Before it was common to make multi-element lenses, astronomers would limit chrom

Question

Before it was common to make multi-element lenses, astronomers would limit chromatic aberration by using long-focus systems. Suppose you are a 17th-century astronomer, and your copper smith can make you a tube for a refractor that will accommodate a focal length of 4.0m. You have a crown glass blank (n=1.52) to grind to a single-element lens of diameter 150mm.

a) If you make a biconvex lens with the same radius of curvature on both sides, what is that radius of curvature? Remember to follow the sign conventions!

b) If you make a plano-convex lens (typically used with the curved side out) what will be the radius of curvature of the curved side?

c) What is the f/# for the system? If you use an eyepiece of focal length 50 mm to look at Mars, what is the angular magnification?

Explanation / Answer

given that focal length f = 4m refractive index of crown glass n = 1.52    diameter d = 150mm = 150 * 10-3 m a) if you use bio convex lens               1/f = ( n-1) ( 1/R1 + 1/R2)         where R1 = R2 = R            1/f = ( n-1) ( 2/R)         R = f ( n-1)2              =4.16m b) if you use plano convex lens            1/f = ( n-1) (1/R1 - 1/)       R1 = f ( n-1)              = 2.16m c) two lens system    eyepiece focal length fe = 50 * 10-3 m angular magnifiacation      m = - f0/ fe          = -4.16m/ 50 * 10-3 m           = -83.2          = -4.16m/ 50 * 10-3 m           = -83.2

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