Reflection and Refraction

🧭 Welcome — Let's Build Your Foundation

Part 1 of 8

Before we dive into equations, we need to understand what light actually does when it meets different materials.

By the end of this part you'll be able to explain, in plain language:

• Why light bends at boundaries
• How fiber optics carry the internet
• Why a straw looks bent in a glass of water

No math yet — just the big picture. Ready?

🥤 Why Does a Straw Look Bent?

Have you ever noticed that a straw in a glass of water looks broken at the surface? Or that you can see your own reflection in a window at night? Or wondered why fiber optic cables can send internet signals thousands of miles without losing data?

The answer lies in how light behaves.

When light travels through different materials — air, water, glass, diamond — it doesn't just pass through unchanged. It bends and bounces.

Understanding these behaviors unlocks the secrets behind:

• 🪞 Mirrors and clear reflections
• 💎 Why diamonds sparkle
• 🌐 The entire internet (fiber optics!)
• 🏊 Why pools look shallower than they are
• 🪟 Why windows act like mirrors at night

In this lesson, you'll master the physics that powers our modern visual world.

💡 What Is Light, Really?

Light is an electromagnetic wave that travels through space at incredible speed.

<div style="text-align: center;">

⚡ Speed of Light in Vacuum

$c = 3.0 \times 10^8$ m/s

</div>

<div style="text-align: center;"> <strong>That's 300,000 kilometers per second!</strong> <br/><br/> ⚡ Fast enough to circle Earth <strong>7.5 times per second</strong> </div>

Next: We'll explore the key properties that make light behave the way it does.

🌟 Key Properties of Light

1️⃣ → Travels in Straight Lines

In a uniform medium like air, light travels in straight lines called rays. This is why shadows have sharp edges and why we can aim lasers precisely.

2️⃣ ↗ Can Change Direction

Light reflects when it hits a surface (like a mirror), and refracts when it enters a new material (like light bending in water). These two behaviors explain almost everything you see!

3️⃣ 🌈 Different Colors = Different Wavelengths

• Red light → ~700 nm (longer wavelength)
• Blue light → ~450 nm (shorter wavelength)

💡 Key Insight: White light is actually a mixture of all the visible colors combined, from red (longest wavelength) to violet (shortest).

What Happens When Light Hits a Boundary?

When light encounters a boundary between two materials (like air and water), it has three options:

The Three Possibilities:

1. Reflect 🪞
Light bounces back into the first material

2. Refract 🌊
Light bends as it enters the second material

3. Do Both! ✨
Most of the time, light partially reflects AND partially refracts

We're going to learn the exact rules that determine what happens in each case.

Ready to see this in action? Let's explore some real-world examples!

✓ Check Your Understanding

Let's make sure you've mastered the basics before moving forward!

Real-World Example #1: Fiber Optic Internet 🌐

🎯 The Challenge

How do we send data thousands of miles without signal loss?

✅ The Solution

Total internal reflection in glass fibers!

• Light enters a thin glass fiber
• Bounces repeatedly off the walls (no signal loss!)
• Travels vast distances at near light speed
• Powers the entire internet backbone

📊 What You'll Learn

How to calculate the critical angle that makes this technology possible.

This same principle is used in medical endoscopes and telecommunications worldwide!

Real-World Example #2: Diamond Sparkle 💎

💬 The Question

Why do diamonds sparkle more than other gems?

💡 The Answer

Diamond's extremely high index of refraction: $n = 2.42$

• Light bends dramatically when entering
• Very small critical angle (only 24.4°)
• Light bounces around inside repeatedly
• Creates the famous brilliance

📊 What You'll Learn

How to predict which materials will sparkle the most based on their optical properties.

Proper diamond cutting uses this physics to maximize sparkle!

Real-World Example #3: Underwater Vision 🏊‍♀️

👀 Ever Noticed?

Objects underwater look closer than they really are!

🔬 The Cause

Light bends (refracts) when exiting water!

• Your brain assumes light travels in straight lines
• But refraction bends the light rays
• This creates optical illusions
• Fish appear closer to the surface than they are

📊 What You'll Learn

How Snell's law predicts which way light bends at the surface — and why that makes underwater objects look shifted from where they actually are.

This same principle explains why pools look shallower than they really are!

Real-World Example #4: Windows That Act Like Mirrors 🪟

👀 Ever Noticed?

In daytime you look straight through a window. At night, with the lights on inside, the same window shows your reflection!

🔬 The Physics

Light at a boundary partly reflects and partly transmits — all the time.

