Solved Problem on Optical Instruments

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The objective of a simple telescope has a focal length of 60 cm, and the eyepiece has a focal length of 1.5 cm. The image of an observed star will form 43.5 cm from the eyepiece. Determine the length of the tube of the telescope.

Image const:ruction:

Using the rule,

A second ray of light parallel to the secondary axis passes through the lens and is refracted, leaving the secondary focus (Figure 2).

The image

Using the rule,

The two rays found above do not determine an image on the observer's side, and to determine, the image it is necessary to extend these rays to the side of the object

Problem data:

- Objective focal length:
*f*_{1}= 60 cm; - Eyepiece focal length:
*f*_{2}= 1.5 cm; - Distance from the image to the eyepiece:
*p'*_{2}= −43.5 cm.

Using the sign convention, on the incident side, we have a positive abscissa for the real object,

Image

Solution

The length of the tube will be the sum of the focal length of the objective,

\[
\begin{gather}
\bbox[#99CCFF,10px]
{\frac{1}{f}=\frac{1}{p}+\frac{1}{p'}}
\end{gather}
\]

applying to the second lens (eyepiece)
\[
\begin{gather}
\frac{1}{f_{2}}=\frac{1}{p_{2}}+\frac{1}{p'_{2}}\\[5pt]
\frac{1}{p_{2}}=\frac{1}{f_{2}}-\frac{1}{p'_{2}}\\[5pt]
\frac{1}{p_{2}}=\frac{1}{1.5}-\frac{1}{(-43.5)}\\[5pt]
\frac{1}{p_{2}}=\frac{1}{1.5}+\frac{1}{43.5}
\end{gather}
\]

writing
\( 1.5=\frac{15}{10} \)
and
\( 43.5=\frac{435}{10} \)
\[
\begin{gather}
\frac{1}{p_{2}}=\frac{1}{\frac{15}{10}}+\frac{1}{\frac{435}{10}}\\[5pt]
\frac{1}{p_{2}}=\frac{10}{15}+\frac{10}{435}
\end{gather}
\]

multiplying and dividing by 29, the first term on the right side of the equation
\[
\begin{gather}
\frac{1}{p_{2}}=\frac{29}{29}\times\frac{10}{15}+\frac{10}{435}\\[5pt]
\frac{1}{p_{2}}=\frac{290+10}{435}\\[5pt]
\frac{1}{p_{2}}=\frac{300}{435}\\[5pt]
p_{2}=\frac{435}{300}\\[5pt]
p_{2}=1.45\;\text{cm}
\end{gather}
\]

the length
\[
\begin{gather}
d=f_{1}+p_{2}\\[5pt]
d=60+1.45
\end{gather}
\]

\[
\begin{gather}
\bbox[#FFCCCC,10px]
{d=61.45\;\text{cm}}
\end{gather}
\]

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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .