Exhaustive Study Notes on Thin Lenses, Optical Instruments, and Human Vision

228 Lecture 4 Overview

Instructor: S-W. Cheong.
Date: Feb. 2, 2026.
Primary Topics: * Thin Lenses (Converging and Diverging). * The Human Eye and Corrective Lenses. * Optical Instruments: The Magnifier (Magnifying Glass), Telescopes, and Microscopes.
Comparison of Imaging Tools: * Optical Microscope: Uses light to resolve features. * Electron Microscope: Uses electrons to achieve much higher resolution. * A scale bar of $5\,\mu m$ is provided in an example comparison between microscope types.

Fundamental Lens Concepts and Definitions

Thin Lenses Definition: A lens is considered "thin" if its physical thickness is small when compared to its radius of curvature.
Lens Types: * Converging Lenses: These are thicker in the center than they are at the edges. They bring parallel rays of light together to a specific focus point. * Diverging Lenses: These are thicker at the edges than they are in the center. They cause parallel light rays to spread out (diverge).
Focal Point ( $F$ ): * For a converging lens, this is the point where parallel rays intersect after passing through the lens. * For a diverging lens, the focal point is the point from which the diverging rays would appear to originate if they were projected backward along their path.

Ray Tracing for Thin Lenses

Ray tracing for thin lenses utilizes three principal rays, similar to the method used for mirrors. The intersection of these rays determines the location and size of the image.

Three Key Rays: 1. Ray 1 (Parallel Ray): A ray travelling parallel to the principal axis. Upon refraction through the lens, it will pass through (or appear to come from) the focal point. 2. Ray 2 (Focal Ray): A ray passing through the focal point. After refraction, it will exit the lens travelling in a direction parallel to the principal axis. 3. Ray 3 (Center Ray): A ray that passes directly through the center of the lens. This ray will continue through the lens undeflected.
Diverging Lens Specifics: The same three rays are used for diverging lenses, typically resulting in an image that is upright and virtual.

Lens Power

Definition: The power of a lens ( $P$ ) is defined as the inverse of its focal length ( $f$ ).
Formula: $\text{P} = \frac{1}{f}$
Units: Lens power is measured in diopters ( $D$ ). $1\,D = 1\,m^{-1}$
Sign Conventions for Power: * Converging lenses have positive power. * Diverging lenses have negative power.

The Thin Lens and Magnification Equations

The Thin Lens Equation: This mirrors the calculation used for mirrors, relating object distance ( $d_o$ ), image distance ( $d_i$ ), and focal length ( $f$ ). $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$
The Magnification Equation: This relates the height of the image ( $h_i$ ) and height of the object ( $h_o$ ) to their respective distances from the lens. $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

Sign Conventions for Lenses

Focal Length ( $f$ ): * Positive (+$) for converging lenses. * Negative (-) for diverging lenses.
Object Distance (d_o): * Positive (+$) when the object is on the same side as the light entering the lens. * Negative ( $-$ ) otherwise.
Image Distance ( $d_i$ ): * Positive (+$) if the image is on the opposite side from the light entering the lens (Real Image). * Negative (-) if the image is on the same side as the light entering the lens (Virtual Image).
Image Height (h_i): * Positive (+$) if the image is upright. * Negative ( $-$ ) otherwise (inverted).

Combinations of Lenses and Problem Solving

Lens Combinations: When using multiple lenses, the image formed by the first lens serves as the object for the second lens. In these scenarios, it is possible for object distances ( $d_o$ ) to be negative.
Problem Solving Strategy: 1. Draw a complete ray diagram. The image is located where the key rays intersect. 2. Explicitly solve for unknowns using the lens and magnification equations. 3. Strictly adhere to the sign convention. 4. Verify that mathematical results are consistent with the visual ray diagram.

The Lensmaker’s Equation

This equation relates the focal length to the material property (index of refraction) and the geometry of the lens surfaces ( $R_1$ and $R_2$ ).

