--- title: "Equivalent focal length and principal points" subject: "Engineering Physics" chapter: 5 key: "semester_1/engineering_physics:two-lenses-ten-four-principal" source: https://www.bctengineer.com/q/semester_1/engineering_physics/two-lenses-ten-four-principal site: BCT Engineering --- # Equivalent focal length and principal points Engineering Physics · Chapter 5: Geometrical Optics - Asked in: 2068 Shrawan, 5 marks - Video: [Effective Focal Length and Principal Plane Explained](https://www.youtube.com/watch?v=Uii-FxBWtSg) — Jordan Edmunds Chetty - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/two-lenses-ten-four-principal - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-5 ## Question Two thin convex lenses having focal lengths 10cm and 4cm are coaxially separated by a distance of 5cm. Find the equivalent focal length of the combination. Determine also the positions of the principal points. ## Short answer Using 1/F = 1/10 + 1/4 - 5/(10×4) gives F = 4.44 cm. The first principal point is 5.56 cm right of the first lens (0.56 cm beyond the second lens); the second principal point is 2.22 cm left of the second lens (2.78 cm from the first lens). ## Answer **Given:** f₁ = 10 cm, f₂ = 4 cm, d = 5 cm (both convex, in air). #### Equivalent focal length ```text 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂) = 1/10 + 1/4 − 5/40 = 0.100 + 0.250 − 0.125 = 0.225 F = 4.44 cm ``` #### Principal points ```text α = F d / f₂ = 4.444 × 5 / 4 = 5.56 cm (H₁ from L1, towards L2) β = F d / f₁ = 4.444 × 5 / 10 = 2.22 cm (H₂ from L2, towards L1) ``` ```text L1 H2 L2 H1 |-----------------.------|--. 0 2.78 5.0 5.56 (cm from L1) ``` H₂ is 5 − 2.22 = 2.78 cm from L1; H₁ lies 0.56 cm beyond L2. **Answer:** F = 4.44 cm; the first principal point is 5.56 cm to the right of L1 (0.56 cm beyond L2); the second principal point is 2.22 cm to the left of L2 (2.78 cm from L1). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is F for f1 = 10, f2 = 4, d = 5 cm? A. 5.56 cm B. 4.44 cm C. 2.5 cm D. 7 cm Answer: B. 1/F = 0.1 + 0.25 − 0.125 = 0.225. (Key point: F) 2. Where is H1? A. 2.78 cm right B. 5.56 cm right of L1, beyond L2 C. 5 cm right D. 2.22 cm right of L1 Answer: B. α = F d / f2 = 4.444 × 5 / 4 = 5.56 cm. (Key point: H1) 3. Where is H2? A. 2.22 cm left of L2, 2.78 cm from L1 B. At L1 C. 5.56 cm left of L2 D. 2.78 cm right of L2 Answer: A. β = F d / f1 = 2.22 cm; 5 − 2.22 = 2.78 cm. (Key point: H2) ## Related in Engineering Physics - [Cardinal points and equivalent focal length](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acardinal-points-equivalent-focal-length) — Engineering Physics, chapter 5 - [Principal points of 3 cm and 4 cm lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-three-four-principal-points) — Engineering Physics, chapter 5 - [Principal points and image, two lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-principal-points-image) — Engineering Physics, chapter 5 - [Final image of convex and concave lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aconvex-concave-lens-final-image) — Engineering Physics, chapter 5 - [Focal lengths of achromatic lens pair](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aachromatic-lenses-equivalent-thirty-values) — Engineering Physics, chapter 5 - [Principal points of achromatic lens pair](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aachromatic-lens-pair-principal-points) — Engineering Physics, chapter 5 ## Related in other subjects - [Thin cylindrical shell: change in length and diameter](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Athin-cyl-shell-4m-25nmm2) — Strength of Materials, chapter 5