Question

It is possible to construct a telescope using only mirrors, not lenses.  Such a "Cassegrain" telescope is shown here.  Analysis of such compound devices proceeds simply by doing one step (i.e. finding the first image) and then the next (use the image as the object for the next mirror). The large parabolic primary mirror has a focal lengh of 50.0 cm.   The hyperbolic secondary is mounted 43.0 cm in front of the large mirror.  The image is formed 3.00 cm behind the large mirror (where the figure shows Cassegrain focus).  

    (A)   What is the radius of the hyperbolic secondary?
    (B)     If you were using the telescope to see something like Saturn, explain why the image won't have a black hole in the middle where the secondary mirror blocks some of the light? 
(Hint:  where might an image of the secondary mirror be formed?)

Answer 1

mirror equation \(\frac{1}{v}+\frac{1}{u}=\frac{1}{f}=\frac{2}{R}\)

where \(u\) and \(v\) are positions of the object and image. \(R\) is radius of the mirror

(a)

Now for hyperbolic mirror, \(v=43+3=46 \mathrm{~cm}, u=-(50-43)=-7 \mathrm{~cm}\)

\(\frac{2}{R}=\frac{1}{46}+\frac{1}{-7} \Rightarrow R=-16.5 \mathrm{~cm}\)

(b)

Since the image distance from the secondary mirror is \(46 \mathrm{~cm}\) and the object distance is \(7 \mathrm{~cm}<46 \mathrm{~cm}\),

since there will not be any shadow beyond \(7 \mathrm{~cm}\) from the mirror, no dark spot will be seen.