Physics Notes: Lesson 1 - Reflection of Light

Reflected Images and Historical Context

When observing the surface of water, a clear reflection is not always visible. Disturbances such as ripples caused by wind or waves from passing boats prevent light from reflecting in a manner that produces a clear image.

Historical Development of Mirrors

Ancient Egypt (approx. 4000 years ago): The Egyptians understood that reflections required a smooth surface. They utilized polished metal mirrors to view their images.
Venice (16th Century): Artisans created mirrors by coating the back of flat glass with a thin sheet of metal.
Jean Foucault (1857): This French scientist developed a method of coating glass with silver, which allowed for the first sharp, well-defined reflected images.
Modern Mirrors: Today, mirrors are manufactured by evaporating aluminum or silver onto highly polished glass. They are essential components in lasers, telescopes, and precise optical systems.

The Law of Reflection

The fundamental principle governing reflection—whether from a still pond, a 16th-century mirror, or a modern laser mirror—is the Law of Reflection.

Definition

The Law of Reflection: The angle that a reflected ray makes as measured from the normal to a reflective surface equals the angle that the incident ray makes as measured from the same normal.

Geometric Components

Normal: An imaginary line that is perpendicular to a surface at the location where light strikes the surface.
Incident Ray: The incoming ray of light approaching the surface.
Reflected Ray: The ray that bounces off the surface.
Planar Relationship: The incident ray, the reflected ray, and the normal all lie in the same plane, which is perpendicular to the surface. Although light travels in three dimensions, the act of reflection is two-dimensional (planar).

The Wave Model of Light

In the wave model, a wavefront of light approaches a surface perpendicularly to the light ray. As each point along the wavefront reaches the surface, it reflects. Because all points travel at the same speed, they cover the same distance in the same amount of time. Consequently, the wavefront as a whole leaves the surface at an angle equal to its incident angle.

Wavelength Independence: The wavelength of light does not affect this process; the surface reflects red, green, and blue light all in the same direction.

Types of Reflection: Specular vs. Diffuse

Specular Reflection

This occurs when light reflects off a smooth surface. A surface is considered smooth if it is not rough on the scale of the wavelength of the light. In specular reflection, parallel incident light rays reflect as parallel reflected rays. Examples include mirrors and still water.

Diffuse Reflection

This occurs when light strikes a rough surface, such as a sheet of paper. While paper may appear smooth to the touch, it is rough on the scale of the wavelength of light.

Scattering: Parallel incident rays strike the surface, but because the normals at different microscopic locations are not parallel, the reflected rays are scattered in different directions.
Applicability of Law: The law of reflection still applies at every individual point on a rough surface; however, the varying orientations of the surface cause the rays to diverge. This explains why we can see a piece of paper from various angles but cannot see our reflection in it.

Objects and Plane-Mirror Images

Definitions

Object: Either a luminous source of light (like a lightbulb) or an illuminated source of light (like a person). Light typically spreads out from the source in all directions.
Plane Mirror: A flat, smooth surface from which light is reflected by specular reflection.

Virtual Images

A plane mirror produces a virtual image, which is a type of image formed by diverging light rays.

The image appears to be on the opposite side of the mirror from the object.
It is called "virtual" because no actual light rays exist at the location of the image; the light only seems to come from that point because the brain processes light as if it travels in a straight path (sight lines).
Multiple light rays from a point on an object strike the mirror and reflect; the image is located where these sight lines converge behind the mirror.

Geometric Properties of Plane-Mirror Images

Image Position and Height

Image Position ( $d_i$ ): With a plane mirror, the image position is equal to the negative of the object position.
- formula: $d_i = -d_o$
- The negative sign indicates that the image is virtual and located behind the mirror.
Image Height ( $h_i$ ): For a plane mirror, the image height is exactly equal to the object height.
- formula: $h_i = h_o$
Congruent Triangles: These properties are derived from geometric models showing that the line segments representing the object and image distances are corresponding sides of congruent triangles.