• A few percent of the light hitting glass reflects; most passes through
• In daytime, bright light from outside passing through the glass swamps the faint reflection of the room
• At night, little light comes in from outside, so the weak reflection of the bright room is easy to see
• The glass didn't change — only which light is brighter

📊 What You'll Learn

The law of reflection, and how to predict the direction of the light that does pass into the new material.

✓ Check Your Understanding

Great progress! Let's verify you understand these real-world applications.

🧩 A Problem-Solving Habit Worth Building

Starting in Part 3, you'll solve real optics problems. Here's the workflow top students use — write these five headers every time:

Given → Find → Formula → Substitute → Interpret

1. Given — list the known values and context
2. Find — state exactly what you need
3. Formula — pick the right equation
4. Substitute — plug in values with correct signs and units
5. Interpret — does the answer make physical sense?

This single habit will prevent most mistakes in the parts ahead.

Before You Move On — let's make sure the foundations are solid.

🗺️ Learning Journey: Tools for Every Optics Problem

Part 2 of 8

Sign conventions, reflection, the index of refraction, Snell's law, total internal reflection and refraction through several surfaces all rely on the same three tools.

This part builds those tools:

• The ray model of light, and when it works
• The normal line, and why every optics angle is measured from it
• The ray diagram, the picture you draw before any calculation

The Ray Model of Light

A ray is a straight line that points in the direction a light wave travels. Rays are drawn perpendicular to the wavefronts (the lines joining the crests of the wave).

• In a uniform material, a ray travels in a straight line.
• A ray changes direction only where the material changes, such as at a boundary between air and glass.
• A laser beam is the closest real thing to a single ray: narrow, one color, and nearly straight.

Two Common Sources

When Rays Are Enough, and When They Are Not

Rays describe light well when mirrors, lenses and openings are much larger than the wavelength. This is geometric optics, and it covers every part of this lesson.

Rays cannot explain light spreading out after a very narrow slit, or the bright and dark fringes made by two slits. Those effects need the wave model, which the interference and diffraction lessons cover.

Check: The Ray Model

The Normal and How Angles Are Measured

Where a ray hits a surface, draw the normal: a line perpendicular to the surface at that point.

• Flat surface: the normal points the same way everywhere on the surface.
• Curved surface (a spherical mirror or a lens): the normal at any point lies along the radius through that point, so its direction changes from point to point.

The angle of incidence $\theta_1$ is the angle between the incoming ray and the normal. The angles of reflection and refraction are measured from the normal too, never from the surface.

Converting From the Surface

Problems sometimes give the angle between the ray and the surface instead. The normal is 90° from the surface, so:

$\theta_{\text{from normal}} = 90° - \theta_{\text{from surface}}$

Example: a ray makes 35° with the surface of a flat mirror.

• Angle of incidence: $90° - 35° = 55°$
• The reflected ray is also 55° from the normal, on the other side.
• The angle between the incoming and reflected rays is $55° + 55° = 110°$.

What Happens at a Boundary

At a boundary between two transparent materials, part of the light reflects and the rest is transmitted into the new material, usually changing direction (refraction).

• A ray that arrives along the normal ($\theta_1 = 0$) is transmitted without bending.
• The frequency of the light never changes at a boundary. Its speed and wavelength can change in the new material.

Check: Measuring From the Normal

Drawing a Ray Diagram

Draw the diagram before any calculation:

1. Draw the boundary (the surface) as a line.
2. Mark the point where the ray hits, and draw the normal there as a dashed line.
3. Draw the incoming ray with an arrow showing its direction of travel.
4. Measure every angle from the normal, and label it.
5. Draw the reflected ray on the other side of the normal at the same angle. Draw any transmitted ray on the far side of the boundary.

Worked Example: Two Mirrors at a Right Angle

A horizontal mirror and a vertical mirror meet at a corner. A ray strikes the horizontal mirror with an angle of incidence of 30°.

• First reflection: the ray leaves the horizontal mirror 30° from that mirror's normal (which is vertical).
• Second mirror: the vertical mirror's normal is horizontal, 90° from the first normal. The ray's angle from it is $90° - 30° = 60°$.
• Second reflection: the ray leaves 60° from the horizontal normal.

Following the arrows shows that the ray leaves traveling exactly opposite to its original direction. This works for any angle of incidence, which is why corner reflectors are used on bicycle reflectors and road signs: they send light straight back toward its source.

