Geometric Optics: Mirrors and Image Formation

Geometric Optics

Introduction

Geometric optics is the branch of physics that studies the propagation of light in terms of rays. It is fundamental for understanding how mirrors and lenses form images, which is essential in many optical devices.

Refraction at a Plane Surface

Basic Principles

• Refraction is the bending of light as it passes from one medium to another with a different refractive index.

• The law of refraction (Snell's Law) is given by: $n_1 \sin \theta_1 = n_2 \sin \theta_2$ where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively.

• At a plane surface, the change in direction is determined by the difference in refractive indices.

Reflection and Image Formation by Spherical Mirrors

Types of Spherical Mirrors

• 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, erect, and reduced images.

Key Terms

• Center of Curvature (C): The center of the sphere of which the mirror is a part.

• Radius of Curvature (R): The radius of that sphere.

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

• Focal Length (f): The distance from the mirror to the focal point; $f = \frac{R}{2}$.

Sign Conventions

• For convex mirrors, both $f$ and $R$ are negative.

• For concave mirrors, $f$ and $R$ are positive (when the object is in front of the mirror).

Principal Rays for Spherical Mirrors

• Three principal rays are used to locate the image:

  1. Ray parallel to the axis reflects through (or appears to come from) the focal point.

  2. Ray through (or toward) the focal point reflects parallel to the axis.

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

Mirror Equation and Magnification

• The mirror equation relates object distance ($s$), image distance ($s'$), and focal length ($f$): $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• The magnification ($m$) is given by: $m = \frac{y'}{y} = -\frac{s'}{s}$ where $y$ is object height and $y'$ is image height.

Image Formation for Different Object Distances (Concave Mirror)

The nature and position of the image depend on the object's distance from the mirror:

Object Distance ($s$)	Image Type	Image Position	Magnification
$s > R$	Real, inverted	Between F and C	$|m| < 1$
$s = R$	Real, inverted	At C	$|m| = 1$
$F < s < R$	Real, inverted	Beyond C	$|m| > 1$
$s = F$	No image (rays parallel)	At infinity	--
$s < F$	Virtual, erect	Behind mirror	$|m| > 1$

Example: Image Formation in a Convex Mirror

• Santa looks at his reflection in a shiny silvered Christmas tree ornament (convex mirror).

• Given: $s = 0.75$ m, ornament diameter $= 7.2$ cm, Santa's height $= 1.6$ m.

• Radius of curvature: $R = -3.6$ cm (negative for convex), $f = -1.8$ cm.

• Image is virtual, erect, and reduced in size.

• Magnification: $m = \frac{y'}{y} = -\frac{s'}{s}$

• Image height: $y' = m y$

Summary Table: Mirror Types and Image Properties

Mirror Type	Focal Length ($f$)	Image Type	Image Orientation
Concave	Positive	Real or Virtual	Inverted or Erect
Convex	Negative	Virtual	Erect

Additional info:

• These notes cover the foundational aspects of geometric optics, focusing on mirrors, ray diagrams, and image formation. Further study would include refraction at spherical surfaces and lens systems.