--- title: "Principal points, object for infinity image" subject: "Engineering Physics" chapter: 5 key: "semester_1/engineering_physics:two-identical-lenses-infinity-object" source: https://www.bctengineer.com/q/semester_1/engineering_physics/two-identical-lenses-infinity-object site: BCT Engineering --- # Principal points, object for infinity image Engineering Physics · Chapter 5: Geometrical Optics - Asked in: 2070 Asar, 5 marks - Video: [Coaxial lens system: Cardinal points (object at infinity)](https://www.youtube.com/watch?v=Bi0giRVpdcU) — Dr.Shailendra Kolhe - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/two-identical-lenses-infinity-object - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-5 ## Question Two thin identical convex lenses of focal length 8 cm each are coaxial and 4 cm apart. Find the principal points and the position of object for which image is formed at infinity. ## Short answer Equivalent focal length F = 5.33 cm. H1 is 2.67 cm right of the first lens, H2 is 2.67 cm left of the second lens. For an image at infinity, the object must be placed at the first focal point, 2.67 cm in front of the first lens. ## Answer **Given:** f₁ = f₂ = 8 cm, d = 4 cm. #### Equivalent focal length ```text 1/F = 1/8 + 1/8 − 4/(8×8) = 1/4 − 1/16 = 3/16 F = 16/3 = 5.33 cm ``` #### Principal points ```text α = F d / f₂ = 5.33 × 4 / 8 = 2.67 cm (H₁: 2.67 cm right of L1) β = F d / f₁ = 5.33 × 4 / 8 = 2.67 cm (H₂: 2.67 cm left of L2) ``` H₂ is 4 − 2.67 = 1.33 cm to the right of L1, so H₂ is nearer to L1 than H₁ is. ```text L1 H2 H1 L2 |---------.---.-----------| 0 1.33 2.67 4.0 (cm from L1) ``` #### Object for an image at infinity An image at infinity is formed when the object is at the **first focal point** F₁ of the system. ```text F₁ from L1 = α − F = 2.67 − 5.33 = −2.67 cm ``` So the object must be **2.67 cm in front of the first lens**. **Check:** for L1, u = −2.67 cm: 1/v₁ = 1/8 − 1/2.67 = −1/4, v₁ = −4 cm. This image is 4 + 4 = 8 cm to the left of L2, which is exactly f₂, so the final rays leave L2 parallel (image at infinity). **Answer:** F = 5.33 cm; H₁ is 2.67 cm to the right of L1; H₂ is 2.67 cm to the left of L2; the object must be 2.67 cm in front of L1. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is F for two 8 cm lenses 4 cm apart? A. 4 cm B. 5.33 cm C. 6 cm D. 8 cm Answer: B. 1/F = 1/4 − 1/16 = 3/16, so F = 16/3 cm. (Key point: Equivalent focal length) 2. Where are H1 and H2? A. Both at the lens centres B. H1 1.33 cm right of L1 C. H1 4 cm right of L1 D. H1 2.67 cm right of L1; H2 2.67 cm left of L2 Answer: D. α = β = F d / f = 2.67 cm. (Key point: Principal points) 3. For image at infinity, the object must be at: A. The principal point H1 B. The second focal point C. The first focal point F1 of the system D. Twice F Answer: C. Rays from F1 emerge parallel. (Key point: Object position) 4. How far in front of L1 is the object? A. 8 cm B. 5.33 cm C. 2.67 cm D. 1.33 cm Answer: C. F1 = α − F = 2.67 − 5.33 = −2.67 cm from L1. (Key point: Result) ## 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 - [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 - [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 - [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 - [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×