Exit Check: Rays, Normals and Diagrams

📏 Sign Convention — The Key to Everything

Part 3 of 8

Mirror and lens problems use one equation for every case, and the signs tell you what kind of image you get. Get the signs right and every formula becomes straightforward.

We'll set up the rule system the AP Physics 2 exam uses:

1. When is a distance positive or negative?
2. When is a focal length positive or negative?
3. When is a height positive or negative?

Once you learn these, you'll assign correct signs in seconds.

The AP Sign Convention: "Real Is Positive"

The AP Physics 2 equation sheet gives one equation for both mirrors and thin lenses:

$\frac{1}{s_i} + \frac{1}{s_o} = \frac{1}{f} \qquad |M| = \left|\frac{h_i}{h_o}\right| = \left|\frac{s_i}{s_o}\right|$

Distances are measured from the mirror or lens, and the signs follow one idea: real is positive, virtual is negative.

Where Real Images Form

• Mirror: in front of the mirror, on the same side as the object (light bounces back toward it).
• Lens: on the far side of the lens from the object (light passes through).

Virtual images form on the opposite side: behind a mirror, or on the same side of a lens as the object.

A plane mirror has its focal point infinitely far away, so $\frac{1}{f} = 0$ and $s_i = -s_o$: the image is virtual and as far behind the mirror as the object is in front.

Two Worked Examples 📐

Example 1 — Concave Mirror

A concave mirror has $f = +10$ cm (converging, so positive). An object sits 30 cm in front of it.

$\frac{1}{s_i} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30} \Rightarrow s_i = +15 \text{ cm}$

Positive $s_i$: a real image 15 cm in front of the mirror. $|M| = 15/30 = 0.5$, so it is half as tall, and a real image from a single mirror is inverted.

Example 2 — Diverging Lens

A concave (diverging) lens has $f = -20$ cm. An object sits 20 cm from it.

$\frac{1}{s_i} = -\frac{1}{20} - \frac{1}{20} = -\frac{1}{10} \Rightarrow s_i = -10 \text{ cm}$

Negative $s_i$: a virtual image 10 cm from the lens on the same side as the object. $|M| = 10/20 = 0.5$, and a virtual image from a single lens is upright.

The Pattern

For a single mirror or lens, the sign of $s_i$ tells you everything:

Some textbooks measure every distance along the direction light travels (a "Cartesian" convention), which makes the object distance negative. The AP exam does not use that convention; stick with "real is positive."

Height Sign Convention

Heights are measured from the principal (optical) axis.

Above the axis → Positive height (+)

Below the axis → Negative height (−)

An upright object drawn above the axis has $h_o > 0$. An inverted image points below the axis, so its height $h_i$ is negative. The equation sheet compares sizes with $|M| = |h_i/h_o|$; use the sign of the height (or real versus virtual) to state the orientation.

Interactive Height Tutorial 📏

Interactive Animation

The optical axis is your reference line. Everything above it is positive, everything below is negative.

Check Your Understanding — Can you assign signs correctly?

🎯 Sign Practice

Choose the sign of each quantity using the AP "real is positive" convention.

Before You Move On — These two traps catch a lot of students.

🪞 Law of Reflection

Part 4 of 8

Time to apply what you know! In this part you'll learn the simplest but most fundamental law in optics:

Angle of incidence = Angle of reflection

We'll also see why mirrors create clear images while paper doesn't, and apply our sign convention to plane mirrors.

Law of Reflection

When light hits a smooth surface, it follows a simple rule:

$\theta_i = \theta_r$

Angle of incidence = Angle of reflection

Key Points:

1. Angles measured from the NORMAL ⊥

• The normal is perpendicular to the surface
• NOT from the surface itself!

2. The rays are COPLANAR 📐

• Incident ray, reflected ray, and normal all lie in the same plane
• Think of it as everything happening on a flat sheet of paper

Specular Reflection 🪞

Mirror-like reflection from smooth surfaces

When light hits a smooth, polished surface, we get specular reflection.

Characteristics:

Smooth Surface ✨

• The surface is flat at a microscopic level
• Examples: mirrors, calm water, polished metal, glass

Parallel Rays Stay Parallel →→→

• When parallel incident rays hit the surface...
• They reflect as parallel rays
• Each ray follows the law: $\theta_i = \theta_r$

Clear Images Form 🖼️

• Because rays stay organized and parallel
• We can see clear reflections
• This is why mirrors work!

Why It Works:

The smoothness means every point on the surface has the same orientation. All the normals (perpendiculars) point in the same direction, so parallel incident rays all reflect at the same angle, maintaining their parallel arrangement.

