---
title: "Thickness of doubly refracting plate"
subject: "Engineering Physics"
chapter: 4
key: "semester_1/engineering_physics:doubly-refracting-plate-thickness"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/doubly-refracting-plate-thickness
site: BCT Engineering
---

# Thickness of doubly refracting plate

Engineering Physics · Chapter 4: Physical Optics

- Asked in: 2074 Asoj, 5 marks
- Video: [Calculate the thickness of (i) a quarter wave plate (ii) a half-wave plate, given that `mu_(e)`=1.53](https://www.youtube.com/watch?v=9gqTnX-un4o) — Doubtnut
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/doubly-refracting-plate-thickness
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-4

## Question

Calculate the thickness of a doubly refracting plate capable of producing a path difference of λ/4 between extraordinary and ordinary rays of wavelength 5890 Å. (Use μo = 1.53 and μe = 1.54)

## Short answer

For a quarter-wave plate, (μe − μo) t = λ/4. Using μe = 1.54, μo = 1.53, λ = 5890 Å, t = λ / [4(μe − μo)] = 1.47 × 10⁻⁵ m (14.7 μm).

## Answer

A plate producing path difference Δ between the e-ray and o-ray has Δ = (μ_e − μ_o) t, where *t* is the plate thickness.

**Given:** Δ = λ/4, λ = 5890 Å = 5.890×10⁻⁷ m, μ_o = 1.53, μ_e = 1.54.

1. Condition: (μ_e − μ_o) t = λ/4
2. Solve for t:

```text
t = λ / [4 (μ_e − μ_o)]
  = 5.890×10⁻⁷ / [4 × (1.54 − 1.53)]
  = 5.890×10⁻⁷ / 0.04
  = 1.4725×10⁻⁵ m
```

**Answer: t ≈ 1.47 × 10⁻⁵ m (14.7 μm); this is a quarter-wave plate for 5890 Å.**

## Check yourself

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

1. What is the path difference between e and o rays through a plate of thickness t?
   A. (μe + μo) t
   B. (μe − μo) t
   C. t / (μe − μo)
   D. (μe − μo) / t
   Answer: B. The optical paths are μe t and μo t, so the difference is (μe − μo)t. (Key point: Path difference relation)

2. For a path difference of λ/4, which formula for t is right?
   A. t = λ / [2(μe − μo)]
   B. t = λ / [4(μe − μo)]
   C. t = 4λ(μe − μo)
   D. t = λ(μe − μo)/4
   Answer: B. Setting (μe − μo)t = λ/4 gives t = λ/[4(μe − μo)]. (Key point: Thickness formula)

3. What is t for λ = 5890 Å, μo = 1.53, μe = 1.54?
   A. About 1.47×10^-5 m
   B. About 1.47×10^-7 m
   C. About 5.9×10^-6 m
   D. About 1.47×10^-3 m
   Answer: A. t = 5.890×10^-7 / (4 × 0.01) = 1.4725×10^-5 m, about 14.7 μm. (Key point: Numerical result)

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