---
title: "Intensity ratio of e and o rays"
subject: "Engineering Physics"
chapter: 4
key: "semester_1/engineering_physics:polarized-intensity-ratio-optic-axis"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/polarized-intensity-ratio-optic-axis
site: BCT Engineering
---

# Intensity ratio of e and o rays

Engineering Physics · Chapter 4: Physical Optics

- Asked in: 2070 Chaitra, 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/polarized-intensity-ratio-optic-axis
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-4

## Question

If the plane of vibration of the incident beam makes an angle of 30° with the optic axis, compare the intensities of extraordinary and ordinary light.

## Short answer

Intensity ratio I_e/I_o = cot²θ. With θ = 30°, cot²30° = 3, so I_e : I_o = 3 : 1.

## Answer

When plane polarized light (amplitude A) falls on a doubly refracting crystal with its plane of vibration at angle θ to the optic axis, it splits into:
- **e-ray** with vibration along the optic axis: amplitude A cos θ
- **o-ray** with vibration perpendicular to the axis: amplitude A sin θ

```text
        optic axis (e)
         |   A
         |  /
         | /θ
         |/_______ o
```

Intensity is proportional to the square of the amplitude:

```text
I_e / I_o = (A cos θ)² / (A sin θ)² = cot² θ
```

**Given:** θ = 30°.

```text
I_e / I_o = cot² 30° = (√3)² = 3
```

**Answer: I_e : I_o = 3 : 1.** The extraordinary ray is three times as intense as the ordinary ray. (Defining θ as the angle to the axis, the e-component is the larger one.)

## Check yourself

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

1. For vibration at angle θ to the optic axis, the e-ray amplitude is:
   A. A sin θ
   B. A cos θ
   C. A tan θ
   D. A
   Answer: B. The e-vibration is along the axis: A cos θ; the o-ray is A sin θ. (Key point: Amplitudes of components)

2. The ratio I_e / I_o equals:
   A. tan^2 θ
   B. cot^2 θ
   C. cos θ / sin θ
   D. sin^2 θ
   Answer: B. Intensity ∝ amplitude^2, so the ratio is cos^2θ/sin^2θ = cot^2 θ. (Key point: Intensity ratio)

3. At θ = 30°, I_e : I_o is:
   A. 1 : 3
   B. 3 : 1
   C. 1 : 1
   D. √3 : 1
   Answer: B. cot^2 30° = (√3)^2 = 3. (Key point: Value at 30°)

## Related in Engineering Physics

- [Single slit intensity at given angle](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asingle-slit-intensity-at-angle) — Engineering Physics, chapter 4
- [Distinguishing plane, circular, elliptical light](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adistinguish-polarized-nicol-wave-plate) — Engineering Physics, chapter 4
- [Production of plane, circular, elliptical light](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Apolarized-light-production-relations) — Engineering Physics, chapter 4, asked 3×
- [Single slit diffraction intensity distribution](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asingle-slit-intensity-distribution) — Engineering Physics, chapter 4, asked 3×
- [Electron on ring axis performs SHM](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-electron-shm-axial) — Engineering Physics, chapter 7
- [Wedge angle from fringe spacing](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Awedge-angle-fringe-spacing) — Engineering Physics, chapter 4

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- [Lateral vibration of a simply supported beam](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Abeam-lateral-vibration-simply-supported) — Theory of Mechanism and Machine II, chapter 10
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