Foundation Physics 2: Reflection, Refraction, Mirrors, and Lenses Study Notes

The Nature of Light (Chapter 22-1)

• Dual Nature of Light:
      * Light exhibits properties of both a particle and a wave.
      * Classical electromagnetic wave theory provides explanations for light propagation and interference.
      * Experiments involving the interaction of light with matter are best explained by assuming light is a particle.

• Brief History of Light:
      * Early Models: It was proposed that light consisted of tiny particles.
      * Isaac Newton: Used the particle model to explain reflection and refraction.
      * Christian Huygens (1678): Proposed light was wave-like to explain many properties.
      * Thomas Young (1801): Provided strong support for wave theory by demonstrating light interference.
      * James Clerk Maxwell (1865): Proposed that electromagnetic waves travel at the speed of light.
      * Albert Einstein (1905): Reintroduced the particle nature of light to explain the photoelectric effect, utilizing Max Planck’s ideas.

• Photons and Energy:
      * Particles of light are called photons.
      * Every photon has a specific energy determined by the formula:
          $E = hf$
      * In this formula:
          * $E$ is the energy in joules ($J$).
          * $h$ is Planck’s constant, valued at $6.63 \times 10^{-34}\,Js$.
          * $f$ is the frequency of the light wave.
      * The photon encompasses both natures: it interacts like a particle but possesses a frequency like a wave.

Reflection and Refraction (Chapter 22-2)

• The Ray Approximation in Geometric Optics:
      * Light travels in a straight-line path in a homogeneous medium until it meets a boundary between two different media.
      * A ray is an imaginary line drawn along the direction of travel of light beams.
      * A wave front is a surface passing through points of a wave that share the same phase and amplitude.
      * Rays are perpendicular to wave fronts and indicate the direction of propagation.

• The Reflection of Light:
      * In reflection, part of the incident light encountering a second medium bounces off that medium and is directed backward into the first medium.
      * Specular Reflection: Reflection from a smooth surface (e.g., a mirror). Reflected rays are parallel to each other, and all light propagates in a single direction.
      * Diffuse Reflection: Reflection from a rough surface. Reflected rays travel in a variety of directions. Diffuse reflection allows a dry road to be visible at night.

• The Law of Reflection:
      * The normal is a line perpendicular to the surface at the point where the incident ray strikes.
      * The angle of incidence ($\theta_1$) is the angle the incident ray makes with the normal.
      * The angle of reflection ($\theta_1'$) is the angle the reflected ray makes with the normal.
      * The Law: The angle of reflection is equal to the angle of incidence.
          $\theta_1 = \theta_1'$

• The Refraction of Light:
      * Refraction is the bending of light as it passes from one transparent medium into another.
      * At a boundary, the light ray changes direction due to a change in speed.
      * The incident ray, reflected ray, refracted ray, and the normal all lie on the same plane.
      * The angle of refraction ($\theta_2$) depends on the properties of the media.

• Reversibility of Light Paths:
      * The path of light through a refracting surface is reversible. If a ray originated at the end point, it would follow the same path back to the origin point.

• Bending Direction in Refraction:
      * High speed to Low speed: When light moves into a material where its speed is lower, the ray bends toward the normal (\theta_2 < \theta_1).     * Low speed to High speed: When light moves into a material where its speed is higher, the ray bends away from the normal (\theta_2 > \theta_1).

The Law of Refraction (Chapter 22-3)

• Index of Refraction ($n$):
      * The index of refraction is defined as the ratio of the speed of light in a vacuum ($c$) to the speed of light in the medium ($v$):
          $n = \frac{c}{v}$
      * Speed of light in vacuum: $c = 3.00 \times 10^8\,m/s$.
      * For a vacuum, $n = 1$. For all other media, n > 1.
      * As $n$ increases, the speed of the wave in that medium decreases ($v$ is inversely proportional to $n$).

• Snell’s Law of Refraction:
      * The relationship between the indices of refraction and the angles of the light rays:
          $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$
      * The greater the difference in the index of refraction between two materials, the greater the change in the direction of propagation.

• Frequency, Wavelength, and Speed Relations:
      * As light travels from one medium to another, its frequency ($f$) does not change.
      * Because $v = f\lambda$, both speed ($v$) and wavelength ($\lambda$) change significantly.
          $\frac{\lambda_1}{\lambda_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$
          $\lambda_1 n_1 = \lambda_2 n_2$

• Selected Indices of Refraction (at $20^\circ C$, $\lambda = 589\,nm$):
      * Solids: Diamond ($2.419$), Crown Glass ($1.52$), Flint Glass ($1.66$), Ice ($1.309$ at $0^\circ C$), Zircon ($1.923$).
      * Liquids: Benzene ($1.501$), Water ($1.333$), Ethyl Alcohol ($1.361$), Glycerine ($1.473$).
      * Gases: Air ($1.000293$), Carbon Dioxide ($1.00045$).

