Homework #7 complete by Friday, October 10

1. 24.X.18 (Use the Rayleigh criterion rather than book approximation.)

2. 24.P.42

3. If we were to send a ruby laser beam (wavelength of 694.3 nm) outward from the barrel of a 2.70-m-diameter telescope, what would be the diameter of the big red spot when the beam hit the Moon 3.84 x 10⁵km away?

4. Assume the range of pupil variation during adaptation of a normal eye is from 2 to 7 mm. What is the corresponding range of distances over which the eye can detect the separation of objects 1 in. apart?

5. The Very Large Array (VLA) is a set of 27 radio telescope dishes in New Mexico. Each dish is 25 meters in diameter. The antennae can be moved apart on railroad tracks, and their combined signals give the resolving power of a synthetic aperture 36.0 km in diameter.

(a) If the detectors are tuned to a frequency of 1.4 x 10⁹ Hz, what is the angular resolution of one of the dishes? Of the array of dishes? Convert your answers from radians to arcsec (1 degree = 60 arcminutes = 3600 arcseconds).
(b) Clouds of hydrogen radiate at this frequency. What must be the separation of two clouds at the center of the galaxy (26,000 lightyears = 2.5 x 10²⁰ m) if they are to be resolved?
(c) Very Long Baseline Interferometry (VLBI) is a method of using telescopes spread across the Earth's surface to observe objects at high resolution. Part of this network is located across the United States. See this link for a map of sites. Called the Very Long Baseline Array ( VLBA) , it consists of 10 telescopes of 25-m-diameter. Such a network allows angular resolutions of better than a milli-arcsecond. Using this resolution, what is the minimum separation of two clouds at the center of the galaxy for them to be resolved by the VLBA? To get a better sense of this distance, convert your answer to units of astronomical units, the Sun-Earth distance, where 1 A.U. = 1.5 x 10¹¹ m.
(d) The VLBA homepage claims that this resolution is equivalent to being able to read a newspaper in New York from the distance of Los Angeles. Check this claim.

6. If a light beam incident on a slit strikes the slit at an angle β from the perpendicular direction, show that the condition for destructive interference must be modified to:
a (sin Θ + sin β) = m λ.

7. (a) Consider a double-slit illuminated with a source of coherent, monochromatic light. The slit separation is d and the distance between the slit and the screen is L . A sheet of transparent plastic with an index of refraction of n and thickness t is placed over the upper slit. As a result , the central maximum of the interference pattern moves upward a distance y' . Find an expression for y' in terms of d, n, t, and L.
(b) Find y' if d = 0.300 mm, L = 1.00 m, n = 1.50, and t = 0.050 mm (about the thickness of a notebook page).

    Several problems extracted from Haliday, Resnick, & Walker, Serway & Beichner, Pedrotti & Pedrotti

Answer Key
1. 72cm
2. ~1500km
3. 240m
4. 83-290m (depends on assumed wavelength)
5. (a) 36 arcmin, 1.5 arcsec (b) 0.19 light-year (c) 1.26 x 10^-4 light-year = 8 AU
7. (b) 8.3cm (!)

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Last updated Oct. 5, 2009