Grade 10 Chapter 7: Lenses

Introduction

Lenses are transparent optical devices that use refraction to focus or disperse light. They are among the most important optical elements, with applications ranging from simple magnifying glasses to complex systems like cameras, microscopes, telescopes, and even the human eye. This chapter explores the principles, types, and applications of lenses.

What are Lenses?

A lens is a transparent optical device with two refracting surfaces, typically made of glass or plastic, that focuses or disperses light rays as they pass through it. The curved surfaces of a lens refract light in a way that can form images of objects.

Types of Lenses

Lenses are broadly classified into two main categories:

1. Convex (Converging) Lenses:
2. Thicker at the center than at the edges
3. Converge parallel light rays to a focus
4. Also called positive lenses

5. Examples: magnifying glass, objective lens of a telescope

6. Concave (Diverging) Lenses:

7. Thinner at the center than at the edges
8. Diverge parallel light rays
9. Also called negative lenses
10. Examples: eyepiece of Galilean telescope, correction lens for nearsightedness

Parts of a Lens

• Optical Center: The point within the lens through which light rays pass without deviation
• Principal Axis: The straight line passing through the optical center and perpendicular to both surfaces of the lens
• Principal Focus: The point at which parallel light rays converge after passing through a convex lens, or from which they appear to diverge after passing through a concave lens
• Focal Length: The distance between the optical center and the principal focus

Refraction Through Lenses

Convex Lens

When light passes through a convex lens: 1. Parallel rays converge at the principal focus after refraction 2. Rays passing through the optical center go straight without deviation 3. Rays passing through the principal focus emerge parallel to the principal axis

Concave Lens

When light passes through a concave lens: 1. Parallel rays diverge after refraction, appearing to come from the principal focus 2. Rays passing through the optical center go straight without deviation 3. Rays directed toward the principal focus emerge parallel to the principal axis

Lens Formula and Magnification

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f):

1/v - 1/u = 1/f

This formula applies to both convex and concave lenses, with appropriate sign conventions.

Sign Convention for Lenses

• Object distance (u) is always negative
• For real images, image distance (v) is positive
• For virtual images, image distance (v) is negative
• For convex lenses, focal length (f) is positive
• For concave lenses, focal length (f) is negative

Magnification

Magnification (m) is the ratio of the height of the image to the height of the object:

m = height of image / height of object = -v/u

Where: - m > 1 indicates a magnified image - m < 1 indicates a diminished image - Positive m indicates an erect image - Negative m indicates an inverted image

Image Formation by Lenses

Image Formation by Convex Lens

The nature, position, and size of the image formed by a convex lens depend on the position of the object:

1. Object at infinity:
2. Image: Real, inverted, highly diminished

3. Position: At the principal focus F₂

4. Object beyond 2F₁:

5. Image: Real, inverted, diminished

6. Position: Between F₂ and 2F₂

7. Object at 2F₁:

8. Image: Real, inverted, same size as object

9. Position: At 2F₂

10. Object between F₁ and 2F₁:

11. Image: Real, inverted, enlarged

12. Position: Beyond 2F₂

13. Object at F₁:

14. Image: Real, inverted, infinitely large

15. Position: At infinity

16. Object between optical center and F₁:

17. Image: Virtual, erect, enlarged
18. Position: On the same side of the lens as the object

Image Formation by Concave Lens

For a concave lens, regardless of the object's position, the image is always: - Virtual - Erect - Diminished - Formed between the optical center and the principal focus on the same side as the object

The only variation is in the degree of diminishment, which decreases as the object moves closer to the lens.

Power of a Lens

The power of a lens is a measure of its ability to converge or diverge light rays. It is defined as the reciprocal of the focal length in meters:

Power (P) = 1/f

Where: - P is measured in diopters (D) - f is the focal length in meters

• Convex lenses have positive power
• Concave lenses have negative power

Combination of Lenses

When two or more lenses are combined (placed in contact):

1. The effective focal length (F) of the combination is given by: 1/F = 1/f₁ + 1/f₂ + ... + 1/fₙ

2. The total power of the combination is the algebraic sum of the individual powers: P = P₁ + P₂ + ... + Pₙ

Lens Defects (Aberrations)

Lenses suffer from various defects that affect image quality:

1. Spherical Aberration: Rays passing through different zones of the lens focus at different points

2. Correction: Use parabolic surfaces or combination of lenses

3. Chromatic Aberration: Different colors focus at different points due to dispersion

4. Correction: Use achromatic doublets (combination of convex and concave lenses of different materials)

5. Astigmatism: Inability to form a point image of a point object

6. Correction: Use cylindrical lenses

7. Coma: Off-axis point objects appear comet-shaped

8. Correction: Use specially designed lens systems

9. Distortion: Straight lines appear curved

10. Correction: Use symmetrical lens arrangements

Applications of Lenses

Simple Microscope (Magnifying Glass)

