Lecture 28: Reflection and Refraction

The Principle of Least Time

Definition: Light travels from one point to another using the principle of least time. This means that light does not necessarily take the shortest distance between two points; rather, it will travel whatever path takes the least amount of time.
Constant Media Constraints:
- In a single medium (such as air throughout the entire journey), the path of least time is identical to the path of least distance because the speed of light remains constant.
- The calculation is based on the formula: $t = \frac{d}{v}$ . If velocity ( $v$ ) is constant, the minimum time ( $t$ ) occurs at the minimum distance ( $d$ ).
Varying Media (The Lifeguard Analogy):
- When light travels through different media where its speed changes, the path of least distance and the path of least time are no longer the same.
- The Lifeguard Problem:
  - A lifeguard on a beach needs to reach a drowning swimmer in the water as quickly as possible.
  - The lifeguard runs faster on sand than she swims in water.
  - A straight-line path (least distance) is sub-optimal because it requires spending too much time in the slower medium (water).
  - An alternative involving minimizing water distance might maximize land distance too much, which is also sub-optimal.
  - The optimal path (least time) involves an optimization: spending more time on the beach and less time in the water without making the overall path excessively long.
Application to Light: When the speed of light changes during its journey from Point A to Point B, the light will bend to create the path of least time, resulting in the phenomenon of refraction.

Reflection and the Law of Reflection

Basic Concept: Reflection occurs when light bounces off a surface.
Selective Reflection: As discussed in previous lectures, the color of opaque objects is determined by selective reflection and absorption. For example, a blue object reflects blue light while absorbing red and green light.
The Law of Reflection:
- Due to the principle of least time and the fact that the speed of light stays constant during reflection within a single medium, the path of least distance equals the path of least time.
- Formal Statement: The angle of reflection is equal to the angle of incidence.
Geometric Measurement:
- Angles are measured with respect to the Normal Line.
- The Normal Line is an imaginary line positioned at a right angle ( $90^\circ$ ) to the mirror's surface at the point of incidence.
- Notation:
  - $\theta_i$ represents the incident angle (measured between the light ray and the normal).
  - $\theta_r$ represents the reflected angle.
  - The relationship is expressed as: $\theta_i = \theta_r$ .
Types of Reflection:
- Specular Reflection: Reflection that occurs off a smooth, polished surface (such as a mirror). Light rays remain organized.
- Diffuse Reflection: Reflection that occurs when a surface is rough. While light still obeys the law of reflection locally, the rays bounce back in scattered directions, preventing the formation of a clear image.

Images and Mirrors

Image Definition: When light reflects, the reflected light has properties that make it appear as though the rays were generated by a specific spot. This perceived source is called an image.
Image Types:
- Virtual Image: An image that appears to be located behind the mirror.
- Real Image: An image that is located in front of the mirror (on the same side as the object).
Planar Mirrors:
- These are flat, smooth reflective surfaces.
- Light obeys the law: $\theta_i = \theta_r$ .
- Tracing reflected rays backward shows they converge at points behind the mirror.
- Properties: The image is virtual, upright, and the same size as the object.
Concave Mirrors (Curved Inward):
- Possess a Focal Point ( $F$ ) where all parallel rays converge.
- Object between Focal Point and Mirror: Creates a virtual image located behind the mirror that is upright and much larger than real life (e.g., makeup or shaving mirrors).
- Object past the Focal Point: The rays converge on the same side of the mirror as the object. This creates a real image that is upside down (inverted) and larger than the object (e.g., funhouse mirror effects).
Convex Mirrors (Curved Outward):
- The focal point is located behind the mirror.
- Properties: Creates a virtual image behind the mirror that is upright and smaller than the object.
- Applications: Used in parking garages for tight corners and as side mirrors in cars. Because images are smaller, they provide a wider field of view, leading to the warning: "Objects in mirror are closer than they appear."
Mirror Composition and One-Way Mirrors:
- Standard Mirror: A smooth reflective metal surface (like silver) covered with protective glass. A gap is visible between a finger and its reflection due to the glass thickness.
- One-Way Mirror: A piece of glass separating a dark area and a light area.
  - On the Light Side: Reflection is visible because reflected light outweighs the small amount of light transmitting from the dark side.
  - On the Dark Side: The observer sees transmitted light from the bright side, making it look like a window. There is no metal reflecting surface, so there is no gap when touching the glass.

Refraction and Snell's Law

Definition: Refraction is the bending of light that occurs when it travels through different media.
Index of Refraction ( $n$ ):
- This is the ratio of the speed of light in free space to the speed of light in a specific medium.
- Formula: $n = \frac{c}{V}$
  - $c = 3 \times 10^8\,\text{m/s}$ (speed of light in free space).
  - $V$ = speed of light in the particular medium.
- Examples: Water ( $n = 1.3$ ), Diamond ( $n = 2.4$ ).
- Bending Behavior:
  - A low index of refraction bends light less; a high index bends light more.
  - From lower to higher $n$ (e.g., Air to Glass): Light bends toward the normal line ( $\theta_{refraction} < \theta_{incidence}$ ).
  - From higher to lower $n$ (e.g., Water to Air): Light bends away from the normal line ( $\theta_{refraction} > \theta_{incidence}$ ).
Snell's Law of Refraction:
- States that the index of refraction of the first medium times the sine of the incident angle equals the index of refraction of the second medium times the sine of the angle of refraction.
- Formula: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$
Practical Examples:
- A straw in a glass of water appearing broken.
- Eyeglasses: Nearsightedness occurs when light focuses in front of the retina; lenses refract light to focus it precisely on the retina.

Dispersion and Total Internal Reflection

Dispersion:
- The index of refraction ( $n$ ) is actually a function of wavelength ( $\lambda$ ). Different colors of light bend by different amounts when passing through a medium.
- Prisms: White light is split into a rainbow because violet light bends the most through glass while red light bends the least.
- This is the physical cause of rainbows in raindrops.
Total Internal Reflection (TIR):
- Occurs when light travels from a high index of refraction to a lower index of refraction.
- Critical Angle ( $\theta_c$ ): The specific incident angle where light refracts at exactly $90^\circ$ along the interface.
- Total Internal Reflection Definition: When the angle of incidence is greater than the critical angle, the light does not refract but instead reflects entirely back into the higher-index medium.
- Applications:
  - Fiber Optics: Light is trapped inside glass fibers by TIR, allowing it to carry telecommunication signals over long distances.
  - Diamonds: Cut at specific angles to ensure light is trapped via TIR, creating sparkle.
  - Underwater observation: The surface of the water looks like a mirror when viewed from underneath at certain angles.

Lenses

Comparison to Mirrors:
- Virtual Image: On the same side of the lens as the object.
- Real Image: On the opposite side of the lens as the object.
Biconcave Lenses (Diverging):
- Both sides are curved inward.
- Parallel light waves bend outward (diverge).
- Possess a virtual focal point in front of the lens.
- Image Properties: Virtual, upright, and smaller than the object.
Biconvex Lenses (Converging):
- Both sides are curved outward.
- Parallel light waves bend inward (converge) to a focal point on the opposite side.
- Object beyond the focal point: Creates a real image on the opposite side that is inverted and larger than the object.
- Object between focal point and lens: Creates a virtual image on the same side that is upright and magnified.
Common Applications: Microscopes, telescopes, eyeglasses, magnifying glasses, and cameras.