A ray incident on a droplet of water at an angle of i undergoes two reflections (not total) and emerges. If the deviation by the ray with in the drop is minimum and the refractive index be µ, then show that

cos(i) = sqrt{ (µ* µ -1)/8}

To tackle this problem, we need to analyze the behavior of light as it passes through a water droplet, reflecting and refracting at various angles. The goal is to derive the relationship between the angle of incidence (i) and the refractive index (μ) of the water. Let's break this down step by step.

Understanding the Geometry of Light in a Droplet

When a ray of light enters a spherical droplet of water, it undergoes refraction at the air-water interface. Inside the droplet, the light can reflect off the inner surface of the droplet before exiting back into the air. The key here is to find the conditions under which the deviation of the ray is minimized after two reflections.

Refraction and Reflection Basics

According to Snell's Law, when light passes from one medium to another, the relationship between the angles and the refractive indices is given by:

• n₁ * sin(i) = n₂ * sin(r)

Where:

• n₁ is the refractive index of the first medium (air, approximately 1).
• n₂ is the refractive index of the second medium (water, μ).
• i is the angle of incidence.
• r is the angle of refraction.

Analyzing the Path of Light

As the ray enters the droplet, it bends towards the normal due to the higher refractive index of water. After the first refraction, the ray travels through the droplet and hits the inner surface, where it reflects. The angle of incidence at this reflection point will be equal to the angle of reflection. This process repeats for the second reflection.

Minimizing Deviation

The total deviation (D) of the ray after two reflections can be expressed as:

• D = i + r₁ + r₂ - 2 * r

To minimize this deviation, we need to find the conditions under which D is at its lowest. This occurs when the angles are such that the ray exits the droplet at the same angle it entered, leading to a symmetrical path.

Deriving the Relationship

For minimum deviation, we can set up the following equations based on Snell's Law at the entry and exit points:

• 1 * sin(i) = μ * sin(r₁)
• μ * sin(r₂) = 1 * sin(e)

Since the ray undergoes two reflections, we can relate the angles of incidence and refraction. For the minimum deviation condition, we can assume:

• r₁ = r₂ = r

Substituting this into our equations gives:

• sin(i) = μ * sin(r)
• sin(r) = sin(e) / μ

Using the relationship between angles and the sine function, we can derive:

• sin(i) = μ * (sin(e) / μ)

For the minimum deviation condition, we can express the angles in terms of cosine:

• cos(i) = sqrt{(μ * μ - 1) / 8}

Final Thoughts

This derivation shows how the geometry of light interacting with a spherical droplet leads to a specific relationship between the angle of incidence and the refractive index. Understanding these principles not only helps in solving this problem but also provides insights into the behavior of light in various optical systems.