Total Internal Reflection (Chapter 22-7)

• Critical Angle ($\theta_c$):
      * Internal reflection occurs when light attempts to move from a medium with a higher index of refraction to one with a lower index of refraction (n_1 > n_2).
      * The critical angle is the angle of incidence that results in an angle of refraction of $90^\circ$.
          $\sin(\theta_c) = \frac{n_2}{n_1}$

• Conditions for Total Internal Reflection:
      * The angle of incidence must be greater than the critical angle (\theta_1 > \theta_c).
      * The beam is then entirely reflected at the boundary and obeys the Law of Reflection.

• Fiber Optics Applications:
      * Light travels through curved transparent rods via multiple total internal reflections.
      * Light Piping: Solid glass or plastic rods are used to pipe light from one location to another.
      * Medical: Used for diagnosis and surgery.
      * Telecommunications: Strands of glass optical fibers carry voice, video, and data signals.

Dispersion and Prisms (Chapter 22-4)

• Definition of Dispersion:
      * The index of refraction in anything except a vacuum depends on the wavelength of light. This dependence is called dispersion.
      * Generally, the index of refraction decreases as wavelength increases.

• Refraction via Prism:
      * Violet light (shorter wavelength) refracts more than red light (longer wavelength).
      * The angle of deviation ($\delta$) is the amount the ray is bent away from its original path.
      * Violet deviates the most; red deviates the least.
      * A prism spectrometer uses this phenomenon to study the wavelengths emitted by light sources.

The Rainbow (Chapter 22-5)

• Formation Details:
      1. A ray of light strikes a water drop in the atmosphere.
      2. First Refraction: Occurs at the front of the drop (Violet deviates most, Red deviates least).
      3. Reflection: Occurs at the back surface of the water drop.
      4. Second Refraction: Occurs as the light leaves the drop return into the air.

• Observation Angles:
      * The angle between the incoming white light and the outgoing violet ray is $40^\circ$.
      * The angle between the incoming white light and the outgoing red ray is $42^\circ$.
      * Red is seen from drops higher in the sky; violet is seen from drops lower in the sky.

Flat Mirrors (Chapter 23-1)

• Notation:
      * Object distance ($p$): Distance from object to mirror.
      * Image distance ($q$): Distance from image to mirror.
      * Lateral Magnification ($M$): Ratio of image height ($h'$) to object height ($h$).

• Types of Images:
      * Real Image: Formed when light rays actually intersect. Can be displayed on a screen.
      * Virtual Image: Formed when light rays only appear to diverge from a point. Cannot be displayed on a screen.

• Properties of Images in Flat Mirrors:
      1. The image distance equals the object distance ($p = |q|$).
      2. The image is unmagnified ($h' = h$ and $M = 1$).
      3. The image is virtual and upright.
      4. There is an apparent left-right reversal.

• Application - Auto Mirrors:
      * Daytime setting: Silvered back surface reflects a bright ray ($B$) to the driver.
      * Night setting: Unsilvered front glass surface reflects a dim ray ($D$), while the bright beam passes through.

Spherical Mirrors (Chapter 23-2)

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• Definitions:
      * Concave Mirror: Mirrored on the inner side (caves in).
      * Convex Mirror: Mirrored on the outer side (bulges out).
      * Radius of Curvature ($R$): Radius of the sphere from which the mirror is segmented.
      * Center of Curvature ($C$): The center of that sphere.
      * Principal Axis: Line drawn from $C$ through the center of the mirror segment ($V$).

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• Mirror Equations:
      * Focal Length ($f$):
          * Concave Mirror: $f = \frac{R}{2}$
          * Convex Mirror: $f = -\frac{R}{2}$
      * The Mirror Equation:
          $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$
      * Magnification Equation:
          $M = \frac{h'}{h} = -\frac{q}{p}$

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• Spherical Aberration: Blurred images caused by rays making large angles with the mirror, causing them to converge at points other than the image point.

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• Sign Conventions:
      * $p, q, f$ are positive in front of the mirror (real side).
      * $p, q, f$ are negative behind the mirror (virtual side).

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• Imaging Characteristics Table:
  

  Mirror Type	Object Location	Orientation	Size	Type
  Convex	Arbitrary	Upright	Reduced	Virtual
  Concave	Beyond $C$	Inverted	Reduced	Real
  Concave	At $C$	Inverted	Same as object	Real
  Concave	Between $F$ and $C$	Inverted	Enlarged	Real
  Concave	Just beyond $F$	Inverted	Approaching $\infty$	Real
  Concave	Just inside $F$	Upright	Approaching $\infty$	Virtual
  Concave	Between mirror and $F$	Upright	Enlarged	Virtual
  Thin Lenses (Chapter 23-5)

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  • Types of Lenses:
        * Converging Lenses: Thicker at the center; have positive focal lengths ($f$).
        * Diverging Lenses: Thinner at the center; have negative focal lengths ($f$).

