Problem 52 - Arihant AIEEE Physics

Question

The human eye can be regarded as a single spherical refractive surface of curvature of cornea \(7.8 \mathrm{~mm}\). If a parallel beam of light comes to focus at \(3.075 \mathrm{~cm}\) behind the refractive surface, the refractive index of the eye is: (a) \(1.34\) (b) 1 (c) \(1.5\) (d) \(1.33\)

Step-by-Step Solution

Answer

The refractive index of the eye is approximately 1.33 (option d).

Step 1: Identify the given data

We know the radius of curvature \( R = 7.8 \) mm \((0.78 \text{ cm})\) and the focal length \( f = 3.075 \text{ cm} \). We need to use these values to find the refractive index \( n \).

Step 2: Use the lensmaker's formula

Using the lensmaker's formula for refraction at a sphere, \[ \frac{n_2}{f} = \frac{n_2 - n_1}{R} \] where \(n_2\) is the refractive index we want to find, \(n_1\) is the refractive index of air (which is 1), and \(R\) is the radius of curvature.

Step 3: Substitute values into the lensmaker's formula

Substitute the given values: \( n_1 = 1 \), \( R = 0.78 \text{ cm} \), \( f = 3.075 \text{ cm} \). The equation becomes: \[ \frac{n_2}{3.075} = \frac{n_2 - 1}{0.78} \].

Step 4: Solve for \( n_2 \)

Multiply through by \( 3.075 \) and rearrange to solve for \( n_2 \): \[ n_2 = 1 + \frac{0.78 \times n_2}{3.075} \]. Simplifying gives \( n_2 \approx 1.3366 \). This approximates to the refractive index value of (d) \(1.33\).

Key Concepts

Lensmaker's Formula

The lensmaker's formula is fundamental in optics. It helps in determining the focal length of a lens based on the refractive index and the curvature of the surfaces. For a lens in the air, it is represented as:

• The formula for refraction at a spherical surface is: \[ \frac{n_2}{f} = \frac{n_2 - n_1}{R} \]
• Where \( n_2 \) is the refractive index of the lens, \( n_1 \) is the refractive index of the surrounding medium (usually air, \( n_1 = 1 \)), \( R \) is the radius of curvature, and \( f \) is the focal length.

This formula is invaluable for designing lenses and understanding how they focus light. In the case of the human eye, it helps calculate how light is focused by the cornea onto the retina.

Refractive Index

The refractive index, symbolized as \( n \), describes how light propagates through a material. It is the ratio of the speed of light in a vacuum to its speed in the material. Understanding the refractive index helps determine:

• How much a ray of light bends when it enters the material.
• The optical properties of the material, such as its ability to focus light.

Refractive indices vary between materials. For example, the refractive index of air is approximately 1.0, whereas water is about 1.33.In the human eye, a refractive index of approximately 1.33 for the eye's aqueous humor and vitreous humor signifies how light refracts through the eye's components.

Curvature of Cornea

The curvature of the cornea is a key factor in determining how the eye focuses light. It is the front surface of the eye, with a radius of curvature that influences how parallel light rays converge. Features of the corneal curvature include:

• A typical curvature radius for the human eye is about 7.8 mm.
• It affects the focusing power; a smaller radius implies a greater refractive power.

The corneal curvature plays a crucial role in vision by bending the light significantly before it passes through the eye's lens, concentrating it onto the retina.

Spherical Refractive Surface

A spherical refractive surface is part of the core principles of geometrical optics, relevant to both lenses and natural optical elements like the cornea. Characteristics of spherical refractive surfaces include:

• They have a consistent curvature radius across the entire surface.
• They alter the path of light by refraction, focusing parallel rays to a point.

In optics, understanding these surfaces aids in predicting how rays will converge or diverge. The human eye acts as a spherical refractive surface, particularly the cornea, by refracting light to produce clear images on the retina.