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Chapter 20: Problem 52

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) \(1.33\)

Short Answer

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

Step by step solution

01

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 \( n \).
02

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.
03

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: \[ \frac{n_2}{3.075} = \frac{n_2 - 1}{0.78} \].
04

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) \(1.33\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The focal length of plano-convex lens, the convex surface of which is silvered is \(0.3 \mathrm{~m}\). If \(\mu\) of the lens is \(7 / 4\), the radius of curvature of the convex surface is : (a) \(0.45 \mathrm{~m}\) (b) \(1.05 \mathrm{~m}\) (c) \(3 \mathrm{~m}\) (d) \(0.9 \mathrm{~m}\)

A ray of light falls on a transparent glass slab of refractive index 1.62. What is the angle of incidence, if the reflected ray and refracted ray are mutually perpendicular? (a) \(\tan ^{-1}(1.62)\) (b) \(\tan ^{-1}\left(\frac{1}{1.62}\right)\) (c) \(\frac{1}{\tan ^{-1}(1.62)}\) (d) None of these

The electric permittivity and magnetic permeability of free space are \(\varepsilon_{0}\) and \(\mu_{0}\), respectively. The index of refraction of the medium, if \(\varepsilon\) and \(\mu\) are the electric permittivity and magnetic permeability in a medium is : (a) \(\frac{\varepsilon \mu}{\varepsilon_{0} \mu_{0}}\) (b) \(\left(\frac{\varepsilon \mu}{\varepsilon_{0} \mu_{0}}\right)^{1 / 2}\) (c) \(\frac{\varepsilon_{0} \mu_{0}}{\varepsilon \mu}\) (d) \(\left(\frac{\varepsilon_{0} \mu_{0}}{\varepsilon \mu}\right)^{1 / 2}\)

The refractive index of a lens material is \(\mu\) and focal length \(f\). Due to some chemical changes in the material, its refractive index has increased by \(2 \%\). The percentage decrease, in focal length for \(\mu=1.5\) will be : (a) \(4 \%\) (b) \(2 \%\) (c) \(6 \%\) (d) \(8 \%\)

Which of the following statements is/are correct? (a) The lens has two principal foci, but may have one focal length (b) A single lens can never bring a beam of white light to a point focus (c) A burning glass brings light rays to same focus as heat radiation (d) Both (a) and (b) are correct
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