Exam 3 Study Guide: Geometric Optics, Wave Nature of Light, and Special Relativity

Chapter 23: Geometric Optics

Ray Model of Light

The ray model of light treats light as traveling in straight lines called rays. This model is useful for understanding reflection and refraction phenomena.

• Light rays represent the direction of light propagation.

• Used to construct ray diagrams for mirrors and lenses.

• Applications: Explains image formation in optical instruments.

Law of Reflection

The law of reflection governs how light bounces off surfaces.

• Law: The angle of incidence equals the angle of reflection.

• Equation: $\theta_i = \theta_r$

• Plane mirrors: Images are virtual, upright, and same size as object.

Reflection by Plane and Spherical Mirrors

• Plane mirrors: Form virtual images behind the mirror.

• Spherical mirrors: Can be concave (converging) or convex (diverging).

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

• Focal length ($f$): Distance from mirror to focal point.

Ray Diagrams for Spherical Mirrors

• Concave mirror: Rays parallel to axis reflect through focal point.

• Convex mirror: Rays parallel to axis reflect as if they came from focal point behind mirror.

• Ray diagram steps:

  1. Draw incident ray parallel to axis.

  2. Draw reflected ray through (or from) focal point.

  3. Draw incident ray through center of curvature; reflects back on itself.

Mirror Equation and Sign Conventions

• Mirror equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ where $f$ is focal length, $d_o$ is object distance, $d_i$ is image distance.

• Sign conventions:

  • Focal length $f$ is positive for concave, negative for convex mirrors.

  • Image distance $d_i$ is positive for real images, negative for virtual images.

Snell's Law (Law of Refraction)

• Snell's Law: Describes how light bends when passing between media. $n_1 \sin \theta_1 = n_2 \sin \theta_2$ where $n_1$, $n_2$ are refractive indices; $\theta_1$, $\theta_2$ are angles to normal.

Ray Diagrams for Thin Lenses

• Convex (converging) lens: Parallel rays refract through focal point on opposite side.

• Concave (diverging) lens: Parallel rays refract as if they came from focal point on same side.

• Ray diagram steps:

  1. Draw ray parallel to axis; refracts through (or from) focal point.

  2. Draw ray through center of lens; passes straight.

Thin Lens Equation and Sign Conventions

• Thin lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Sign conventions:

  • Focal length $f$ is positive for convex, negative for concave lenses.

  • Image distance $d_i$ is positive for real images (opposite side), negative for virtual images (same side as object).

Ray Diagrams for Combination Lens Systems

• Draw image from first lens; treat as object for second lens.

• Apply thin lens equation sequentially.

• Applications: Used in microscopes and telescopes.

Chapter 24: Wave Nature of Light

Huygens' Principle

Huygens' Principle states that every point on a wavefront acts as a source of secondary wavelets. The new wavefront is the envelope of these wavelets.

• Explains reflection, refraction, and diffraction.

Young's Double Slit Experiment

Demonstrates the interference of light, showing its wave nature.

• Bright spots (constructive interference): Occur where path difference is integer multiple of wavelength.

• Dark spots (destructive interference): Occur where path difference is half-integer multiple of wavelength.

• Location of bright fringes: $d \sin \theta = m \lambda$ where $d$ is slit separation, $\theta$ is angle, $m$ is integer, $\lambda$ is wavelength.

• Location of dark fringes: $d \sin \theta = (m + 0.5) \lambda$

Visible Spectrum and Dispersion

• Visible spectrum: Range of wavelengths visible to human eye (about 400–700 nm).

• Dispersion: Separation of light into colors by prism due to wavelength-dependent refractive index.

Single Slit Diffraction

• Minima location: $a \sin \theta = m \lambda$ where $a$ is slit width, $m$ is integer (excluding zero), $\lambda$ is wavelength.

• Central maximum is brightest and widest.

Diffraction Grating

• Multiple slits produce sharp, bright fringes.

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

• Used for spectral analysis.

Polarization

• Polarization: Restriction of light vibration to a single plane.

• Occurs via reflection, transmission through polarizing filters, or scattering.

• Applications: Sunglasses, photography, LCD screens.

Chapter 26: Special Theory of Relativity

Postulates of Special Relativity

• First postulate: Laws of physics are the same in all inertial frames.

• Second postulate: Speed of light in vacuum is constant for all observers, regardless of motion.

Time Dilation

• Moving clocks run slower: $\Delta t = \gamma \Delta t_0$ where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$, $\Delta t_0$ is proper time, $v$ is relative velocity, $c$ is speed of light.

• Example: Muons created in atmosphere reach Earth's surface due to time dilation.

Length Contraction

• Moving objects appear shorter: $L = L_0 / \gamma$ where $L_0$ is proper length, $L$ is observed length.

Relativistic Momentum

• Equation: $p = \gamma m v$ where $m$ is mass, $v$ is velocity.

• Momentum increases rapidly as $v$ approaches $c$.

Mass-Energy Relationship

• Equation: $E = m c^2$ where $E$ is energy, $m$ is mass, $c$ is speed of light.

• Shows mass and energy are equivalent.

Relativistic Addition of Velocities

• Equation: $u' = \frac{u + v}{1 + \frac{uv}{c^2}}$ where $u$ is velocity in one frame, $v$ is velocity of frame, $u'$ is velocity in another frame.

• Ensures no object exceeds speed of light.

Additional info: Academic context and equations have been expanded for completeness and clarity.