NCEA Level 2 Physics Waves Study Notes

FUNDAMENTAL WAVE AND OPTICS FORMULAE

• Lens and Mirror Equations (Descartes' Method)

  • Focal Length: $\frac{1}{f} = \frac{1}{d_i} + \frac{1}{d_o}$
  • Where $f$ is focal length (m), $d_o$ is object distance (m), and $d_i$ is image distance (m).

• Magnification Formulae

  • Basic Magnification: $m = \frac{h_i}{h_o}$
  • Ratio of Distances: $m = \frac{d_i}{d_o}$
  • Using Newton's Method components: $m = \frac{f}{S_o} = \frac{S_i}{f}$
  • Where $h_i$ is image height (m) and $h_o$ is object height (m).

• Newton's Method (Focal Point Distances)

  • Equation: $S_i S_o = f^2$
  • Where $S_o$ is the object-focal distance (m) and $S_i$ is the image-focal distance (m).

• Refraction and Snell's Law

  • Law of Refraction: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$
  • Refractive Index Relationships: $\frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2} = \frac{\sin(\theta_1)}{\sin(\theta_2)}$
  • Definition of Refractive Index: $n = \frac{c}{v}$
  • Speed of Light in a vacuum ($c$): $3.00 \times 10^8\,ms^{-1}$

• General Wave Equations

  • Velocity, Frequency, and Wavelength: $v = f \lambda$
  • Period and Frequency: $f = \frac{1}{T}$
  • Velocity, Distance, and Time: $v = \frac{d}{t}$

THE NATURE OF LIGHT

• Electromagnetic Wave Definition: Light is a transverse electromagnetic wave consisting of fluctuating electric and magnetic fields.

• Wavelength Categorization: Behavior depends on wavelength, divided into 7 groups. Named examples include:

  • Radio/Communication: AM, FM, WiFi, Bluetooth, Radar.
  • Thermal/Control: Microwave ovens, TV remotes.
  • High Energy/Medical: Curing resin, Sterilisation, Radiotherapy, Imaging (PET scans).

• Propagation Rules:

  • Velocity: Light travels at exactly $300,000,000\,ms^{-1}$ ($3 \times 10^8\,ms^{-1}$) in a vacuum.
  • Medium Requirements: Unlike sound, light does not require a medium (matter) to travel through.

REFLECTION BASICS AND IMAGE NATURE

• Law of Reflection: Waves reflect symmetrically. Mathematically, the angle of incidence equals the angle of reflection ($\theta_i = \theta_r$). These angles are measured relative to the "Normal Line" (a line perpendicular to the surface).

• Describing the Nature of Images: Images are characterized in three specific ways:

  • Size: Diminished (smaller than object), Same size, or Enlarged (larger than object).
  • Orientation: Upright (pointing the same way as the object) or Inverted (flipped upside down).
  • Type:
  • Real image: Rays actually meet at a point; the image can be projected onto a screen.
  • Virtual image: Rays do not meet but only appear to meet; the image cannot be projected.

CURVED MIRRORS AND RAY DIAGRAMS

• Types of Mirrors:

  • Concave Mirrors: Converging mirrors that focus incoming parallel rays.
  • Convex Mirrors: Diverging mirrors that spread incoming rays apart.
  • Parabolic Mirrors: Specifically designed to focus rays to a single, precise point. Spherical mirrors do this only approximately.

• Ray Diagram Components:

  • Principle Axis: The horizontal line passing through the center of the mirror/lens.
  • Focal Point ($f$): The point where parallel rays converge or appear to diverge from.
  • Centre of Curvature ($C$): Located at double the focal length ($C = 2f$).

• Ray Construction Rules:

  • Ray 1: Drawn through or towards the focal point, then it reflects/refracts parallel to the principle axis.
  • Ray 2: Drawn parallel to the axis, then it reflects/refracts through or towards the focal point.
  • Ray 3: Drawn symmetrically through the middle (point where axis meets mirror/lens).

• Image Characteristics by Mirror Type:

  • Convex Mirrors: Always produce Virtual, Upright, and Diminished images.
  • Concave Mirrors:
  • Object outside $f$: Real, Inverted image.
  • Object at $f$: No image formed (rays are parallel).
  • Object inside $f$ (between mirror and $f$): Virtual, Upright, and Enlarged image.

MATHEMATICAL MODELLING OF MIRRORS AND LENSES

• Descartes' Sign Convention:

  • Focal Length ($f$): Positive for Concave mirrors/Convex lenses (Converging). Negative for Convex mirrors/Concave lenses (Diverging).
  • Image Distance ($d_i$): Positive for Real images (in front of mirror/behind lens). Negative for Virtual images (behind mirror/in front of lens).

