Astronomers are proud of their telescopes and often say they can "resolve a dime at so many miles." How many miles from a $\text { 102-cm-diameter }$ telescope mirror can you place a coin of diameter 3.0 cm and just resolve the coin using light of wavelength $389 \mathrm {~nm} \text { ? }$ (1 mi = 1.609 km)
A)19 mi
B)9.7 mi
C)40 mi
D)190,000,000 mi

Question

Quizplus · Accepted Answer

The Rayleigh criterion states that two objects are just resolvable when the center of the diffraction pattern of one lies directly over the first minimum of the diffraction pattern of the other. For a circular aperture, the angular separation of the first minimum from the central maximum is given by:$$\theta = 1.22 \frac{\lambda}{D}$$where $\theta$ is the angular separation, $\lambda$ is the wavelength of light, and D is the diameter of the aperture.In this problem, we have a telescope mirror with a diameter of 102 cm and a coin with a diameter of 3.0 cm. We want to find the distance at which the coin can be placed from the telescope mirror and just be resolved.First, we need to calculate the angular separation of the first minimum from the central maximum:$$\theta = 1.22 \frac{\lambda}{D} = 1.22 \frac{389 \times 10^{-9} \text{ m}}{1.02 \text{ m}} = 4.53 \times 10^{-6} \text{ rad}$$Next, we need to use the small-angle approximation to convert the angular separation to a linear separation:$$d = R\theta$$where d is the linear separation, R is the distance from the aperture to the object, and $\theta$ is the angular separation.In this problem, we have:$$d = 3.0 \times 10^{-2} \text{ m}$$$$\theta = 4.53 \times 10^{-6} \text{ rad}$$$$R = \frac{d}{\theta} = \frac{3.0 \times 10^{-2} \text{ m}}{4.53 \times 10^{-6} \text{ rad}} = 6622 \text{ m}$$Converting to miles:$$R = 6622 \text{ m} \times \frac{1 \text{ mi}}{1.609 \text{ km}} \times \frac{1000 \text{ m}}{1 \text{ km}} = 4.12 \text{ mi}$$Therefore, the coin can be placed 4.12 miles from the telescope mirror and just be resolved.

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Astronomers Are Proud of Their Telescopes and Often Say They $\text { 102-cm-diameter }$