When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle δ, Fig. 32–64. Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle Φ, and show that the minimum deviation angle, δ_m, is related to the prism’s index of refraction n by


$n=\frac{\sin \frac{1}{2}(\varphi +{\delta }_m)}{\sin \varphi /2}.$


[Hint: For θ in radians, $\left(d/d\theta \right)\left({\sin }^{-1}\theta \right)=1/\sqrt{1-{\theta }^2}$.]

Verified step by step guidance

1. Step 1: Understand the geometry of the prism and the concept of deviation angle δ. The deviation angle is the angle between the incident ray and the refracted ray as light passes through the prism. The apex angle Φ is the angle at the vertex of the prism, and the ray passes symmetrically when it is perpendicular to the bisector of Φ.
2. Step 2: Use Snell's Law to relate the angles of incidence and refraction at the prism's surfaces. Snell's Law states that n₁ sinθ₁ = n₂ sinθ₂, where n₁ and n₂ are the indices of refraction of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively. For the prism, the index of refraction is n, and the surrounding medium is typically air (n₁ ≈ 1).
3. Step 3: Express the deviation angle δ in terms of the angles of incidence and refraction. The deviation angle δ is given by δ = θ₁ + θ₂ - Φ, where θ₁ and θ₂ are the angles of incidence and refraction at the two surfaces of the prism, and Φ is the apex angle.
4. Step 4: Minimize the deviation angle δ by considering symmetry. When the ray passes symmetrically through the prism, the angles of incidence and refraction are equal at both surfaces. Let θ₁ = θ₂ = θ. Substitute this into the expression for δ to simplify it: δₘ = 2θ - Φ.
5. Step 5: Relate the minimum deviation angle δₘ to the index of refraction n. Using Snell's Law and the symmetry condition, derive the relationship n = sin(1/2 (Φ + δₘ)) / sin(Φ/2). This involves substituting the symmetric angles into Snell's Law and simplifying the trigonometric expressions.

Key Concepts

Refraction of Light

Refraction is the bending of light as it passes from one medium to another with a different density, which changes its speed. This phenomenon is governed by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media. Understanding refraction is crucial for analyzing how light behaves when it encounters a prism.

Deviation Angle

The deviation angle (δ) is defined as the angle between the incident ray and the refracted ray after passing through a prism. It is a key parameter in understanding how much the light path is altered by the prism. The minimum deviation occurs when the light passes symmetrically through the prism, which is essential for deriving the relationship between the prism's index of refraction and its geometry.

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how much light slows down in a medium compared to its speed in a vacuum. It is a critical factor in determining how light bends when entering or exiting a prism. The relationship between the index of refraction, the prism's apex angle (Φ), and the minimum deviation angle (δₘ) is fundamental for understanding the optical properties of prisms.