• Uses a single convex lens with short focal length
• Object is placed between optical center and focus
• Forms a virtual, erect, and magnified image
• Magnification = 1 + D/f (where D is the least distance of distinct vision, typically 25 cm)

Compound Microscope

• Uses two convex lenses: objective (short focal length) and eyepiece
• Object is placed just beyond the focus of the objective
• Objective forms a real, inverted, magnified image
• This image acts as the object for the eyepiece
• Eyepiece forms a final virtual, inverted, and further magnified image
• Total magnification = Magnification of objective × Magnification of eyepiece

Telescope

• Used to view distant objects
• Astronomical telescope:
• Uses two convex lenses: objective (long focal length) and eyepiece (short focal length)
• Forms a final image that is virtual, inverted, and magnified

• Magnification = Focal length of objective / Focal length of eyepiece

• Terrestrial telescope:

• Similar to astronomical telescope but includes an erecting lens or prism system
• Forms a final image that is virtual, erect, and magnified

Human Eye

• The eye is a natural optical system with a lens
• The cornea and lens together act as a converging lens system
• The retina acts as the screen where the image is formed
• Accommodation: The ability to adjust focal length by changing the curvature of the lens
• Near point: The closest point at which an object can be placed and still form a clear image
• Far point: The farthest point at which an object can be placed and still form a clear image

Defects of Vision and Their Correction

1. Myopia (Nearsightedness):
2. Can see near objects clearly but not distant objects
3. Far point is closer than infinity
4. Caused by elongated eyeball or too powerful lens

5. Corrected using concave lenses

6. Hypermetropia (Farsightedness):

7. Can see distant objects clearly but not near objects
8. Near point is farther than normal
9. Caused by shortened eyeball or less powerful lens

10. Corrected using convex lenses

11. Presbyopia:

12. Age-related loss of accommodation
13. Difficulty in seeing nearby objects
14. Caused by hardening of the lens

15. Corrected using convex lenses (reading glasses)

16. Astigmatism:

17. Inability to focus on both horizontal and vertical lines simultaneously
18. Caused by irregular curvature of cornea or lens
19. Corrected using cylindrical lenses

Lens Maker's Formula

The lens maker's formula relates the focal length of a lens to its refractive index and the radii of curvature of its surfaces:

1/f = (n-1) × [1/R₁ - 1/R₂]

Where: - f is the focal length - n is the refractive index of the lens material relative to the surrounding medium - R₁ is the radius of curvature of the first surface - R₂ is the radius of curvature of the second surface

Sign conventions for radii of curvature: - R is positive if the center of curvature is on the side from which light is coming - R is negative if the center of curvature is on the side to which light is going

Experimental Determination of Focal Length

For Convex Lens

1. Distant Object Method:
2. Focus a distant object (effectively at infinity) on a screen
3. Measure the distance between the lens and the screen

4. This distance equals the focal length

5. u-v Method:

6. Place an object at various distances (u) from the lens
7. For each position, find the image distance (v) by focusing on a screen
8. Plot a graph of 1/v versus 1/u
9. The y-intercept gives 1/f

For Concave Lens

Since a concave lens always forms a virtual image, indirect methods are used:

1. Combination Method:
2. Combine the concave lens with a powerful convex lens
3. Find the focal length of the combination

4. Use the formula 1/F = 1/f₁ + 1/f₂ to calculate the focal length of the concave lens

5. Auxiliary Lens Method:

6. Use a convex lens to form a real image of an object
7. Place the concave lens between the convex lens and this image
8. The new image position helps calculate the focal length of the concave lens

Summary

Lenses are fundamental optical elements that use refraction to manipulate light. They come in two main types: convex (converging) and concave (diverging). The behavior of light through lenses is governed by the lens formula and follows specific patterns depending on the type of lens and the position of the object.

Lenses find applications in numerous optical instruments, from simple magnifying glasses to complex systems like microscopes and telescopes. They also play a crucial role in vision, both in the natural lens of the eye and in corrective eyewear.

Understanding the principles of lenses is essential for fields ranging from physics and engineering to medicine and astronomy.

Key Terms

• Lens
• Convex lens
• Concave lens
• Optical center
• Principal axis
• Principal focus
• Focal length
• Lens formula
• Magnification
• Power of a lens
• Diopter
• Aberrations
• Myopia
• Hypermetropia
• Presbyopia
• Astigmatism