Optics

Misconceptions

Key Concepts

Reflection

Law of Reflection:

• •Angle of incidence = Angle of reflection
• •theta_i = theta_r
• •Angles measured from the normal (perpendicular to surface)

Types of Reflection:

• •Specular reflection: Smooth surfaces, parallel rays remain parallel
• •Diffuse reflection: Rough surfaces, parallel rays scatter in many directions

Refraction

Snell's Law:

• •n_1 * sin(theta_1) = n_2 * sin(theta_2)
• •n = refractive index (n = c/v where c is speed of light in vacuum)
• •Light bends toward the normal when entering a denser medium (higher n)
• •Light bends away from the normal when entering a less dense medium (lower n)

Common Refractive Indices:

• •Vacuum: n = 1.00
• •Air: n = 1.00 (approximately)
• •Water: n = 1.33
• •Glass: n = 1.50 (typical)
• •Diamond: n = 2.42

Total Internal Reflection

Critical Angle:

• •Occurs when light travels from higher n to lower n medium
• •At critical angle, refracted ray travels along the boundary (theta_2 = 90 degrees)
• •sin(theta_c) = n_2 / n_1 (where n_1 > n_2)
• •For angles greater than critical angle: total internal reflection

Applications:

• •Fiber optics
• •Diamond brilliance
• •Prisms in binoculars

Mirrors

Plane Mirrors:

• •Image is virtual, upright, same size
• •Image distance = object distance (behind mirror)
• •Magnification = +1

Concave Mirrors (Converging):

• •Center of curvature (C) at distance R from mirror
• •Focal point (F) at distance f = R/2 from mirror
• •Real, inverted images when object beyond F
• •Virtual, upright, magnified images when object inside F

Convex Mirrors (Diverging):

• •Focal point behind mirror (virtual focus)
• •Always produce virtual, upright, diminished images
• •Used for wide field of view (car side mirrors)

Lenses

Converging (Convex) Lenses:

• •Thicker in middle than at edges
• •Real focus on opposite side from object
• •Similar behavior to concave mirrors

Diverging (Concave) Lenses:

• •Thinner in middle than at edges
• •Virtual focus on same side as object
• •Always produce virtual, upright, diminished images

Mirror and Lens Equation

The Thin Lens/Mirror Equation:

• •1/f = 1/d_o + 1/d_i
• •f = focal length
• •d_o = object distance (always positive for real objects)
• •d_i = image distance

Sign Conventions:

• •Mirrors: d_i positive = in front of mirror (real), d_i negative = behind mirror (virtual)
• •Lenses: d_i positive = opposite side from object (real), d_i negative = same side as object (virtual)
• •Converging: f positive
• •Diverging: f negative

Magnification

Magnification Equation:

• •m = -d_i / d_o = h_i / h_o
• •h_i = image height, h_o = object height
• •|m| > 1: magnified
• •|m| < 1: diminished
• •m > 0: upright (virtual image)
• •m < 0: inverted (real image)

Image Characteristics

Real Images:

• •Form where light rays actually converge
• •Can be projected onto a screen
• •Always inverted (for single lens/mirror)
• •d_i is positive

Virtual Images:

• •Form where light rays appear to diverge from
• •Cannot be projected onto a screen
• •Always upright (for single lens/mirror)
• •d_i is negative

Equations

[1] theta_i = theta_r (law of reflection)
[2] n_1 * sin(theta_1) = n_2 * sin(theta_2) (Snell's law)
[3] sin(theta_c) = n_2 / n_1 (critical angle, n_1 > n_2)
[4] 1/f = 1/d_o + 1/d_i (mirror/lens equation)
[5] m = -d_i / d_o (magnification)
[6] m = h_i / h_o (magnification from heights)
[7] f = R/2 (focal length of spherical mirror)
[8] n = c/v (refractive index definition)

Worked Examples

Example 1: Snell's Law

Problem: Light travels from air (n=1.00) into glass (n=1.50) at an angle of 30 degrees from the normal. What is the angle of refraction?

Solution:

1. •Use Snell's law: n_1 * sin(theta_1) = n_2 * sin(theta_2)
2. •1.00 * sin(30) = 1.50 * sin(theta_2)
3. •0.50 = 1.50 * sin(theta_2)
4. •sin(theta_2) = 0.333
5. •theta_2 = 19.5 degrees

Example 2: Critical Angle

Problem: What is the critical angle for light traveling from water (n=1.33) to air (n=1.00)?

Solution:

1. •Use: sin(theta_c) = n_2 / n_1
2. •sin(theta_c) = 1.00 / 1.33 = 0.752
3. •theta_c = 48.8 degrees

Example 3: Concave Mirror

Problem: An object is placed 30 cm from a concave mirror with focal length 20 cm. Find the image distance and magnification.

Solution:

1. •Use mirror equation: 1/f = 1/d_o + 1/d_i
2. •1/20 = 1/30 + 1/d_i
3. •1/d_i = 1/20 - 1/30 = 3/60 - 2/60 = 1/60
4. •d_i = 60 cm (positive, so real image)
5. •m = -d_i/d_o = -60/30 = -2 (inverted, magnified)

Example 4: Converging Lens

Problem: A converging lens has a focal length of 15 cm. An object is placed 10 cm from the lens. Where is the image and what are its characteristics?

Solution:

1. •Use lens equation: 1/f = 1/d_o + 1/d_i
2. •1/15 = 1/10 + 1/d_i
3. •1/d_i = 1/15 - 1/10 = 2/30 - 3/30 = -1/30
4. •d_i = -30 cm (negative, so virtual image on same side as object)
5. •m = -d_i/d_o = -(-30)/10 = +3 (upright, magnified)
6. •Image is virtual, upright, and magnified

Example 5: Magnification

Problem: An object 5 cm tall is placed 25 cm from a lens. The image forms 50 cm from the lens on the opposite side. What is the image height?

Solution:

1. •Calculate magnification: m = -d_i/d_o = -50/25 = -2
2. •Use m = h_i/h_o: h_i = m * h_o = -2 * 5 = -10 cm
3. •The negative sign indicates the image is inverted
4. •The image is 10 cm tall and inverted

Explanation Patterns

1. •Draw a ray diagram - Always start with a sketch showing object, optical element, and principal rays
2. •Identify the type - Is it a mirror or lens? Converging or diverging?
3. •Apply sign conventions - Be careful with positive/negative for f, d_o, d_i
4. •Use appropriate equations - Mirror/lens equation for distances, magnification for size/orientation
5. •Interpret the results - Positive/negative d_i tells you real/virtual, sign of m tells you orientation
6. •Check reasonableness - Does the image location and type match the ray diagram?

Common Problem Types

1. •Snell's law calculations: Finding refraction angles
2. •Critical angle problems: Calculating when TIR occurs
3. •Mirror problems: Finding image distance and magnification
4. •Lens problems: Finding image distance and magnification
5. •Image characterization: Determining if real/virtual, upright/inverted, magnified/diminished
6. •Combined problems: Multi-step problems involving both equations
7. •Ray diagram interpretation: Understanding image formation from diagrams