## Focal Lengths of Lenses of Refracting Telescope of Given Length and Magnification

In normal adjustment, the length of a refracting telescope is equal to the sum of the foal lengths of the primary (objective) and secondary (eyepiece) lens. Light rays approach from infinity and are focused by the primary lens to a point in the common focal plane of the primary and secondary lens, and are then made to form an image at infinity by the secondary lens.
The magnification is given by
$M=\frac{f_{PRIMARY}}{f_{SECONDARY}}$

For a telescope of length
$l$
$f_{PRIMARY}+f_{SECONDARY}=l$

Suppose
$m=50, \: l=M$
, then
$50=\frac{f_{PRIMARY}}{f_{SECONDARY}}$
and
$f_{PRIMARY}+f_{SECONDARY}=1m$

From the first of these equations
$f_{PRIMARY}=50 f_{SECONDARY}$
and substituting this into the second gives
$50 f_{SECONDARY}+f_{SECONDARY}= 1 \rightarrow f_{SECONDARY}=1/51 \simeq 0.0196 m$
then
$f_{PRIMARY}=50 f_{SECONDARY} \simeq 50 \times 0.0196 =0.98 m$