The angle of deviation (δ) is a fundamental concept in optics, representing how much a ray of light bends away from its original direction as it passes through an optical medium, such as a prism. The derivation of this angle involves understanding the geometry of light refraction at each surface of the prism and the overall shape of the prism itself.
Key Components
The derivation relies on understanding the various angles involved:
- Angle of Prism (A): The angle between the two refracting surfaces of the prism.
- Angle of Incidence (i₁): The angle between the incoming light ray and the normal to the first refracting surface.
- Angle of Refraction (r₁): The angle between the refracted light ray inside the prism and the normal to the first refracting surface.
- Angle of Incidence at Second Surface (r₂): The angle between the light ray inside the prism and the normal to the second refracting surface.
- Angle of Emergence (i₂ or e): The angle between the outgoing light ray and the normal to the second refracting surface.
- Angle of Deviation (δ): The total angle between the direction of the incident ray and the direction of the emergent ray.
Step-by-Step Derivation
The derivation of the angle of deviation formula for a prism proceeds in several logical steps, combining principles of refraction and basic geometry.
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Deviation at Each Surface:
When a ray of light enters the prism at the first surface, it deviates by an angle δ₁ = i₁ - r₁. Similarly, when it emerges from the second surface, it deviates by an angle δ₂ = i₂ - r₂ (where i₂ is the angle of emergence). -
Total Deviation:
The total angle of deviation (δ) of the light ray as it passes through the prism is the sum of the deviations at the two surfaces. As stated in the reference, "As seen in the above image, the angle of deviation is represented by δ In the △MPQ by exterior angle theorem, we get δ = δ₁ + δ₂" [1]. Substituting the individual deviations, we get:
δ = (i₁ – r₁) + (i₂ – r₂) [2] -
Rearranging the Deviation Formula:
This equation can be rearranged to group the angles of incidence/emergence and the angles of refraction:
δ = (i₁ + i₂) – (r₁ + r₂) [3] (Note: The reference uses 'e' for i₂, so it appears as δ = (i₁ + e) - (r₁ + r₂) [3]) -
Geometric Relationship Inside the Prism:
Now, we need to find a relationship between the angles of refraction (r₁, r₂) and the angle of the prism (A). Consider the quadrilateral formed by the two normals to the refracting surfaces and the points where the ray enters and exits the prism. Let O be the point where the two normals intersect inside the prism, and let N and P be the points on the prism surfaces where the ray enters and exits, respectively. The sum of angles in the triangle formed by the intersection of the ray with the normals (△ONP) is 180°.
∠ONP + r₁ + r₂ = 180° [4]
Using θ to represent ∠ONP, this is written as θ + r₁ + r₂ = 180° [5].
Also, in the quadrilateral formed by the vertex of the prism (A), the points N and P, and the intersection of the normals (O), two angles are 90° (between the normal and the surface). The sum of angles in a quadrilateral is 360°. Thus, A + 90° + ∠ONP + 90° = 360°, which simplifies to A + ∠ONP = 180°, or A + θ = 180° [6]. -
Relating Angles of Refraction to Prism Angle:
From the two geometric relationships (θ + r₁ + r₂ = 180° [5] and A + θ = 180° [6]), we can see that both (r₁ + r₂) and A are equal to 180° - θ. Therefore:
r₁ + r₂ = A [7] -
Final Angle of Deviation Formula:
Finally, substitute the geometric relationship (r₁ + r₂ = A) into the rearranged deviation formula (δ = (i₁ + i₂) – (r₁ + r₂)):
δ = (i₁ + i₂) - A
This is the standard formula for the angle of deviation of a light ray passing through a prism.
Significance
This derivation shows that the angle of deviation depends on:
- The angles of incidence and emergence (i₁ and i₂). These angles are related to the refractive index (n) of the prism material and the surrounding medium via Snell's Law (n = sin i / sin r [8]), and thus depend on the wavelength of light.
- The angle of the prism (A).
Understanding this derivation is crucial for analyzing how prisms disperse light into its constituent colors and for applications in spectrometers and other optical instruments.