• Newton's Method Nuances:

  • Uses distances from the focal point ($S_o$ and $S_i$) rather than the surface.
  • Negative values have no meaning in the Newton equation ($S_i S_o = f^2$). To determine image nature, a ray diagram is required.

• Magnification Interpretation:

  • Magnification ($m$) is the image height/distance expressed as a fraction of the object height/distance.
  • Value examples: $0.25$ (Diminished), $1.0$ (Same size), $1.75$ (Enlarged).

REFRACTION AND TOTAL INTERNAL REFLECTION

• Refraction Definition: When light enters a different medium and changes speed, it changes direction. The extent of bending is determined by the refractive index ($n$).

• Total Internal Reflection (TIR):

  • Occurs when moving from a "slow" (high $n$) medium to a "fast" (low $n$) medium.
  • As the incident angle ($\theta_1$) increases, the refracted ray bends further from the normal.
  • Critical Angle ($\theta_c$): The specific incident angle where the refracted ray travels exactly along the boundary ($\theta_2 = 90^{\circ}$).
  • Calculation: $\sin(\theta_c) = \frac{n_2}{n_1}$, or $\theta_c = \sin^{-1}(\frac{n_2}{n_1})$.
  • If $\theta_i > \theta_c$, no refraction occurs; all light is reflected back into the first medium (TIR).

• Apparent Depth:

  • Objects submerged in water appear shallower than they actually are.
  • Light rays speed up as they exit water to air, bending away from the normal.
  • Human eyes assume light travels in straight lines and require two eyes to perceive the depth where the projected rays meet.

• Dispersion:

  • Refractive index is wavelength-dependent. Shorter wavelengths refract at greater angles.
  • "Violet refracts more violently."
  • White light striking a prism separates into a rainbow due to this effect.

WAVE MOTION AND CLASSIFICATION

• Definition: A wave is a propagating disturbance that relocates energy without relocating matter.

• Mechanical Waves: Require a medium (e.g., water, sound).

• Electromagnetic Waves: Do not require a medium (e.g., light).

• Wave Types:

  • Transverse Waves: Oscillate at $90^{\circ}$ to the direction of travel (e.g., water, light). Features Peaks and Troughs.
  • Longitudinal Waves: Oscillate in the same direction as travel (e.g., sound). Features Compressions (Peaks) and Expansions (Troughs).

• Sound Properties:

  • Represented as a transverse pressure wave for convenience.
  • Pitch corresponds to frequency. Frequency is inversely proportional to wavelength ($f \propto \frac{1}{\lambda}$).
  • Speed depends on density: Air ($340\,ms^{-1}$), Water ($1500\,ms^{-1}$), Iron ($3000\,ms^{-1}$).

WAVEFRONT DYNAMICS AND INTERFERENCE

• Wavefront Reflection and Refraction:

  • Wavefront lines represent wave peaks.
  • When drawing, the ray should be drawn first, with wavefronts kept perpendicular to the ray.
  • In refraction, the wavelength (distance between wavefronts) changes with velocity.

• Diffraction: The bending of waves around boundaries.

  • Large wavelengths bend more than short wavelengths.
  • Passing through gaps: A small gap allows less light but creates more radial (circular) diffraction. A large gap allows more light but exhibits less diffraction at the center.

• Phase and Superposition:

  • Phase: A way of comparing points in a wave cycle ($360^{\circ}$ or $2\pi$ per cycle).
  • Superposition: When waves occupy the same position, their amplitudes add together.
  • Constructive Interference: Waves combine to form a larger amplitude (In phase, peaks meet peaks).
  • Destructive Interference: Waves combine to form a smaller amplitude or cancel out ($180^{\circ}$ out of phase, peaks meet troughs).

• Standing Waves: Produced when identical waves travelling in opposite directions overlap (e.g., a reflected wave).

  • Nodes: Points of destructive interference/cancellation (zero amplitude).
  • Antinodes: Points of constructive interference (maximum amplitude).

• 2D Interference Patterns:

  • Two adjacent sources create Antinodal lines (constructive) and Nodal lines (destructive).
  • Path Difference: The difference in distance from a point to each source, expressed in wavelengths ($\lambda$).
  • Antinodal Line: Path difference is a whole number ($0, 1, 2...$\lambda).
  • Nodal Line: Path difference is a half-number ($0.5, 1.5, 2.5...$\lambda).
  • Example: Path difference of $|17\lambda - 14.5\lambda| = 2.5\lambda$ indicates a Nodal line.