. We can write for the phases

because the sources are identical. For constructive interference,

m 0, 1, 2, 3, …

(b) For m 0 the wavelength for constructive interference is

For m 1 and 2, and . Red and violet together give a purplish color.

23.37. Model: Two objects are marginally resolvable if the angular separation between the objects, as seen from the lens, is .

Solve: Let y be the separation between the two light bulbs, and let L be their distance from a telescope. Thus,

23.73. Model: Two objects are marginally resolved if the angular separation between the objects, as seen from your eye lens, is .

Solve: Let y be the separation between the two headlights of the incoming car and let L be the distance of these lights from your eyes. We will assume the wavelength of the light to be 600 nm. Then,

Assess: The two headlights are not resolvable if L 11.5 km, marginally resolvable at 11.5 km, and resolvable at L 11.5 km.

23.74. Model: Two objects are marginally resolved if the angular separation between the objects is .

Visualize: /

Solve: (a) The angular separation between the sun and Jupiter is

(b) The sun is vastly brighter than Jupiter, which is much smaller and seen only dimly by reflected light. In theory it may be possible to resolve Jupiter and the sun, but in practice the extremely bright light from the sun will overwhelm the very dim light from Jupiter.