Telescopes (a) If you were stranded on an island with only a pair of 3.5 -D reading glasses, could you make a telescope? If so, what would be the length of the telescope and what would be the best possible angular magnification? (b) Answer the same questions if you also had a pair of 1.3 -D reading glasses.

The angular magnification (M) of a telescope is given by the formula: M = (-f_obj) / (f_eye) where f_obj is the focal length of the objective lens and f_eye is the focal length of the eyepiece lens.

The power (P) of a lens is given by the formula: P = 1/f where f is the focal length of the lens. In the first part of the exercise, we are given a pair of 3.5 -D reading glasses. We can calculate the focal length (f_eye) using the formula: f_eye = 1/P = 1/(-3.5 D) = -1/3.5 m In the second part of the exercise, we also have a pair of 1.3 -D reading glasses. We can similarly calculate the focal length (f_obj) for these glasses: f_obj = 1/P = 1/(-1.3 D) = -1/1.3 m

(a) If we only have a pair of 3.5 -D reading glasses, we can use these as both the eyepiece and the objective lens. Therefore, f_eye = f_obj and the angular magnification becomes: M = (-f_obj) / (f_eye) = (-(-1/3.5 m))/(1/3.5 m) = 1 This means that the best possible angular magnification with only 3.5 -D reading glasses is 1 (no magnification). Therefore, it is not possible to make a telescope that provides any magnification using only the 3.5 -D reading glasses. In this case, the length of the telescope would be approximately the distance between the lenses, which is given as the sum of their focal lengths: Length = f_obj + f_eye = (-1/3.5 m) + (-1/3.5 m) = -2/3.5 m ≈ -0.57 m (b) If we also have a pair of 1.3 -D reading glasses, we can use these as the objective lens while still using the 3.5 -D reading glasses as the eyepiece lens. In this case, the angular magnification becomes: M = (-f_obj) / (f_eye) = (-(-1/1.3 m))/(1/3.5 m) = 3.5/1.3 ≈ 2.69 This means that the best possible angular magnification with both 3.5 -D and 1.3 -D reading glasses is approximately 2.69. In this case, the length of the telescope would be approximately the distance between the lenses, which is given as the sum of their focal lengths: Length = f_obj + f_eye = (-1/1.3 m) + (-1/3.5 m) = (-1/1.3) + (-1/3.5) m ≈ -0.92 m