---
title: "Separation of two achromatic lenses"
subject: "Engineering Physics"
chapter: 5
key: "semester_1/engineering_physics:two-lenses-equivalent-focal-fifty"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/two-lenses-equivalent-focal-fifty
site: BCT Engineering
---

# Separation of two achromatic lenses

Engineering Physics · Chapter 5: Geometrical Optics

- Asked in: 2073 Shrawan, 5 marks
- Video: [Lecture - 1 ACHROMATIC COMBINATION OF TWO THIN LENSES SEPERATED BY A DISTANCE](https://www.youtube.com/watch?v=GoTkXgRTJL0) — ANR Karanam
- 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-equivalent-focal-fifty
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-5

## Question

Two thin lenses of focal length f1 and f2 separated by a distance having an equivalent focal length 50 cm. The combination satisfies the condition for no chromatic aberration and minimum spherical aberration. Find the separation between the two lenses if both lenses are of same materials.

## Short answer

For same material, achromatism and minimum spherical aberration require x = (f1+f2)/2 = f1 - f2, giving f1 = 3f2. Equivalent focal length F = 1.5 f2, so with F = 50 cm, f2 = 33.3 cm, f1 = 100 cm, separation x = 66.7 cm.

## Answer

**Conditions for a two-lens combination of the same material**

1. **No chromatic aberration (achromatism):** ω/f₁ + ω/f₂ = 2xω/(f₁f₂), which gives **x = (f₁ + f₂) / 2**.
2. **Minimum spherical aberration:** **x = f₁ − f₂** (with f₁ > f₂).

**Step 1: relate f₁ and f₂**
```text
(f₁ + f₂)/2 = f₁ − f₂
f₁ + f₂ = 2f₁ − 2f₂
f₁ = 3 f₂
```

**Step 2: equivalent focal length**
```text
1/F = 1/f₁ + 1/f₂ − x/(f₁f₂)
F   = f₁f₂ / (f₁ + f₂ − x)
    = f₁f₂ / ((f₁ + f₂)/2)       (since x = (f₁+f₂)/2)
    = 2 f₁f₂ / (f₁ + f₂)
```
With f₁ = 3f₂: F = 2 × 3f₂² / 4f₂ = 1.5 f₂.

**Step 3: numbers (F = 50 cm)**
```text
f₂ = 50/1.5 = 33.33 cm
f₁ = 3 f₂   = 100 cm
x  = (f₁ + f₂)/2 = (100 + 33.33)/2 = 66.67 cm
```
**Check:** x = f₁ − f₂ = 66.67 cm, and F = 100 × 33.33 / (133.33 − 66.67) = 50 cm.

**Answer: the two lenses must be 66.7 cm (about 2/3 m) apart (f₁ = 100 cm, f₂ = 33.3 cm).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For two lenses of the same material, which separation gives no chromatic aberration?
   A. x = f1 − f2
   B. x = (f1 + f2)/2
   C. x = f1 + f2
   D. x = f1 f2
   Answer: B. Putting ω1 = ω2 in the achromatism condition gives x = (f1 + f2)/2. (Key point: Achromatic condition)

2. Equating (f1 + f2)/2 with f1 − f2 gives:
   A. f1 = 2f2
   B. f1 = 3f2
   C. f1 = f2
   D. f1 = 4f2
   Answer: B. f1 + f2 = 2f1 − 2f2, so f1 = 3f2. (Key point: Relation of focal lengths)

3. With x = (f1 + f2)/2 and f1 = 3f2, what is F?
   A. F = 0.5 f2
   B. F = 1.5 f2
   C. F = 3 f2
   D. F = 2 f2
   Answer: B. F = 2f1f2/(f1 + f2) = 6f2²/4f2 = 1.5 f2. (Key point: Equivalent focal length)

4. For F = 50 cm, what is x?
   A. 33.3 cm
   B. 50 cm
   C. 66.7 cm
   D. 100 cm
   Answer: C. f2 = 33.3 cm, f1 = 100 cm, x = (100 + 33.3)/2 = 66.7 cm. (Key point: Separation)

## Related in Engineering Physics

- [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
- [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
- [Condition for achromatism of two lenses](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aachromatism-condition-dispersive-powers) — Engineering Physics, chapter 5
- [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
- [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×
- [Chromatic aberration and its minimization](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Achromatic-aberration-single-lens-minimize) — Engineering Physics, chapter 5

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