Optics

Class 11 Physics

Chapter 16

Refraction Through Prisms

Class 11 Physics – Refraction Through Prisms Notes PDF

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Chapter Overview

A prism is a transparent refracting medium bounded by two plane refracting surfaces inclined to each other. When a ray passes through a prism it is refracted twice and generally emerges deviated from its original direction.

This chapter focuses on the angle of deviation, the special condition of minimum deviation, the relation between prism angle, minimum deviation and refractive index, and the approximate deviation produced by a small-angle prism.

Main Quantities

Angle of prism A, angle of incidence i, angle of emergence e, internal refraction angles r₁ and r₂, deviation δ, and refractive index μ.

Central Result

At minimum deviation the path through the prism is symmetric: i = e and r₁ = r₂ = A/2.

1. Refraction Through a Prism

Important Terms

Refracting faces: the two plane surfaces through which light enters and leaves the prism.
Refracting edge: the line where the two refracting faces meet.
Angle of prism, A: the angle between the two refracting faces.
Base: the face opposite the refracting edge in the prism’s cross-section.
Angle of deviation, δ: the angle between the direction of the incident ray produced forward and the emergent ray.

Diagram 1 — Ray Passing Through a Triangular Prism

The ray refracts at both prism faces and emerges deviated toward the base.

Geometrical Relation Inside the Prism

The angle between the normals equals the prism angle, giving:

A = r₁ + r₂

Total Angle of Deviation

Deviation at the first face is i − r₁; deviation at the second face is e − r₂.

δ = (i − r₁) + (e − r₂)

Using A = r₁ + r₂:

δ = i + e − A

Diagram 2 — Geometry of Prism Angles

For a prism, the sum of the two internal refraction angles equals the prism angle.

2. Minimum Deviation

For a given prism and wavelength, the angle of deviation changes as the angle of incidence is changed. The deviation first decreases, reaches a least value, and then increases. The least possible deviation is called the angle of minimum deviation, denoted by δ_m.

Diagram 3 — Deviation Versus Angle of Incidence

The deviation–incidence curve has a minimum. On either side of the minimum, two incidence angles can produce the same deviation.

Condition for Minimum Deviation

At minimum deviation, the path of the ray through the prism becomes symmetric:

i = e

r₁ = r₂

Since A = r₁ + r₂:

r₁ = r₂ = A/2

From δ = i + e − A and i = e:

δ_m = 2i − A

i = (A + δ_m)/2

Diagram 4 — Symmetric Ray Path at Minimum Deviation

At minimum deviation, the internal ray is symmetrically placed with respect to the two refracting faces.

Exam tip: The minimum-deviation condition is one of the most important results in this chapter. In derivations, write both i = e and r₁ = r₂ = A/2.

3. Relation Between Prism Angle, Minimum Deviation and Refractive Index

For refraction at the first face, Snell’s law gives:

μ = sin i / sin r₁

At minimum deviation:

i = (A + δ_m)/2 and r₁ = A/2

Substitution gives the standard prism formula:

μ = sin[(A + δ_m)/2] / sin(A/2)

Diagram 5 — Minimum-Deviation Formula Map

The minimum-deviation condition reduces Snell’s law to the standard prism refractive-index formula.

4. Deviation Produced by a Small-Angle Prism

A prism is called a small-angle prism when its prism angle is sufficiently small that the relevant angles are small and measured in radians.

For small angles, sinθ ≈ θ. At the first face:

μ ≈ i/r₁ ⇒ i ≈ μr₁

At the second face:

μ ≈ e/r₂ ⇒ e ≈ μr₂

Using δ = i + e − A and A = r₁ + r₂:

δ ≈ μ(r₁+r₂) − A

δ ≈ (μ − 1)A

Diagram 6 — Small-Angle Prism

For a thin prism and small angles in radians, deviation is approximately proportional to the prism angle.

Validity: The formula δ = (μ−1)A is an approximation for a small-angle prism and small angular values. The exact minimum-deviation relation should be used when this approximation is not justified.

5. Important Relationships at a Glance

Relationship	Formula	Use
Internal prism geometry	A = r₁ + r₂	Relates the two internal refraction angles to prism angle
Total deviation	δ = i + e − A	General prism ray path
Minimum-deviation symmetry	i = e	Minimum-deviation condition
Internal symmetry	r₁ = r₂ = A/2	Minimum-deviation condition
Incidence at minimum deviation	i = (A + δₘ)/2	Useful in derivation and numericals
Refractive index of prism	μ = sin[(A+δₘ)/2] / sin(A/2)	Exact minimum-deviation formula
Small-angle prism	δ ≈ (μ−1)A	Small-angle approximation, angles in radians

6. Solved Numerical Examples

Example 1 — General Deviation

Question: A prism has A = 60°, i = 48° and e = 42°. Find the deviation.