Result: You see a clear, sharp image - like looking in a mirror!

Diffuse Reflection 🌫️

Scattered reflection from rough surfaces

When light hits a rough surface, we get diffuse reflection.

Characteristics:

Rough Surface 🏔️

• The surface is bumpy at a microscopic level
• Examples: paper, walls, unpolished wood, fabric, your skin

Parallel Rays Scatter →↗→↖→

• When parallel incident rays hit the surface...
• They reflect in many different directions
• Each ray STILL follows $\theta_i = \theta_r$ at its local surface!

No Clear Images 🌁

• Rays scatter in all directions
• Information about the original scene gets scrambled
• You can see the surface, but not a reflection in it

Why It Happens:

The roughness means each point on the surface faces a different direction. The normals point in many different directions, so even parallel incident rays reflect at different angles because they're measured from different normals.

Result: You can see the object itself (because light scatters to your eyes), but you don't see mirror-like reflections!

Key Insight:

Both specular and diffuse reflection follow the same law: $\theta_i = \theta_r$

The difference is just whether the surface is smooth (specular) or rough (diffuse) at the microscopic scale!

Plane Mirror Sign Convention

Let's apply the AP sign convention to plane mirrors (flat mirrors with specular reflection).

Ray Tracing Animation

Watch how we locate the virtual image using ray tracing:

Applying the Sign Convention

Use the AP "real is positive" convention from Part 3.

Object in front of the mirror (where light comes from):

• Object distance $s_o$ is positive, as it is for every real object.

Image behind the mirror:

• No light actually travels behind the mirror; the reflected rays only appear to come from there.
• So the image is virtual, and its image distance $s_i$ is negative.

The Relationship

A plane mirror is flat, so its focal point is infinitely far away and $\frac{1}{f} = 0$. The mirror equation then gives

$\frac{1}{s_i} + \frac{1}{s_o} = 0 \quad\Rightarrow\quad s_i = -s_o$

The image is exactly as far behind the mirror as the object is in front, and $|M| = |s_i/s_o| = 1$: same size and upright.

Key Point: The minus sign does not mean "to the left." It means the image is virtual.

Quick Check: Reflection ✓

Test your understanding of reflection concepts!

Before You Move On — Two common traps with reflection.

Real-World Check — A driver sees broad glare from a wet rough road but a crisp reflection from a mirror road sign. Why?

🌊 Index of Refraction

Part 5 of 8

Why does light slow down in water? Why do diamonds sparkle? The answer is a single number: the index of refraction.

In this part you'll learn to:

• Calculate $n$ from the speed of light in a material
• Predict what changes (speed, wavelength) and what stays the same (frequency)
• Use $n$ as a sanity check — if you ever get $n < 1$, something's wrong!

Index of Refraction 🌊

Light slows down when entering a material!

The Index

Index of refraction ($n$):

$n = \frac{c}{v}$

where:

• $c$ = speed of light in vacuum = $3.0 \times 10^8$ m/s
• $v$ = speed of light in the material

Key Properties:

• $n \geq 1$ (light can't go faster than $c$!)
• No units (dimensionless)
• Higher $n$ → slower light → more bending

Common Values:

What Changes, What Doesn't:

When light enters a new medium:

Changes:

• ✅ Speed: $v = c/n$
• ✅ Wavelength: $\lambda = \lambda_0/n$

Stays the Same:

• ❌ Frequency: $f$ constant
• ❌ Color (determined by frequency)

Why This Matters:

The relationship $c = \lambda f$ still holds, but in a medium:

$v = \lambda f$

Since $f$ stays constant: $\frac{c}{n} = \frac{\lambda_0}{n} \cdot f$

This confirms $v$ and $\lambda$ both decrease by factor $n$!

Practice: Index of Refraction ✓

Calculate and apply!

Before You Move On — Two traps with index of refraction.

Real-World Check — An engineer compares two lens materials: A has n = 1.40 and B has n = 1.70.

⚡ Snell's Law — The Master Equation

Part 6 of 8

This is the equation you'll use more than any other in optics:

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

It tells you exactly how much light bends at any boundary. By the end of this part, you'll solve refraction problems in a few clean lines.

Snell's Law

When light crosses a boundary between two materials, it changes direction! This phenomenon is called refraction.

Snell's Law

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

This powerful equation predicts exactly how much light will bend at any interface.