Image Orientation and Reversal

Orientation: The image is upright (same orientation as the object).
Front-to-Back Reversal: A plane mirror does not actually reverse left and right. Instead, it reverses the image such that it faces the opposite direction of the object (front-to-back reversal). When a person extends their right hand, the image's hand on the same side is extended, but since the image faces the person, that hand appears to be the image's left hand.

Example Problem: Changing the Angle of Incidence

Problem: A light ray strikes a plane mirror at an angle of $52.0^\circ$ to the normal. The mirror then rotates $35.0^\circ$ around the point of incidence so that the angle of incidence decreases. The axis of rotation is perpendicular to the plane of the rays. What is the angle between the initial and final reflected ray?

1. Analyze and Sketch:

Initial angle of incidence ( $\theta_{i, \text{initial}}$ ) = $52.0^\circ$
Change in mirror angle ( $\Delta \theta_{\text{mirror}}$ ) = $35.0^\circ$ (clockwise rotation to reduce the angle).

2. Solve:

New angle of incidence ( $\theta_{i, \text{final}}$ ) = $52.0^\circ - 35.0^\circ = 17.0^\circ$ .
By the Law of Reflection, the final angle of reflection ( $\theta_{r, \text{final}}$ ) also equals $17.0^\circ$ .
The initial reflected ray was at $52.0^\circ$ from the original normal. As the mirror rotates clockwise by $35.0^\circ$ , the normal also rotates clockwise by $35.0^\circ$ .
The reflected ray rotates by twice the angle of the mirror rotation.
Angle of rotation of reflected ray = $2 \times 35.0^\circ = 70.0^\circ$ .

Practice and Review Questions

Ground Glass: Explain why the reflection of light off ground glass changes from diffuse to specular if you spill water on it.
- Explanation: Water fills in the microscopic irregularities of the ground glass, creating a smooth surface relative to the wavelength of light.
Angle Calculation: What is the angle of incidence of a light ray reflected off a plane mirror at an angle of $35^\circ$ to the normal?
- Answer: $35^\circ$ .
Incident Ray Scenario: Suppose the angle of incidence is $42^\circ$ .
- a. What is the angle of reflection? ( $42^\circ$ )
- b. What is the angle the incident ray makes with the mirror? ( $90^\circ - 42^\circ = 48^\circ$ )
Laser Strike: A laser strikes a mirror at $38^\circ$ to the normal. If the angle of incidence increases by $5^\circ$ , what is the new angle of reflection?
- Calculation: $38^\circ + 5^\circ = 43^\circ$ .
Right Angle Mirrors: You position two plane mirrors at right angles. A light ray strikes one at $30^\circ$ to the normal and reflects to the second. What is the angle of reflection off the second mirror?
- Logic: The ray will strike the second mirror at $90^\circ - 30^\circ = 60^\circ$ . The reflection angle is $60^\circ$ .
Retro-reflector Challenge: Designing a reflector to reflect a laser beam by $180^\circ$ independent of direction. What should the angle between the mirrors be?
- Answer: $90^\circ$ .

Questions & Discussion

Focus Question: Why does light reflected from a mirror make an image while light reflected from a piece of paper does not?
- Response: A mirror has a smooth surface that allows for specular reflection, where light rays stay parallel and can converge to form a distinct image. Paper has a rough surface causing diffuse reflection, which scatters light in many directions, preventing image formation.
Critical Thinking: Explain how diffuse reflection of light off an object enables you to see that object from any angle.
- Response: Because the light is scattered in all directions (diffuse reflection), some reflected light rays will reach your eyes regardless of where you are standing relative to the object.
Image Properties: A dog looks at its image in a plane mirror. What are the position, height, and type of image?
- Response: Position ( $d_i$ ) is equal and opposite to the dog's distance from the mirror ( $-d_o$ ); height ( $h_i$ ) is equal to the dog's height ( $h_o$ ); the image is virtual.