--- title: "Principal points of 3 cm and 4 cm lenses" subject: "Engineering Physics" chapter: 5 key: "semester_1/engineering_physics:two-lenses-three-four-principal-points" source: https://www.bctengineer.com/q/semester_1/engineering_physics/two-lenses-three-four-principal-points site: BCT Engineering --- # Principal points of 3 cm and 4 cm lenses Engineering Physics · Chapter 5: Geometrical Optics - Asked in: 2072 Kartik, 5 marks - 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-three-four-principal-points - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-5 ## Question Two thin converging lenses of focal lengths 3 cm and 4 cm respectively are placed coaxially in air separated by a distance of 2 cm. An object is placed at 4 cm in front of the first lens. Locate the positions of the principal points and final image. ## Short answer Equivalent focal length F = 2.4 cm. The first principal point H1 is 1.2 cm to the right of the first lens; the second principal point H2 is 1.6 cm to the left of the second lens. The final image is located 2.86 cm to the right of the second lens (real, inverted). ## Answer Use the Cartesian sign convention with all distances measured from the lens centre. **Given:** f₁ = 3 cm, f₂ = 4 cm, d = 2 cm, object at u = −4 cm from L1. #### Equivalent system and principal points ```text 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂) = 1/3 + 1/4 − 2/12 = 5/12 F = 2.4 cm α = F d / f₂ = 2.4 × 2 / 4 = 1.2 cm (H₁ is 1.2 cm right of L1) β = F d / f₁ = 2.4 × 2 / 3 = 1.6 cm (H₂ is 1.6 cm left of L2) ``` H₂ lies 2 − 1.6 = 0.4 cm to the right of L1. ```text L1 H2 H1 L2 |----.---------.------------| 0 0.4 1.2 2.0 (cm from L1) ``` #### Final image (lens by lens) ```text Lens 1: 1/v₁ = 1/3 − 1/4 = 1/12, v₁ = +12 cm Lens 2: object I₁ is 12 − 2 = 10 cm beyond L2, u₂ = +10 cm 1/v₂ = 1/4 + 1/10 = 7/20, v₂ = +2.857 cm ``` **Check using the principal points:** the object is 4 + 1.2 = 5.2 cm from H₁ (U = −5.2 cm). 1/V = 1/2.4 − 1/5.2 gives V = 4.457 cm from H₂, which is 4.457 − 1.6 = 2.857 cm from L2. **Answer:** F = 2.4 cm; H₁ is 1.2 cm to the right of L1; H₂ is 1.6 cm to the left of L2 (0.4 cm to the right of L1); the final image is 2.86 cm to the right of L2 (real, inverted). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is F for f1 = 3, f2 = 4, d = 2 cm? A. 5 cm B. 2.4 cm C. 1.71 cm D. 3.5 cm Answer: B. 1/F = 1/3 + 1/4 − 2/12 = 5/12, so F = 2.4 cm. (Key point: Equivalent focal length) 2. Where is H1? A. 1.6 cm right of L1 B. 1.2 cm right of L1 C. 2.4 cm right of L1 D. 1.2 cm left of L1 Answer: B. α = F d / f2 = 2.4 × 2 / 4 = 1.2 cm. (Key point: H1) 3. Where is H2? A. 1.2 cm left of L2 B. 1.6 cm right of L2 C. 1.6 cm left of L2, i.e. 0.4 cm right of L1 D. 0.4 cm left of L1 Answer: C. β = F d / f1 = 1.6 cm toward L1, and 2 − 1.6 = 0.4 cm from L1. (Key point: H2) 4. Where is the final image from L2? A. 12 cm right B. 10 cm right C. 2.86 cm left, virtual D. 2.86 cm right, real Answer: D. I1 is a virtual object at u2 = +10 cm; 1/v2 = 1/4 + 1/10 gives 2.857 cm. (Key point: Final image) ## Related in Engineering Physics - [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 - [Image by two lenses 30 cm and 40 cm](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-thirty-forty-image) — Engineering Physics, chapter 5, asked 3× - [Image and magnification, 3 cm and 4 cm lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-three-four-lateral-magnification) — Engineering Physics, chapter 5, asked 3× - [Equivalent focal length and principal points](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-lenses-ten-four-principal) — Engineering Physics, chapter 5 - [Principal points, object for infinity image](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Atwo-identical-lenses-infinity-object) — 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