Question

In a refracting telescope, the focal length of the objective is 50 times the focal length of the eyepiece. When the final image is formed at infinity, the length of the tube is 102 cm. Find the focal lengths of the two lenses.

Hint:

In a refracting telescope, the total tube length is the sum of the focal lengths of the objective and eyepiece lenses. If the objective's focal length is 50 times the eyepiece's focal length, you can use this relationship to find the focal lengths.

Answer 1

We know that the total length of the telescope tube is the sum of the focal lengths of the objective lens and the eyepiece lens. Let:
- \( f_o \) be the focal length of the objective lens,
- \( f_e \) be the focal length of the eyepiece lens. We are given: \[ f_o = 50 f_e \] and \[ f_o + f_e = 102 \, \text{cm} \] Substituting \( f_o = 50 f_e \) into the second equation: \[ 50 f_e + f_e = 102 \] \[ 51 f_e = 102 \] \[ f_e = \frac{102}{51} = 2 \, \text{cm} \] Now, using \( f_o = 50 f_e \): \[ f_o = 50 \times 2 = 100 \, \text{cm} \] Thus, the focal lengths of the two lenses are:
- \( f_o = 100 \, \text{cm} \) (objective lens),
- \( f_e = 2 \, \text{cm} \) (eyepiece lens).