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solutions4

# solutions4 - Homework Set 4 Due Wednesday 07/20 Problem 1 A...

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Unformatted text preview: Homework Set 4 Due Wednesday, 07/20 Problem 1: A room has two open windows on the same wall, at a distance of 4 meters apart. A loud car horn is sounded outside. An observer along the opposite wall, 15 meters away, can hear the sound loudly when he is directly opposite the point between the two windows, but if he moves to the point directly across from one of the windows, the sound becomes much quieter, growing louder again as the observer moves farther away from the center. What is the main frequency of the car horn? Solution: The point directly across from the window is the first point where destructive interference occurs. The path length difference there is 1/2 λ . From the diagram above, the path lengths from the two windows are x 1 = 15 m and x 2 = (6 2 +15 2 ) 1/2 m = 16.16 m . The path length difference is half the wavelength, and is equal to 1.16 m . Thus the wavelength is 2.32 m . This corresponds to a frequency of f = v / λ = 330 m/s / 2.32 m = 142 hz . Problem 2: Laser light of wavelength l = 520 nm is incident on a CD at an angle of 30 degrees to the normal, as shown in the diagram. The reflected light is projected onto a screen. The diffraction fringes are found to lie in directions 1.2 degrees apart. What is the separation between the tracks on the CD? Hint: the laser light arriving at the different tracks already has a path length difference from the laser, in addition to the path length difference to the screen. Solution: The CD has a bunch of parallel tracks that reflect light and act as a diffraction grating. Since the diffraction grating produces the same diffraction angles as a double slit with the same spacing, let us look at a pair of nearby tracks and see what is the path length difference for the light reflected off these two tracks: Suppose the light comes in at an angle of θ = 30 o , and we look at a point on the screen in the direction φ = θ + δ . Note that we use this angle because some light will be diffracted away by an angle δ from the central maximum given by the law of reflection. Beam 1 accumulates an additional path length ∆ x 1 = d sin( θ + δ ) after reflection, while beam 2 has an additional path length ∆ x 2 = d sin θ prior to reflection. The path length difference is ∆ x 1- ∆ x 2 = d (sin( θ + δ ) - sin θ ) = n λ for constructive interference. The n = 1 diffraction maximum is separated from the central maximum by δ = 1.2 o , so Problem 3: In the year 2265, United Earth spaceships are equipped with 100GW laser cannons, firing infra-red laser beams with a wavelength of 1.0 μ m . The laser beams have a diameter of 25 cm when they emerge from the laser cannon. Aliens from Alpha Centauri are invading. The alien spaceships have shields that can withstand infra-red light intensities of up to 1GW / m 2 . What is the effective range of the laser cannons against the alien spaceships?...
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solutions4 - Homework Set 4 Due Wednesday 07/20 Problem 1 A...

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