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Ray Optics & Dispersion

Refraction through a Prism

Explore how light refracts, deviates, and disperses as it passes through a triangular glass prism. Toggle between Newton's rainbow experiment, the minimum deviation laser lab, and right-angled total internal reflection.

Triangular Prism Laboratory ⓘ

Observe refraction angles, rainbow dispersion bands, and right-angled TIR retroreflectors.

Normal Refraction

Optical Telemetry

A + δ = i + e ⇒ 60° + 38.6° = 45° + 53.6°
Refractive Index (n)
1.52 (Crown Glass)
Incident Angle (i)
45.0°
Emergence Angle (e)
53.6°
Angle of Deviation (δ)
38.6°
Critical Angle (θc)
41.1°

Refraction and Deviation in a Prism

An optical prism is a transparent solid body bound by flat refracting surfaces that intersect at an angle called the refracting angle (or apex angle, A). Unlike flat glass panes where opposing surfaces are parallel (causing rays to emerge parallel to their original path), the non-parallel faces of a prism cause a net change in direction.

This change in direction is measured by the angle of deviation (δ), which is the angle between the original incident ray and the final emergent ray.

Prism Refraction Geometry

As light enters a prism, it refracts at the first boundary, bending toward the normal inside the glass, and refracts again at the exit boundary, bending away from the normal.

The geometry of these refractions is bound by two fundamental equations:
• A = r1 + r2 (The apex angle equals the sum of internal refraction angles).
• A + δ = i + e (The sum of the apex angle and deviation angle equals the sum of the incident and emergence angles).

Angle of Minimum Deviation

When the angle of incidence is adjusted, the deviation angle decreases to a certain minimum value (Dm) and then rises again. At this minimum deviation point:

  • The path of the light ray inside the prism is perfectly symmetrical (i = e and r1 = r2 = A/2).

Prism Equation

n = sin((A + Dm)/2) / sin(A/2)

Newton's Dispersion of White Light

In 1666, Sir Isaac Newton demonstrated that white light is not a simple entity, but a mixture of different colors. When white light enters a prism:

  • Wavelength-Dependent Refraction: The refractive index (n) of the glass is higher for shorter wavelengths (Violet) than for longer wavelengths (Red).
  • Spectrum Splitting: Snell\'s Law dictates that the higher refractive index causes violet light to bend more sharply than red light, dispersing the beam into its spectrum (ROYGBIV).

Step-by-Step Solved Problems

Examine these solutions to master calculations involving prism geometry, index values, and minimum deviation.

Example 1 Problem Statement

A light ray is incident at an angle of 45° on one face of an equilateral glass prism of refracting angle 60°. If the ray emerges at an angle of 55°, calculate the angle of deviation produced by the prism.

View Step-by-Step Prism Solution
  1. Identify the given values: Apex/Refracting angle of the prism A = 60° (since it is an equilateral prism), Angle of incidence i = 45°, Angle of emergence e = 55°.
  2. Recall the fundamental prism relation: A + δ = i + e.
  3. Rearrange the equation to solve for the angle of deviation δ: δ = i + e - A.
  4. Substitute the values: δ = 45° + 55° - 60°.
  5. Calculate the final deviation: δ = 100° - 60° = 40°.

Final Derived Answer: Angle of Deviation δ = 40°.

Example 2 Problem Statement

Calculate the refractive index of a glass prism having an apex angle of 60° if the angle of minimum deviation is measured to be 38.6°.

View Step-by-Step Prism Solution
  1. Identify the parameters: Apex angle A = 60°, Minimum deviation Dm = 38.6°.
  2. Recall the prism formula for refractive index n: n = sin((A + Dm)/2) / sin(A/2).
  3. Calculate the numerator angle: (A + Dm)/2 = (60° + 38.6°)/2 = 98.6° / 2 = 49.3°.
  4. Calculate the denominator angle: A/2 = 60° / 2 = 30°.
  5. Substitute values into the trigonometric ratios: n = sin(49.3°) / sin(30°).
  6. Evaluate the sine values (using a calculator): sin(49.3°) ≈ 0.758, sin(30°) = 0.500.
  7. Solve for n: n = 0.758 / 0.500 ≈ 1.516.

