Reflection and Refraction of Light

Reflection and Refraction of Light: Study Summary

1. Light as an Electromagnetic Wave and the Electromagnetic Spectrum

• Light is an **electromagnetic (E-M) wave**.
• The frequency (f) of an electromagnetic wave is related to its wavelength (λ) and the speed of light (c) by the equation:

  c = λf

• The **electromagnetic spectrum** encompasses a wide range of wavelengths/frequencies, including:
  • Radio waves
  • Microwaves (e.g., radar)
  • Infrared
  • **Visible Light** (400 nm to 700 nm)
  • Ultraviolet
  • X-rays
  • Gamma rays

2. The Ray Model of Light

• Light often travels in straight lines.
• For geometric optics, light is represented using **rays**, which are straight lines emanating from an object.

3. Reflection; Image Formation by a Plane Mirror

Law of Reflection

• The angle of reflection (θr) equals the angle of incidence (θi). Both angles are measured with respect to the **normal** (a line perpendicular to the surface at the point of incidence).

Types of Reflection

• **Diffuse Reflection:** Occurs from a rough surface. The law of reflection still holds for each microscopic point, but since the surface normal varies, reflected rays scatter in many directions. Your eye sees reflected light at all angles.
• **Specular Reflection:** Occurs from a smooth, mirror-like surface. Reflected rays leave in a predictable, concentrated direction. Your eye must be in the correct position to see the reflection.

Image Formation by a Plane Mirror

• An image formed by a plane (flat) mirror is:
  • **Virtual:** Light does not actually pass through the image location.
  • **Upright:** The image is oriented the same way as the object.
  • **Same size** as the object.
  • Located **behind the mirror**.
  • The **image distance** (di) is equal to the **object distance** (do) from the mirror.
• To see your whole body in a plane mirror, the minimum height of the mirror must be half your height, and its lower edge must be half the distance from your eyes to your feet.

4. Formation of Images by Spherical Mirrors

Types of Spherical Mirrors

• **Concave Mirror:** Reflective surface is on the inside of the sphere (curves inward).
• **Convex Mirror:** Reflective surface is on the outside of the sphere (curves outward).

Focal Point and Spherical Aberration

• **Focal Point (F):** For a spherical mirror with small curvature, parallel rays striking the mirror converge to a single point after reflection (for concave mirrors) or appear to diverge from a single point (for convex mirrors).
• **Spherical Aberration:** If the curvature of a spherical mirror is large, parallel rays do not all converge at exactly the same place, leading to a blurry image. This can be avoided by using parabolic reflectors.
• **Focal Length (f):** The distance from the mirror to the focal point. For spherical mirrors, it is half the radius of curvature (r):

  f = r/2

Ray Diagrams for Spherical Mirrors

To determine image location, use at least two (and ideally three) key rays from a point on the object:

1. A ray parallel to the principal axis; after reflection, it passes through the focal point (F).
2. A ray through the focal point (F); after reflection, it is parallel to the principal axis.
3. A ray perpendicular to the mirror (passing through the center of curvature, C); it reflects back on itself.

The intersection of these rays (or their extensions) gives the image location.

Mirror Equation and Magnification

• **Mirror Equation:** Relates object distance (do), image distance (di), and focal length (f):

  1/do + 1/di = 1/f

• **Magnification (m):** Ratio of image height (hi) to object height (ho):

  m = hi/ho = -di/do

Sign Conventions for Spherical Mirrors

• **Object/Image Distance (do, di):**
  • Positive if on the reflective side of the mirror (real object/image).
  • Negative if behind the mirror (virtual object/image).
• **Focal Length (f):**
  • Positive for concave mirrors (focal point is real).
  • Negative for convex mirrors (focal point is virtual).
• **Radius of Curvature (r):**
  • Positive for concave mirrors (center of curvature on reflective side).
  • Negative for convex mirrors (center of curvature behind mirror).
• **Magnification (m):**
  • Positive if image is upright.
  • Negative if image is inverted.

