Question

An astronomical telescope has objective and eyepiece of focal lengths $40\, cm$ and $4 \,cm$ respectively. To view an object $200\, cm$ away from the objective, the lenses must be separated by a distance :

• A. 46.0 cm
• B. 50.0 cm
• C. 54.0 cm
• D. 37.3 cm

Answer 1

The Correct Option is C

Lens Formula and Tube Length Calculation 

Using the lens formula for the objective lens:

\(\frac{1}{v_0} - \frac{1}{u_0} = \frac{1}{f_0} \Rightarrow \frac{1}{v_0} = \frac{1}{f_0} + \frac{1}{u_0}\)

Substitute the given values: - \( f_0 = 40 \, \text{cm} \) (focal length of the objective lens), - \( u_0 = -200 \, \text{cm} \) (object distance, negative because the object is real).

\(\frac{1}{v_0} = \frac{1}{40} + \frac{1}{-200} = \frac{+5 - 1}{200}\)

Simplifying:

\(\frac{1}{v_0} = \frac{4}{200}\)

Thus, solving for \( v_0 \), we get:

\(v_0 = 50 \, \text{cm}\)

Tube Length Calculation

The tube length \( \ell \) is the sum of the absolute value of the image distance and the focal length of the eyepiece:

\(\ell = |\text{v}_0| + f_e = 50 + 4 = 54 \, \text{cm}\)

Conclusion:

The tube length is \( 54 \, \text{cm} \).