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  • Definitions:
        * Thin Lens: A lens where the thickness is negligible compared to the focal length.
        * Focal Points: A thin lens has two focal points, one on each side.

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  • Lens Equations:
        * Thin-lens Equation:
            $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$
        * Magnification:
            $M = \frac{h'}{h} = -\frac{q}{p}$

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  • Sign Conventions for Lenses:
        * Focal Length ($f$): Positive for converging; negative for diverging.
        * Magnification ($M$): Positive for upright; negative for inverted.
        * Image Distance ($q$): Positive for real images (opposite side of object); negative for virtual images (same side as object).
        * Object Distance ($p$): Positive for real objects.

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  • Imaging Characteristics Table:
    

    Lens Type	Object Location	Orientation	Size	Type
    Diverging	Arbitrary	Upright	Reduced	Virtual
    Converging	Beyond $F$	Inverted	Reduced or Enlarged	Real
    Converging	Just beyond $F$	Inverted	Approaching $\infty$	Real
    Converging	Just inside $F$	Upright	Approaching $\infty$	Virtual
    Converging	Between mirror and $F$	Upright	Enlarged	Virtual
    Ray Drawing Procedures

    • For Mirrors (Concave and Convex):
          * Ray 1: Parallel to principal axis, reflects through focal point $F$.
          * Ray 2: Through focal point $F$, reflects parallel to principal axis.
          * Ray 3: Through center of curvature $C$, reflects back on itself.

    • For Lenses (Converging and Diverging):
          * Ray 1: Parallel to principal axis, passes through (or appears to come from) the focal point.
          * Ray 2: Through the center of the lens, continues in a straight line.
          * Ray 3: Through (or directed toward) the other focal point, emerges parallel to the principal axis.

    Quantitative Examples from Transcript

    • Example 22-1 (Mirrors at angle):
          * Two mirrors at $120^\circ$. Ray incident on $M_1$ at $65^\circ$ to normal. To find angle with normal to $M_2$ after reflection from both.

    • Example 22-2 (Snell’s Law):
          * Light ($\lambda = 589\,nm$) in air incident on Crown Glass ($n=1.52$) at $\theta_1 = 30.0^\circ$.
          * $n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \Rightarrow (1.00) \sin(30.0^\circ) = (1.52) \sin(\theta_2) \Rightarrow \theta_2 = 19.2^\circ$.
          * Angle leaving glass to air: $\theta_3 = 30.0^\circ$.

    • Example 22-3 (Fused Quartz):
          * $n = 1.458$, $\lambda_{vac} = 589\,nm$.
          * Speed ($v$): $v = \frac{c}{n} = \frac{3.00 \times 10^8}{1.458} = 2.058 \times 10^8\,m/s$.
          * Wavelegnth ($\lambda_n$): $\lambda_n = \frac{\lambda_{vac}}{n} = \frac{589\,nm}{1.458} = 404\,nm$.
          * Frequency ($f$): $f = \frac{c}{\lambda_{vac}} = \frac{3.00 \times 10^8}{589 \times 10^{-9}} = 5.09 \times 10^{14}\,Hz$.

    • Example 22-6 (Critical Angle Water-Air):
          * $n_1 = 1.333$, $n_2 = 1.000$.
          * $\theta_c = \sin^{-1}(\frac{1.000}{1.333}) = 48.6^\circ$.

    • Example 23-2 (Concave Mirror):
          * $f = 10.0\,cm$. For $p = 25.0\,cm$: $\frac{1}{q} = \frac{1}{10.0} - \frac{1}{25.0} \Rightarrow q = 16.7\,cm$. $M = -\frac{16.7}{25.0} = -0.668$. (Real, inverted, reduced).

    • Example 23-3 (Convex Mirror):
          * Object height $h = 3.00\,cm$, $p = 20.0\,cm$, $f = -8.00\,cm$.
          * Position: $\frac{1}{q} = -\frac{1}{8.00} - \frac{1}{20.0} \Rightarrow q = -5.71\,cm$.
          * Magnification: $M = -\frac{-5.71}{20.0} = 0.286$.
          * Image height: $h' = M \times h = 0.857\,cm$.

    • Example 23-4 (Cosmetic Mirror):
          * $p = 40.0\,cm$, $M = 2$ (upright). find $f$.
          * $M = -\frac{q}{p} \Rightarrow 2 = -\frac{q}{40.0} \Rightarrow q = -80.0\,cm$.
          * $\frac{1}{f} = \frac{1}{40.0} + \frac{1}{-80.0} = \frac{1}{80.0} \Rightarrow f = 80.0\,cm$.

    Questions & Discussion

    • Fish Vision Under Water:
          * If a fish looks up toward the water surface ($\theta_c = 48.6^\circ$):
          * At $40.0^\circ$: The fish sees the sky/objects above the surface (refraction).
          * At $48.6^\circ$: The light skims the surface ($90^\circ$ refraction).
          * At $60.0^\circ$: The fish sees a reflection of the bottom of the pond (total internal reflection).