Internal-focusing telescope: focal length of the focusing lens A telescope has an objective (object-glass) of focal length 14 cm located 20 cm from the diaphragm (cross-hair plane). When focusing on a staff 16.50 m away, the focusing lens is midway between the objective and the diaphragm. Determine the focal length of the focusing lens.

Introduction / Context:
Many surveying telescopes use an internal focusing lens placed between the objective and the diaphragm to bring objects at finite distances into sharp focus at the cross-hairs. When the focusing lens position is known for a specific object distance, lens formulae allow calculation of its focal length.

Given Data / Assumptions:

• Objective focal length, f_o = 14 cm.
• Distance between objective and diaphragm = 20 cm.
• Focusing lens is midway between them for this focus → its position is 10 cm from both objective and diaphragm.
• Object distance (from objective) u_o = 16.50 m = 1650 cm.
• Thin-lens, paraxial approximation applies.

Concept / Approach:

First, find where the objective alone would form an image of the staff. Then treat that image as the (virtual) object for the focusing lens, which must form the final image at the diaphragm. Apply the thin-lens formula 1/f = 1/v − 1/u for the focusing lens, using consistent Cartesian sign conventions (image distances positive to the right of the lens; object distances positive to the left; a virtual object on the image side has negative u).

Step-by-Step Solution:

Verification / Alternative check:

A direct ray diagram confirms that a converging lens of ≈3 cm focal length placed midway shifts the objective’s intermediate image at x ≈ 14.12 cm to the diaphragm plane at x = 20 cm.

Why Other Options Are Wrong:

4.9, 6.9, and 8.9 cm do not satisfy the lens equation with u_f ≈ −4.12 cm and v_f = +10 cm; they would place the final focus away from the diaphragm.

Common Pitfalls:

Using the wrong sign for a virtual object; assuming the focusing lens is afocal; neglecting that the “midway” condition fixes the geometry and hence f_f tightly near 3 cm.

Final Answer:

2.9 cm (approximately)