What you see:

• Light traveling from air (less dense) into water (more dense)
• The ray bends toward the normal when entering the denser medium
• Angle decreases: 45° → 32°

Understanding the Parameters

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

Where:

• $n_1$ = index of refraction of first medium (where light is coming from)
• $\theta_1$ = angle from normal in first medium
• $n_2$ = index of refraction of second medium (where light is going)
• $\theta_2$ = angle from normal in second medium

Critical Points:

1. Angles measured from the NORMAL ⊥

• The normal is perpendicular to the surface
• NOT from the surface itself!

2. Light direction matters

• Light travels from medium 1 → medium 2
• We follow the light ray's path

3. Both angles always positive

• We measure angles as magnitudes (0° to 90°)
• Direction of bending tells us the physics

Bending Rules: Which Way Does Light Bend?

The direction light bends depends on the relative density of the two media.

The Pattern 🎯

When light crosses a boundary:

• Entering denser medium ($n_2 > n_1$): Bends TOWARD normal
• Entering less dense medium ($n_2 < n_1$): Bends AWAY FROM normal
• Same medium ($n_1 = n_2$): No bending (continues straight)

Memory Aid 💡

"Fast → Slow: bend toward"
"Slow → Fast: bend away"

Think of it like a car going from pavement to sand at an angle—one wheel hits the sand first and slows down, causing the car to turn toward the normal!

Let's see this in action with visual examples...

Light Bending TOWARD Normal 📉

Entering Denser Medium ($n_2 > n_1$)

When light enters a denser medium (higher index of refraction), it slows down and bends toward the normal.

Key Observations:

✓ Angle decreases: $\theta_2 < \theta_1$ (50° → 30°)
✓ Light slows down: Moves from faster medium to slower medium
✓ Bends toward normal: Gets closer to perpendicular

Examples:

• Air → Water
• Air → Glass
• Water → Diamond

Why it happens: Light slows down in denser materials, causing the wavefront to pivot toward the normal—like a marching band turning when one side slows down!

Light Bending AWAY FROM Normal 📈

Entering Less Dense Medium ($n_2 < n_1$)

When light enters a less dense medium (lower index of refraction), it speeds up and bends away from the normal.

Key Observations:

✓ Angle increases: $\theta_2 > \theta_1$ (30° → 42°)
✓ Light speeds up: Moves from slower medium to faster medium
✓ Bends away from normal: Gets farther from perpendicular

Examples:

• Water → Air
• Glass → Air
• Diamond → Air

Why it happens: Light speeds up in less dense materials, causing the wavefront to pivot away from the normal—like a marching band spreading out when one side speeds up!

Important: This is where Total Internal Reflection can occur if the angle is too large!

Snell's Law Example 1: Air to Water 💧

Problem:

Light travels from air into water at an angle of 45° from the normal. Find the refraction angle.

Given:

• $n_1 = 1.0$ (air)
• $\theta_1 = 45°$
• $n_2 = 1.33$ (water)
• $\theta_2 = ?$

Step-by-Step Solution:

Step 1: Set up the coordinate system

        Air ($n_{1}$ = 1.0)
    ────┼────  45° from normal
        │
────────┴────────  ← water surface
        │ $\theta_{2}$ = ?
        ┼────
    Water ($n_{2}$ = 1.33)

Light travels downward, from air into water.

Step 2: Apply Snell's Law

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

Step 3: Solve for $\theta_2$

$\sin\theta_2 = \frac{n_1 \sin\theta_1}{n_2}$

$\sin\theta_2 = \frac{(1.0)\sin(45°)}{1.33}$

$\sin\theta_2 = \frac{(1.0)(0.707)}{1.33} = 0.531$

$\theta_2 = \sin^{-1}(0.531) = 32.1°$

Step 4: Interpret the result

• $\theta_2 = 32.1° < 45° = \theta_1$ ✓
• Light bent toward normal ✓
• Makes sense! Entering denser medium ($n_2 > n_1$)

Visual Result:

The light ray bends toward the normal when entering water!

Snell's Law Example 2: Water to Air 🌊➜🌤️

Problem:

Light travels from water to air at 30° from the normal. Find the refraction angle.

Given:

• $n_1 = 1.33$ (water)
• $\theta_1 = 30°$
• $n_2 = 1.0$ (air)
• $\theta_2 = ?$

Step-by-Step Solution:

Step 1: Set up the coordinate system

    Water ($n_{1}$ = 1.33)
        ┼────  30° from normal
        │
────────┴────────  ← water surface
        │
    ────┼────  $\theta_{2}$ = ?
        Air ($n_{2}$ = 1.0)

Light travels upward, from water into air.