Image Characteristics for Different Mirror Types

• **Concave Mirror:**
  • Object far beyond C: Real, inverted, smaller.
  • Object between C and F: Real, inverted, larger.
  • Object inside F: Virtual, upright, larger.
• **Convex Mirror:**
  • Always forms a **virtual, upright, and smaller** image, regardless of object position. Used as rearview mirrors.
• **Real Image:** Light rays actually converge and pass through the image location. Can be projected onto a screen.
• **Virtual Image:** Light rays appear to diverge from the image location, but do not actually pass through it. Cannot be projected.

5. Index of Refraction

• **Index of Refraction (n):** A dimensionless number that describes how fast light travels through a medium. It is the ratio of the speed of light in vacuum (c) to the speed of light in the medium (v):

  n = c/v

• Since v ≤ c, the index of refraction n ≥ 1.
• Examples: Vacuum (1.0000), Air (1.0003), Water (1.33), Glass (1.46-1.58), Diamond (2.42).

6. Refraction: Snell's Law

• **Refraction:** The bending of light as it passes from one medium to another. This change in direction occurs because light changes speed.
• The angle the outgoing ray makes with the normal is called the **angle of refraction** (θ2).
• Snell's Law (Law of Refraction): Relates the indices of refraction of the two media (n1, n2) and the angles of incidence and refraction (θ1, θ2):

  n1 sin θ1 = n2 sin θ2

• **Direction of Bending:**
  • If n2 > n1 (e.g., air to water), the ray bends **towards** the normal.
  • If n1 > n2 (e.g., water to air), the ray bends **away** from the normal.
• Refraction causes optical illusions, such as objects half-submerged in water appearing bent or shortened, or the apparent depth of an object underwater being shallower than its real depth.

7. Visible Spectrum and Dispersion

• **Visible Light:** Wavelengths range from approximately 400 nm (violet) to 700 nm (red).
• **Dispersion:** The phenomenon where the index of refraction of a material varies slightly with the wavelength of light.
• When white light (a mixture of all visible wavelengths) passes through a prism or water droplet, different wavelengths are bent to varying degrees, separating the colors into a spectrum.
• **Violet light** (shorter wavelength) is bent the most because its index of refraction is typically higher.
• **Red light** (longer wavelength) is bent the least because its index of refraction is typically lower.
• Dispersion is responsible for the formation of **rainbows**.
• The color we perceive for an object depends on its intrinsic properties and the medium it's in. The frequency of light remains constant across different media, but the wavelength changes. Our eyes perceive color based on frequency.

8. Total Internal Reflection; Fiber Optics

• **Critical Angle (θC):** When light travels from a medium with a higher index of refraction (n1) to one with a lower index (n2), there is a specific angle of incidence for which the angle of refraction is 90°.

  sin θC = n2/n1

• **Total Internal Reflection (TIR):** If the angle of incidence exceeds the critical angle, no light is refracted into the second medium; instead, all light is reflected back into the first medium.
• **Applications of TIR:**
  • **Binoculars:** Use prisms to achieve 100% reflection, which is superior to even the best mirrors.
  • **Fiber Optics:** Light signals are transmitted through thin optical fibers by repeatedly undergoing total internal reflection, allowing for very small signal losses over long distances.
• When viewing from underwater (water to air), there's a critical angle. Beyond this angle, one sees reflections of objects within the water, creating a "window" of the outside world surrounded by reflected images of the bottom or sides of the pool.

9. Refraction at a Spherical Surface

• Rays from a single point object can be focused by a spherical interface between two media with different indices of refraction, provided the angles are small.
• The relationship between object distance (do), image distance (di), indices of refraction (n1, n2), and radius of curvature (R) for a spherical refracting surface is:

  n1/do + n2/di = (n2 - n1)/R

• **Sign Conventions for Spherical Refracting Surfaces:**
  • R is positive if the center of curvature is on the side to which light is refracted (convex surface as seen by incident light).
  • R is negative if the center of curvature is on the side from which light is incident (concave surface as seen by incident light).
  • di is positive if the image is formed on the side to which light is refracted (real image).
  • di is negative if the image is formed on the side from which light is incident (virtual image).
• For a concave spherical interface, rays will diverge from a virtual image.