Study Notes for Chapter Nine: Ray Optics and Optical Instruments

Chapter Nine: Ray Optics and Optical Instruments

9.1 Introduction

Nature has endowed the human eye (retina) with the sensitivity to detect electromagnetic waves within a small range of the electromagnetic spectrum. Electromagnetic radiation belonging to this region of the spectrum (wavelength of about 400 nm to 750 nm) is called light. It is mainly through light and the sense of vision that we know and interpret the world around us.

There are two essential observations about light based on common experience:

1. Speed of Light: Light travels with enormous speed.

2. Straight-Line Propagation: It travels in straight lines.

It took time for people to realize that the speed of light is finite and measurable. Its presently accepted value in vacuum is
$c = 2.99792458 imes 10^8 ext{ m s}^{-1}$
For many purposes, we can take
$c = 3 imes 10^8 ext{ m s}^{-1}$.
The speed of light in vacuum is the highest speed attainable in nature. The intuitive notion that light travels in a straight line seems to contradict the wave nature of light discussed in Chapter 8. However, the small wavelength of light compared to the size of ordinary objects allows us to consider light traveling from one point to another in a straight line (known as a ray of light). A bundle of such rays is referred to as a beam of light.

In this chapter, we consider the phenomena of reflection, refraction, and dispersion of light using the ray picture and study the image formation by plane and spherical reflecting and refracting surfaces.

9.2 Reflection of Light by Spherical Mirrors

The laws of reflection state:

1. The angle of reflection (the angle between the reflected ray and the normal to the surface) equals the angle of incidence (the angle between the incident ray and the normal).

2. The incident ray, reflected ray, and the normal to the reflecting surface at the point of incidence lie in the same plane.

These laws are valid for all reflecting surfaces, whether plane or curved. We focus on curved surfaces, specifically spherical surfaces.

9.2.1 Sign Convention

To derive formulas for reflection by spherical mirrors and refraction by spherical lenses, we adopt the Cartesian sign convention. According to this convention:

• Distances are measured from the pole of the mirror or the optical center of the lens.

• Distances in the same direction as the incident light are positive, while those in the opposite direction are negative.

• Heights measured upwards are positive, and those measured downwards are negative.

9.2.2 Focal Length of Spherical Mirrors

9.2.2.1 Focal Point

When a parallel beam of light is incident on:

• A concave mirror, the reflected rays converge at a point called the principal focus, F.

• A convex mirror, the reflected rays appear to diverge from point F.

The distance between the focus F and the pole P of the mirror is called the focal length (f) of the mirror, given by the formula:
$f = rac{R}{2}$ where R is the radius of curvature of the mirror.

9.2.2.2 Derivation of Focal Length

Using geometry of reflection:

• Consider C as the center of curvature and an incident ray parallel to the principal axis striking the mirror at point M. The angle of incidence at M is denoted by θ.

• For small angles, we derive the relationships:

1. $an θ ext{ is approximated by } θ$ for small θ.

2. The relationship resulting from the triangles formed leads us to conclude $f = rac{R}{2}$.

9.2.3 The Mirror Equation

The mirror equation relates object distance (u), image distance (v), and focal length (f):
$rac{1}{f} = rac{1}{v} + rac{1}{u}$
This equation indicates the relationship between the distances. For a real image formed by a concave mirror, the magnification (m) is defined as the ratio of the height of the image (h') to the height of the object (h):
$m = rac{h'}{h} = - rac{v}{u}$

Examples and cases demonstrate the behaviors of images formed by concave and convex mirrors.

Example 9.1

Covering half of a concave mirror would not reduce the size of the image but would decrease its intensity.

Example 9.2

Conclude that the distortion of the image depends on the placement of the phone against the mirror.

9.3 Refraction

When a beam of light interacts with another transparent medium, part of the light reflects while the rest refracts into the new medium.

9.3.1 Snell's Law

Snell experimentally formulated:

1. The incident ray, refracted ray, and the normal at the point of incidence are coplanar.

2. The ratio of sines of the angle of incidence (i) to the angle of refraction (r) is a constant:
   $n_{21} = rac{ ext{sin } i}{ ext{sin } r}$
   where n21 is the refractive index of the second medium with respect to the first medium.

9.4 Total Internal Reflection

Light traveling from a denser medium to a rarer medium can undergo total internal reflection when the angle of incidence exceeds the critical angle (ic). The relationship is defined by:
$ext{sin } i</em>c = n<em>{21}$ When i > ic, the light is completely reflected, hence no transmission occurs.

9.4.1 Practical Applications

1. Optical fibers utilize total internal reflection for transmission of signals.

2. Prisms can bend light utilizing total internal reflection principles as well.

9.5 Refraction at Spherical Surfaces and by Lenses

Refraction at a spherical interface can be analyzed using geometric principles similar to those for spherical mirrors.

9.5.1 Lens Maker’s Formula

The relationship between the refractive indices of two media and the radii of curvature of the two surfaces of a lens is given in the lens maker’s equation:
$rac{1}{f} = (n<em>2 - n</em>1) \bigg( rac{1}{R<em>1} - rac{1}{R</em>2} \bigg)$
Where f is the focal length, R1 and R2 are the radii of curvature of the lens surfaces.

9.5.2 Thin Lens Formula

For a lens, using the previous definitions we have:
$rac{1}{v} = rac{1}{u} + rac{1}{f}$

9.7 Optical Instruments

Optical instruments are based on the principles derived from the behavior of light. Common examples include:

1. Microscopes: Utilize lenses to magnify objects.

2. Telescopes: Used to magnify distant objects.

3. Periscopes, Kaleidoscopes, and various other optical devices.

9.7.1 Microscope

A simple microscope uses a single converging lens to provide magnification by allowing the object to be positioned closer than the focal length.

Magnification for Compound Microscope

• Objective: Generates a real, inverted image.

• Eyepiece: Functions similarly to a simple microscope producing the final image.

9.7.2 Telescope

Uses an objective lens to focus light from distant sources, leading to a final image produced by an eyepiece.

Summary

1. Reflection and refraction follow specific laws.

2. The mirror equation connects the object distance, image distance, and focal length.

3. Total internal reflection is vital for technologies like optical fibers.

4. Microscopes and telescopes derive observational functions from these optical principles.

Exercises

Exercises at the end reinforce concepts from reflection, refraction, optical devices, and their calculations.