Geometric Optics: Mirrors and Lenses (Chapter 24 Study Notes)

Geometric Optics

Introduction

Geometric optics is the branch of physics that studies the behavior of light in terms of rays. This chapter focuses on the reflection and refraction of light at plane and spherical surfaces, as well as the formation of images by mirrors and lenses. Understanding these principles is essential for analyzing optical instruments and everyday phenomena involving light.

Reflections at Plane Surfaces

Key Terms and Concepts

• Object: The source of light rays.

• Image: The reproduction of an object formed by reflected or refracted rays.

• Real Image: Formed when rays actually converge at a point; can be projected onto a screen.

• Virtual Image: Formed when rays only appear to diverge from a point; cannot be projected.

• Distance to Image (s'): The distance from the mirror to the image.

• Distance to Object (s): The distance from the mirror to the object.

• Magnification (m): The ratio of image height to object height.

• Upright: Image has the same orientation as the object.

• Inverted: Image is upside down relative to the object.

Reflection at a Plane Mirror

• Light rays reflect off a plane mirror such that the angle of incidence equals the angle of reflection.

• The image formed is always virtual, upright, and the same size as the object.

• The image appears as far behind the mirror as the object is in front of it.

Example: Standing in front of a bathroom mirror, your image appears to be the same distance behind the mirror as you are in front.

Refractions and Virtual Focus

• When the eye follows reflected rays back to the mirror, the brain perceives a virtual image behind the mirror.

• This is why objects appear to be located behind the mirror surface.

Sign Conventions for Images and Objects

• The position of the object and image determines the sign convention used in calculations.

• For mirrors, object distances are positive if the object is in front of the mirror (real object).

• Image distances are positive if the image is on the same side as the outgoing rays (real image).

Magnification

• Magnification is given by:

$m = \frac{y'}{y} = \frac{s'}{s}$

• For a plane mirror, magnification is 1 (the image and object are the same size).

Inverted vs. Erect Images

• An image is erect if it has the same orientation as the object.

• An image is inverted if it is upside down relative to the object.

Left-Right Reversal in Plane Mirrors

• Plane mirrors reverse images left-to-right, which is why text appears reversed in a mirror.

• This effect is commonly seen on emergency vehicles, where words are written in reverse so they appear correctly in a rear-view mirror.

Spherical Mirrors

Types and Properties

• Concave Mirror: Curved inward like the inside of a sphere; can form real or virtual images depending on object position.

• Convex Mirror: Curved outward; always forms virtual, upright, and reduced images.

• The radius of curvature (R) determines the mirror's focal length:

$f = \frac{R}{2}$

Principal Rays for Mirror Imaging

• Ray tracing is used to locate images formed by spherical mirrors.

• Principal rays include:

  • Ray parallel to axis reflects through the focal point.

  • Ray through the focal point reflects parallel to the axis.

  • Ray through the center of curvature reflects back on itself.

• The mirror equation relates object distance (s), image distance (s'), and focal length (f):

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

Applications of Spherical Mirrors

• Concave mirrors are used in telescopes and shaving mirrors for magnified images.

• Convex mirrors are used for security and automotive rear-view mirrors due to their wide field of view.

Refraction at Spherical Surfaces and Thin Lenses

Refraction at Spherical Surfaces

• When light passes from one medium to another with a different refractive index, it bends (refracts).

• The refraction at a spherical surface is governed by:

$\frac{n_1}{s} + \frac{n_2}{s'} = \frac{n_2 - n_1}{R}$

• Where $n_1$ and $n_2$ are the refractive indices, $s$ and $s'$ are object and image distances, and $R$ is the radius of curvature.

Examples: Glass Rods in Air or Water

• Light bends differently depending on the refractive indices of the materials involved (e.g., air, water, glass).

• This explains phenomena such as the apparent bending of a straw in a glass of water.

Optical Illusions from Refraction

• Refraction can cause objects to appear displaced from their actual positions, such as a straw appearing bent in water or a swimmer appearing closer to the surface than they are.

Thin Lenses

Converging (Convex) Lenses

• A biconvex lens focuses parallel rays to a point called the focal point.

• Converging lenses can form real or virtual images depending on object position.

Diverging (Concave) Lenses

• A biconcave lens causes parallel rays to diverge as if they originated from a focal point on the same side as the object.

• Diverging lenses always form virtual, upright, and reduced images.

Principal Rays for Thin Lenses

• Ray tracing for lenses uses similar rules as for mirrors:

  • Ray parallel to axis refracts through (or appears to come from) the focal point.

  • Ray through the center of the lens passes straight without deviation.

  • Ray through the focal point emerges parallel to the axis.

• The thin lens equation relates object distance (s), image distance (s'), and focal length (f):

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

Magnification for Lenses

• Magnification is given by:

$m = \frac{y'}{y} = -\frac{s'}{s}$

• The negative sign indicates image inversion.

Lenses and Left-Right Reversal

• Lenses invert images top-to-bottom but do not reverse left and right, unlike mirrors.

Diverse Lens Shapes

• Lenses can be constructed in various shapes (meniscus, plano-convex, double convex, plano-concave, double concave) to suit different optical applications.

Applications and Examples

• Thin lens analysis is used in eyeglasses, cameras, microscopes, and telescopes.

• Worked examples illustrate how to apply the lens and mirror equations to solve for image position, size, and orientation.

Summary Table: Mirror and Lens Equations

Type	Equation	Image Properties
Plane Mirror	$s' = -s$ $m = 1$	Virtual, upright, same size
Spherical Mirror	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$ $f = \frac{R}{2}$	Real or virtual, inverted or upright
Thin Lens	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$	Real or virtual, inverted or upright
Magnification	$m = \frac{y'}{y} = -\frac{s'}{s}$	Negative: inverted, Positive: upright

Additional info: The above notes expand on the brief slide points, providing definitions, equations, and context for a comprehensive understanding of geometric optics as covered in a college physics course.