Study Guide: Light, Ray Optics, and Wave Optics

Ch 15: Traveling Waves and Light (Selected Topics)

Wave Model of Light

The wave model describes light as a transverse electromagnetic wave, which helps explain many of its properties, such as interference and diffraction.

• Wave Properties: Light exhibits wavelength (λ), frequency (f), speed (v), amplitude, and phase.

• Electromagnetic Waves: Light is an oscillation of electric and magnetic fields, propagating through space.

• Speed of Light: In a vacuum, light travels at $c = 3.00 \times 10^8\ \mathrm{m/s}$.

• Relationship: $c = \lambda f$

• Example: Red light with $\lambda = 700\ \mathrm{nm}$ has $f = c/\lambda$.

Graphical and Mathematical Representations of Waves

Waves can be represented graphically (as sinusoidal curves) and mathematically (using equations).

• Wave Equation: $y(x, t) = A \sin(kx - \omega t + \phi)$

• Parameters:

  • Amplitude (A): Maximum displacement

  • Wave number (k): $k = \frac{2\pi}{\lambda}$

  • Angular frequency (\omega): $\omega = 2\pi f$

  • Phase (\phi): Initial angle at $t=0$

• Example: A wave with $A = 2\ \mathrm{cm}$, $\lambda = 0.5\ \mathrm{m}$, $f = 2\ \mathrm{Hz}$ is $y(x, t) = 2 \sin(4\pi x - 4\pi t)$.

Ch 16: Superposition and Standing Waves (Section 1)

Principle of Superposition

When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the displacements due to each wave.

• Constructive Interference: Waves add to produce a larger amplitude.

• Destructive Interference: Waves add to produce a smaller (or zero) amplitude.

• Standing Waves: Formed by the superposition of two waves traveling in opposite directions with the same frequency and amplitude.

• Example: Two sound waves of equal amplitude and frequency traveling in opposite directions create nodes and antinodes.

Ch 17: Wave Optics (Sections 1-4)

Models of Light

Light can be described using three models, each useful in different contexts:

• Ray Model: Light travels in straight lines; explains reflection and refraction.

• Wave Model: Light as a wave; explains interference and diffraction.

• Particle Model: Light as photons; explains photoelectric effect.

Propagation of Light

• In Vacuum: Light travels at $c = 3.00 \times 10^8\ \mathrm{m/s}$.

• In Media: Speed decreases: $v = \frac{c}{n}$, where $n$ is the index of refraction.

• Index of Refraction: $n = \frac{c}{v}$

• Example: In water ($n = 1.33$), $v = 2.26 \times 10^8\ \mathrm{m/s}$.

Double-Slit Interference

When light passes through two slits, it produces an interference pattern of bright and dark fringes.

• Condition for Bright Fringes (Constructive): $d \sin \theta = m \lambda$

• Condition for Dark Fringes (Destructive): $d \sin \theta = (m + 0.5) \lambda$

• Variables: $d$ = slit separation, $\theta$ = angle to fringe, $m$ = order (integer), $\lambda$ = wavelength

• Example: For $d = 0.5\ \mathrm{mm}$, $\lambda = 600\ \mathrm{nm}$, $m = 1$, $\sin \theta = 0.0012$

Diffraction Gratings

Diffraction gratings consist of many closely spaced slits, producing sharp interference maxima.

• Grating Equation: $d \sin \theta = m \lambda$

• Application: Used in spectroscopy to separate light into its component wavelengths.

• Example: A grating with 1000 lines/mm, $d = 1 \mu\mathrm{m}$.

Thin Film Interference

Interference occurs when light reflects from the top and bottom surfaces of a thin film, such as oil on water.

• Path Difference: $2 t n$ (where $t$ is film thickness, $n$ is refractive index)

• Constructive Interference: $2 t n = m \lambda$ (with phase change considered)

• Destructive Interference: $2 t n = (m + 0.5) \lambda$

• Example: Soap bubbles show colors due to thin film interference.

Ch 18: Ray Optics (Sections 1-7)

Ray Model of Light and Ray Diagrams

The ray model treats light as straight lines (rays) that can be reflected or refracted at surfaces.

• Ray Diagrams: Used to locate images formed by mirrors and lenses.

• Example: Drawing rays for a converging lens to find the image position.

Specular Reflection and Law of Reflection

Reflection occurs when light bounces off a surface. Specular reflection is from smooth surfaces.

• Law of Reflection: $\theta_i = \theta_r$ (angle of incidence equals angle of reflection)

• Example: A mirror reflects a ray at the same angle as it arrives.

Refraction and Snell's Law

Refraction is the bending of light as it passes from one medium to another.

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Total Internal Reflection: Occurs when light cannot exit a medium and is reflected entirely inside.

• Critical Angle: $\sin \theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$)

• Example: Light in water ($n=1.33$) to air ($n=1.00$): $\theta_c = 48.8^\circ$

Ray Tracing for Lenses and Mirrors

Ray tracing is a graphical method to determine the location and size of images formed by lenses and mirrors.

• Principal Rays: Standard rays used for construction (e.g., parallel to axis, through center, through focus).

• Example: For a converging lens, parallel rays pass through the focal point after refraction.

Real vs. Virtual Images and Sign Conventions

Images can be real (formed by actual convergence of rays) or virtual (apparent, formed by extensions of rays).

• Real Image: Can be projected onto a screen; rays converge.

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

• Sign Convention: Positive and negative values for object/image distances and focal lengths, depending on lens/mirror type.

• Example: A plane mirror forms a virtual image behind the mirror.

Magnification

Magnification describes how much larger or smaller the image is compared to the object.

• Magnification Equation: $M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

• Variables: $h_i$ = image height, $h_o$ = object height, $d_i$ = image distance, $d_o$ = object distance

• Example: If $d_i = -20\ \mathrm{cm}$, $d_o = 40\ \mathrm{cm}$, $M = 0.5$

Thin-Lens Equation

The thin-lens equation relates object distance, image distance, and focal length for lenses and spherical mirrors.

• Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Variables: $f$ = focal length, $d_o$ = object distance, $d_i$ = image distance

• Example: For $f = 10\ \mathrm{cm}$, $d_o = 30\ \mathrm{cm}$, $d_i = 15\ \mathrm{cm}$

Ch 19: Optical Instruments (Section 2)

The Eye and Vision Correction

The human eye is an optical instrument that forms images on the retina. Vision defects can be corrected with lenses.

• Myopia (Nearsightedness): Eye focuses images in front of the retina; corrected with diverging (concave) lenses.

• Hyperopia (Farsightedness): Eye focuses images behind the retina; corrected with converging (convex) lenses.

• Presbyopia: Age-related loss of accommodation; corrected with reading glasses (converging lenses).

• Astigmatism: Irregular curvature of the cornea or lens; corrected with cylindrical lenses.

• Example: A person with myopia uses glasses with $f = -2.0\ \mathrm{m}$ to see distant objects clearly.

Summary Table: Key Equations in Light and Optics

Concept	Equation	Description
Speed of Light	$c = 3.00 \times 10^8\ \mathrm{m/s}$	Speed in vacuum
Wave Equation	$y(x, t) = A \sin(kx - \omega t + \phi)$	General wave form
Index of Refraction	$n = \frac{c}{v}$	Ratio of speeds
Snell's Law	$n_1 \sin \theta_1 = n_2 \sin \theta_2$	Refraction at interface
Double-Slit Interference	$d \sin \theta = m \lambda$	Bright fringe condition
Thin-Lens Equation	$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$	Image formation
Magnification	$M = -\frac{d_i}{d_o}$	Image/object size ratio