Spherical Mirrors and Wave Mechanics Study Notes

Spherical and Parabolic Mirrors

• Spherical Mirrors: Formed from silvered spherical glass.
  • Convex (Diverging): Silvered on the inside.
  • Concave (Converging): Silvered on the outside.
• Key Terms:
  • Centre of curvature ($C$): The center of the original sphere.
  • Pole ($P$): The center of the mirror surface.
  • Principal axis: Line connecting $C$ and $P$.
  • Principal focus ($F$): Converging point for reflected rays (real for concave, virtual for convex).
  • Radius of curvature ($r$): Distance from $P$ to $C$.
  • Focal length ($f$): Distance from $P$ to $F$.
• Parabolic Mirrors: Used to produce wide parallel beams or converge light to a single point; common in car headlights and spotlights.

Image Formation by Spherical Mirrors

• Ray Diagram Rules: Images are located using the intersection of two rays:
  • Ray parallel to the principal axis (reflects through $F$ or appears to).
  • Ray through $C$ (reflects back along its own path).
  • Ray through $F$ (reflects parallel to the principal axis).
• Concave Mirror Image Cases:
  • Object at infinity: Real, inverted, diminished image at $F$.
  • Object at $C$: Real, inverted, same size image at $C$.
  • Object behind $C$: Real, inverted, diminished image between $C$ and $F$.
  • Object between $F$ and $C$: Real, inverted, magnified image behind $C$.
  • Object at $F$: Image at infinity.
  • Object between $F$ and $P$: Virtual, erect, magnified image behind the mirror.
• Convex Mirror Image Cases: Always formed behind the mirror; always virtual, erect, and diminished.

Applications and Magnification

• Applications: Used in satellite dishes, shaving mirrors, telescopes, and driving mirrors.
• Magnification ($M$): The ratio of image size to object size, or image distance to object distance.
  • $M = \frac{\text{height of the image}}{\text{height of the object}}$
  • $M > 1$ is magnified; $M < 1$ is diminished.
• Quantitative Examples:
  • Case 1 (Concave): Object height $5.0\,cm$, $f = 15\,cm$, distance $35\,cm$. Image location is $27\,cm$ in front of mirror, size is $3.75\,cm$ (real and inverted).
  • Case 2 (Convex): Object height $5\,cm$, $f = 15\,cm$, distance $10\,cm$. Image location is $6.0\,cm$ behind mirror, size is $3.0\,cm$ (virtual and erect). $M = 0.6$.

Introduction to Waves

• Wave: A disturbance moving through a medium.
• Electromagnetic Waves: Can travel through a vacuum (e.g., radio, X-rays, gamma rays, UV rays).
• Mechanical Waves: Require a material medium to transfer (e.g., water, sound waves).
• Wave Types:
  • Transverse Waves: Displacement is perpendicular to the direction of travel (consist of crests and troughs).
  • Longitudinal Waves: Particles vibrate parallel to the direction of travel (consist of compressions and rarefactions).

Properties and Measurement of Waves

• Characteristics:
  • Wavelength ($\lambda$): Distance between two successive points in a wave, measured in metres ($m$).
  • Frequency ($f$): Number of waves passing a point in one second, measured in Hertz ($Hz$).
  • Period ($T$): Time for a complete wave to pass; $T = \frac{1}{f}$.
  • Amplitude: Maximum displacement from the rest position.
• Wave Equation: $v = f \lambda$
• Calculation Practice:
  • Example 1: For a rope at $3\,Hz$ with a wavelength of $0.8\,m$, speed $v = 3 \times 0.8 = 2.4\,m/s$.
  • Example 2: In water with $v = 2\,ms^{-1}$, displacement-time graph yields amplitude $= 0.4\,cm$, period $T = 0.20\,seconds$, frequency $f = 5\,Hz$, and wavelength $\lambda = 0.4\,m$.