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 :
- 46.0 cm
- 50.0 cm
- 54.0 cm
- 37.3 cm
The Correct Option is C
Solution and Explanation
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 \( 54 \, \text{cm} \).
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Concepts Used:
Ray Optics and Optical Instruments
Optics, deals with the determination of behaviour and the properties of light, along with its interactions with the matter and also with the instruments that are used to detect it.
Ray optics is also known as the geometrical optics and it is a branch of science which describes light propagation.
Reflection is the change in direction of light at an interface in-between two different media so that the wave-front returns into a medium from which it was originated.
Speed of light is the rate at which the light travels in free space.
A phenomenal change in image formed when the light is passed from one medium to another which is called Refraction.
Total Internal Reflection is the reflection of light when the light ray enters into a rarer medium from a denser medium and the angle of incidence is higher than the critical angle of incidence then that light ray will be reflected back to the denser medium.
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