Geometric Optics: Reflection at Plane and Spherical Surfaces

Chapter 24: Geometric Optics

Introduction to Geometric Optics

Geometric optics is the study of light propagation in terms of rays. It explains how images are formed by mirrors and lenses, focusing on the laws of reflection and refraction. This chapter covers the principles of image formation by plane and spherical mirrors.

Reflection at a Plane Surface

Image Formation by Plane Mirrors

When light rays reflect off a plane mirror, they obey the law of reflection: the angle of incidence equals the angle of reflection. The image formed is virtual, upright, and the same size as the object.

• Law of Reflection: The angle of incidence ($heta_i$) equals the angle of reflection ($heta_r$): $heta_i = heta_r$

• Image Characteristics:

  • Virtual (cannot be projected onto a screen)

  • Upright

  • Same size as the object

  • Laterally reversed (left-right reversal)

• Object and Image Distances:

  • Object distance ($s$): Distance from object to mirror

  • Image distance ($s'$): Distance from image to mirror

  • For a plane mirror: $s' = -s$

• Example: If you stand 2 m in front of a plane mirror, your image appears 2 m behind the mirror.

Lateral Magnification in Plane Mirrors

Lateral magnification ($m$) describes the ratio of image height to object height.

• Definition: $m = rac{y'}{y}$ where $y'$ is image height and $y$ is object height.

• For Plane Mirrors: $m = 1$ (image is same size as object)

• Sign Convention: If $m$ is negative, the image is inverted; if positive, it is upright.

Special Cases and Conceptual Questions

• Minimum Mirror Height: To see your full body, the minimum mirror height is half your height, with the bottom edge halfway between your eyes and the floor.

• Obstructed Mirror: If the top half of a mirror is covered, you still see the full image of an object (e.g., a pencil), because rays from all parts of the object can reach the eye via reflection from the uncovered part.

• Multiple Mirrors: Multiple images can be formed when two mirrors are placed at an angle, each producing its own image.

Reflection at a Spherical Surface

Types of Spherical Mirrors

• Concave Mirror: Reflecting surface is on the inner side of the sphere. Can form real or virtual images depending on object position.

• Convex Mirror: Reflecting surface is on the outer side of the sphere. Always forms virtual, upright, and diminished images.

Formation of Images by Spherical Mirrors

Image formation depends on the position of the object relative to the mirror's focal point and center of curvature.

• Principal Axis: The straight line passing through the center of curvature ($C$) and the vertex ($V$) of the mirror.

• Focal Point ($F$): Point where parallel rays converge (concave) or appear to diverge from (convex).

• Focal Length ($f$): $f = rac{R}{2}$ where $R$ is the radius of curvature.

Spherical Mirror Equation

The relationship between object distance ($s$), image distance ($s'$), and focal length ($f$) is given by:

• $rac{1}{s} + rac{1}{s'} = rac{1}{f}$

• For a concave mirror, $f > 0$; for a convex mirror, $f < 0$.

Lateral Magnification for Spherical Mirrors

• $m = rac{y'}{y} = - rac{s'}{s}$

• If $m > 0$, image is upright; if $m < 0$, image is inverted.

Sign Conventions

• Object Distance ($s$): Positive if object is in front of mirror (real object).

• Image Distance ($s'$): Positive if image is in front of mirror (real image), negative if behind (virtual image).

• Radius of Curvature ($R$): Positive for concave, negative for convex.

Focal Point of a Concave Spherical Mirror

• Any incoming ray parallel to the optic axis is reflected through the focal point.

• Any incoming ray passing through the focal point is reflected parallel to the optic axis.

Examples and Applications

• Example 1: A lamp is placed 10 cm in front of a concave mirror that forms an image on a screen 30 cm from the mirror. The radius of curvature is calculated as $R = 19.4 ext{ cm}$. The lateral magnification is $m = -3$, meaning the image is inverted and three times larger than the object.

• Example 2: A convex mirror forms an upright, virtual, diminished, reversed image. For distant objects, the image is always virtual and diminished.

Summary Table: Properties of Plane and Spherical Mirrors

Mirror Type	Image Nature	Image Orientation	Magnification	Equation
Plane	Virtual	Upright, reversed	1	$s' = -s$
Concave (object outside focal point)	Real	Inverted	$|m| > 1$ or $|m| < 1$	$rac{1}{s} + rac{1}{s'} = rac{1}{f}$
Concave (object inside focal point)	Virtual	Upright	$|m| > 1$	$rac{1}{s} + rac{1}{s'} = rac{1}{f}$
Convex	Virtual	Upright, diminished	$|m| < 1$	$rac{1}{s} + rac{1}{s'} = rac{1}{f}$

Key Equations

• Law of Reflection: $heta_i = heta_r$

• Plane Mirror Image Distance: $s' = -s$

• Lateral Magnification: $m = rac{y'}{y}$

• Spherical Mirror Equation: $rac{1}{s} + rac{1}{s'} = rac{1}{f}$

• Focal Length: $f = rac{R}{2}$

• Lateral Magnification (spherical): $m = - rac{s'}{s}$

Additional info:

• These notes cover the foundational concepts of geometric optics relevant to college-level physics, focusing on image formation by plane and spherical mirrors, including sign conventions, magnification, and practical examples.