δ = i + e − A = 48° + 42° − 60° = 30°

Answer: 30°.

Example 2 — Refractive Index at Minimum Deviation

Question: A prism of angle 60° has minimum deviation 40°. Find its refractive index.

μ = sin[(A+δₘ)/2] / sin(A/2) = sin50° / sin30° ≈ 0.7660 / 0.5 ≈ 1.53

Answer: μ ≈ 1.53.

Example 3 — Minimum Deviation

Question: A prism has A = 60° and refractive index μ = 1.50. Find δₘ.

μ = sin[(A+δₘ)/2] / sin(A/2)

1.50 = sin[(60°+δₘ)/2] / 0.5

sin[(60°+δₘ)/2] = 0.75

(60°+δₘ)/2 ≈ 48.59° ⇒ δₘ ≈ 37.18°

Answer: δₘ ≈ 37.2°.

Example 4 — Small-Angle Prism

Question: A thin prism has A = 5° and μ = 1.60. Estimate its deviation.

δ ≈ (μ−1)A = (1.60−1)(5°) = 3°

Answer: approximately 3°.

Example 5 — Internal Angles at Minimum Deviation

Question: A prism has angle A = 50° and is adjusted for minimum deviation. Find r₁ and r₂.

r₁ = r₂ = A/2 = 25°

Answer: r₁ = r₂ = 25°.

Important Exam Questions

Short-Answer Questions

Define a prism and the angle of prism.
What is meant by the angle of deviation?
Write the relation between A, r₁ and r₂.
Show that δ = i + e − A.
What is minimum deviation?
State the conditions satisfied by a prism at minimum deviation.
Why is the ray path symmetric at minimum deviation?
Write the relation connecting μ, A and δₘ.
What is a small-angle prism?
State the small-angle prism deviation formula and its condition of validity.

Long-Answer Questions

Draw a labelled ray diagram for refraction through a prism and derive δ = i + e − A.
Explain the minimum-deviation condition with the help of the δ–i graph.
Derive the relation μ = sin[(A+δₘ)/2] / sin(A/2).
Derive the expression δ = (μ−1)A for a small-angle prism.
Explain why the deviation first decreases and then increases as the angle of incidence is varied.

Derivations to Practice

A = r₁ + r₂.
δ = i + e − A.
Minimum-deviation conditions i = e and r₁ = r₂ = A/2.
μ = sin[(A+δₘ)/2] / sin(A/2).
δ ≈ (μ−1)A for a small-angle prism.

Numerical Questions

A prism has A = 60°, i = 50° and e = 45°. Find δ.
A 60° prism has δₘ = 38°. Calculate its refractive index.
A prism has μ = 1.52 and A = 60°. Calculate the minimum deviation.
A small prism has A = 4° and μ = 1.5. Estimate its deviation.
A prism is at minimum deviation with A = 54°. Find r₁ and r₂.

Diagram Questions

Draw and label a ray passing through a triangular prism.
Draw the geometry showing A = r₁ + r₂.
Draw the graph of deviation versus angle of incidence and mark δₘ.
Draw the symmetric ray path through a prism at minimum deviation.
Draw a small-angle prism and show the direction of deviation.

One-Minute Revision

A prism has two plane refracting faces inclined at angle A.
A light ray through a prism is generally deviated toward the base.
The internal angles satisfy A = r₁ + r₂.
Total deviation is δ = i + e − A.
Deviation changes with the angle of incidence.
The least deviation is called minimum deviation δₘ.
At minimum deviation, the ray path is symmetric.
At minimum deviation, i = e.
At minimum deviation, r₁ = r₂ = A/2.
Therefore i = (A+δₘ)/2.
The exact prism formula is μ = sin[(A+δₘ)/2] / sin(A/2).
For a small-angle prism, sinθ ≈ θ when angles are in radians.
Small-angle prism deviation is δ ≈ (μ−1)A.

Diagram Practice

Redraw the complete prism ray diagram and label i, r₁, r₂, e, A and δ.
Draw the geometrical construction proving A = r₁ + r₂.
Draw the δ–i curve and clearly mark the minimum point.
Draw the symmetric ray path at minimum deviation and label i = e and r₁ = r₂.
Draw a thin small-angle prism and show the incident direction and deviated emergent direction.
Write the exact minimum-deviation formula below your diagram from memory.

Syllabus Coverage Checklist

NEB/CDC Chapter 16 scope	Covered
16.1 Minimum deviation condition	Yes — definition, graph, symmetry and conditions
16.2 Relation between angle of prism, angle of minimum deviation and refractive index	Yes — full derivation and numericals
16.3 Deviation in a small-angle prism	Yes — derivation and applicability
Supporting prism geometry and ray terminology	Yes
Related numerical and conceptual questions	Yes

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