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 diamete $\mathrm { r } 3.0 \mathrm {~cm}$ and just resolve the coin using light of wavelength $389 \mathrm {~nm} ?$ (1 mi = 1.609 km)

Question

Quizplus · Accepted Answer

The Rayleigh criterion states that two objects are just resolvable when the center of the diffraction pattern of one object is directly over the first minimum of the diffraction pattern of the other object. The angular separation of these two objects 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 telescope mirror.In this problem, we are given that the diameter of the telescope mirror is $102 \text{ cm}$ and the wavelength of light is $389 \text{ nm}$. We want to find the distance at which a coin of diameter $3.0 \text{ cm}$ can be just resolved.First, we need to find the angular separation of the two objects. Using the Rayleigh criterion, we have:$$\theta = 1.22 \frac{\lambda}{d} = 1.22 \frac{389 \text{ nm}}{102 \text{ cm}} = 4.6 \times 10^{-6} \text{ rad}$$Now, we can use the small angle approximation to find the distance at which the two objects can be just resolved. The small angle approximation states that:$$\tan \theta \approx \sin \theta \approx \theta$$for small angles.Using this approximation, we have:$$\theta = \frac{r}{d}$$where $r$ is the distance between the two objects and $d$ is the diameter of the telescope mirror.Solving for $r$, we get:$$r = d \theta = (102 \text{ cm})(4.6 \times 10^{-6} \text{ rad}) = 4.7 \times 10^{-4} \text{ cm}$$Converting this to miles, we get:$$r = 4.7 \times 10^{-4} \text{ cm} \left(\frac{1 \text{ m}}{100 \text{ cm}}\right) \left(\frac{1 \text{ km}}{1000 \text{ m}}\right) \left(\frac{1 \text{ mi}}{1.609 \text{ km}}\right) = 19 \text{ mi}$$Therefore, the correct answer is A.

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

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