Formula: $\frac{1}{f} = (n - 1) \times \left( \frac{1}{R_1} + \frac{1}{R_2} \right)$
Parameters: * $n$ : Index of refraction of the lens material. * $R_1, R_2$ : Radii of curvature of the two lens surfaces.

The Human Eye and Vision Correction

Anatomy Comparison: The eye functions similarly to a camera, featuring an adjustable lens, an iris (aperture), and a retina (sensor/film).
Refractive Indices within the Eye: * Liquid/Humor: $n = 1.336$ * Crystalline Lens: $n = 1.437$
Focusing Mechanism: Most refraction occurs at the cornea's surface. The internal lens makes minor adjustments to focus on objects at various distances (accommodation). * Relaxed Lens: Focused on distant objects.
Vision Benchmarks: * Near Point: The closest distance at which the eye can focus clearly. For a normal eye, this is approximately $25\,cm$ . * Far Point: The farthest distance at which an object can be seen clearly. For a normal eye, this is at infinity ( $\infty$ ).
Common Vision Impairments: * Nearsightedness (Myopia): The far point is too close; distant objects are blurry. Corrected with a diverging lens. * Farsightedness (Hyperopia): The near point is too far away; close objects are blurry. Corrected with a converging lens (often a converging meniscus lens).
Vision Underwater: Refraction is significantly reduced when light enters the eye from water rather than air, causing blurry vision. This is corrected by wearing goggles to maintain an air-cornea interface.

Magnifying Glass and Angular Magnification

Definition: A simple magnifier is a converging lens that allows an object to be placed closer than the standard near point, creating a larger image on the retina.
Angular Magnification ( $M$ ): Defined as the ratio of the angle subtended by the image ( $\theta'$ ) to the angle subtended by the object at the near point ( $\theta$ ). $M = \frac{\theta'}{\theta}$
Relaxed Eye Magnification: When the eye is focused at infinity, the magnification is given by: $M = \frac{\theta'}{\theta} = \frac{h/f}{h/N} = \frac{N}{f}$ * Where $N = 25\,cm$ (standard near point).

Telescopes

Structure: A refracting telescope consists of two lenses at opposite ends of a tube. * Objective Lens: The lens closest to the object being viewed. * Eyepiece: The lens closest to the user's eye.
Telescope Magnification Formula: $M = \frac{\theta'}{\theta} = -\frac{f_o}{f_e}$ * $f_o$ : Focal length of the objective lens. * $f_e$ : Focal length of the eyepiece.

Questions and Discussion

Question 1: Image Description Which pair of terms most accurately describes the image when the object distance is greater than the focal length (d_o > f), image height is negative ( $h_i < 0$ ), and image distance is positive ( $d_i > 0$ )? * Answer: b) real, inverted.
Question 2: Image Size Calculation What is the size of the image of a $7.6\text{-cm}$ -high leaf placed $1.00\,m$ from a $+50.0\text{-mm}$ -focal-length lens? * Calculation Steps: * Convert units: $h_o = 7.6\,cm$ , $d_o = 1000\,mm$ , $f = 50\,mm$ . * Use $\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}$ . * $m = -\frac{d_i}{d_o}$ . * Answer: b) $-0.40\,cm$ .
Question 3: Farsightedness Correction Sue is farsighted with a near point of $100\,cm$ . Reading glasses must have what lens power so she can read a newspaper at $25\,cm$ ? (Assume lens is very close to the eye). * Logic: The lens must take an object at $d_o = 25\,cm$ and create a virtual image at her near point ( $d_i = -100\,cm$ ). * Calculation: $\frac{1}{f} = \frac{1}{25\,cm} + \frac{1}{-100\,cm} = \frac{4}{100\,cm} - \frac{1}{100\,cm} = \frac{3}{100\,cm} = 0.03\,cm^{-1}$ . * Converted to meters: $3.0\,m^{-1}$ . * Answer: c) $+3\,D$ .