Final Derived Answer: Refractive Index of the Prism material n ≈ 1.52.

Example 3 Problem Statement

A crown glass right-angled periscope prism has angles of 45°-90°-45° and a refractive index of 1.52. Verify if a ray incident normally on one of the shorter faces will undergo total internal reflection at the hypotenuse face.

View Step-by-Step Prism Solution
  1. Determine the angle of incidence at the hypotenuse interface: A ray entering normally (at 90° to the surface) passes undeflected. Geometry shows it hits the internal hypotenuse face at an angle of i = 45° relative to the normal.
  2. Calculate the critical angle θc for the crown glass-air boundary: θc = arcsin(1 / n).
  3. Substitute the refractive index n = 1.52: θc = arcsin(1 / 1.52) ≈ arcsin(0.658) ≈ 41.1°.
  4. Compare the incidence angle to the critical angle: i = 45° and θc = 41.1°. Since 45° > 41.1° (i > θc), total internal reflection must occur.
  5. Conclude the result: The light ray is completely reflected at 90°, exiting the other face normally. The prism retroreflects successfully.

Final Derived Answer: Yes, Total Internal Reflection occurs because the angle of incidence (45°) is greater than the critical angle (41.1°).

Self-Check Questions

Question 1

What happens to the angle of deviation as the angle of incidence on a prism is continuously increased from a small value?

Show Answer & Explanation

As the angle of incidence increases, the angle of deviation initially decreases, reaches a minimum value called the angle of minimum deviation (Dm), and then increases again. This forms a characteristic parabolic curve on an i-δ graph.

Question 2

State the conditions under which a light ray undergoes minimum deviation through a prism.

Show Answer & Explanation

A light ray undergoes minimum deviation when: (1) The ray passes symmetrically through the prism, meaning the angle of incidence equals the angle of emergence (i = e). (2) The angle of refraction at the first face equals the angle of refraction at the second face (r_1 = r_2 = A/2). (3) The refracted ray inside the prism is parallel to the base of an isosceles or equilateral prism.

Question 3

Explain why violet light deviates more than red light when white light passes through a glass prism.

Show Answer & Explanation

According to Cauchy's relation (n = B + C/λ²), the refractive index of glass is higher for shorter wavelengths. Since violet light has a shorter wavelength than red light, glass has a higher refractive index for violet (n_violet > n_red). According to Snell's law, a higher refractive index results in greater bending at the interfaces, causing violet light to deviate more.

Question 4

A right-angled isosceles prism is placed in water (n = 1.33). Will a light ray incident internally at 45° still undergo total internal reflection if the prism is made of glass (n = 1.50)?

Show Answer & Explanation

For a glass-water interface, the critical angle is θ_c = arcsin(n_water / n_glass) = arcsin(1.33 / 1.50) ≈ arcsin(0.887) ≈ 62.5°. Since the angle of incidence (45°) is less than this new critical angle (62.5°), total internal reflection will NOT occur. Light will refract out into the water instead.

Question 5

Write down the algebraic relationship between the refracting angle of a prism, the angles of refraction at the first and second faces, and explain its derivation.

Show Answer & Explanation

The relationship is A = r_1 + r_2. In the quadrilateral formed by the apex of the prism, the normals, and the points of incidence, the sum of opposite angles is 180°. This relates the apex angle A to the angle between the normals. In the triangle formed by the ray path and the normals, the angles are r_1, r_2, and the supplement of A, which yields A = r_1 + r_2.

Question 6

Can a prism produce dispersion without showing any net average deviation of the beam? Explain.

Show Answer & Explanation

Yes. By combining two prisms made of different glasses (e.g., crown glass and flint glass) with their refracting angles pointing in opposite directions, the dispersion of the first prism can be combined with the counter-dispersion of the second. If designed correctly, their deviations cancel out while leaving a net dispersion. This is called a direct-vision spectroscope.