Step 2: Apply Snell's Law

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

Step 3: Solve for $\theta_2$

$\sin\theta_2 = \frac{n_1 \sin\theta_1}{n_2}$

$\sin\theta_2 = \frac{(1.33)\sin(30°)}{1.0}$

$\sin\theta_2 = \frac{(1.33)(0.5)}{1.0} = 0.665$

$\theta_2 = \sin^{-1}(0.665) = 41.7°$

Step 4: Interpret the result

• $\theta_2 = 41.7° > 30° = \theta_1$ ✓
• Light bent away from normal ✓
• Makes sense! Entering less dense medium ($n_2 < n_1$)

Visual Result:

The light ray bends away from the normal when exiting water!

Key Difference: Same angle in water (30°), but now we're going the opposite direction → larger angle in air!

Check Your Understanding — Predict which way light bends.

Computation Drill (5-Step Style)

Light goes from air ($n_1=1.00$) into glass ($n_2=1.50$) at $\theta_1=30^\circ$.

Enter in order (to 3 significant figures where applicable):

1. $\sin\theta_2$

2. $\theta_2$ in degrees

3. Direction phrase: toward or away

Before You Move On — Two common Snell's Law mistakes.

💎 Total Internal Reflection

Part 7 of 8

What happens when light tries to leave a dense material and the angle is too steep? It can't escape — it reflects completely back inside. This is total internal reflection (TIR).

TIR powers fiber optic internet and makes diamonds sparkle. Let's see how it works.

Total Internal Reflection

Something special happens when light tries to go from dense → less dense!

The Critical Angle

When going from higher $n$ to lower $n$ (e.g., water → air):

There's a maximum incident angle called the critical angle ($\theta_c$)

At $\theta_c$: Refracted ray travels along the boundary ($\theta_2 = 90°$)

Beyond $\theta_c$ ($\theta_1 > \theta_c$): Light completely reflects back!

Three Scenarios:

1. θ < θc: Normal refraction occurs
     Water (n = 1.33)
        ↓  
     ────┴────  → (light exits to air)
         Air

2. θ = θc: Critical angle - light along boundary
     Water
        ↓  
     ────┴────→ (grazes surface)
         Air

3. θ > θc: Total Internal Reflection
     Water
        ↓ ↗  (100% reflection!)
     ────┴────
         Air (no light escapes)

Finding the Critical Angle

Start with Snell's Law at the critical condition:

$n_1 \sin\theta_c = n_2 \sin(90°)$

$n_1 \sin\theta_c = n_2 (1)$

$\sin\theta_c = \frac{n_2}{n_1}$

$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$

Important: Only exists when $n_1 > n_2$ (why?)

If $n_1 < n_2$, then $\frac{n_2}{n_1} > 1$ → no solution! (sin can't exceed 1)

What Happens at Total Internal Reflection

When total internal reflection occurs:

• No refracted ray (it's reflected instead)
• Reflected ray follows law of reflection
• All energy stays in original medium

Example: Water-Air Interface

$\theta_c = \sin^{-1}\left(\frac{1.0}{1.33}\right) = \sin^{-1}(0.752) = 48.8°$

If light in water hits surface at > 48.8°: 100% reflection!

Applications of Total Internal Reflection 🌟

TIR isn't just theory—it powers modern technology!

1. Fiber Optics 💡

How it works:

• Glass or plastic fiber with high $n$
• Light enters at one end
• Repeatedly reflects off walls (TIR)
• Travels long distances with minimal loss!

Light in ─┐  ╱╲  ╱╲  ╱╲  ╱╲  ╱╲ ┌─ Light out
         └─┘  └─┘  └─┘  └─┘  └─┘
        [Fiber optic cable]

Applications:

• Internet (fiber optic cables)
• Medical endoscopes
• Telecommunications

Why it's amazing:

• Almost no signal loss
• Immune to electromagnetic interference
• Can carry huge amounts of data

2. Diamonds Sparkle 💎

Why diamonds sparkle so much:

Diamond has $n = 2.42$ (very high!)

$\theta_c = \sin^{-1}\left(\frac{1.0}{2.42}\right) = 24.4°$

• Very small critical angle!
• Light easily undergoes TIR
• Bounces around inside, creating sparkle
• Proper cut maximizes this effect

3. Binoculars and Periscopes 🔭

Prisms use TIR instead of mirrors:

    Light in ─→ ╱│
               ╱ │
              ╱  ↓ (TIR at 45°)
             ╱___│
                 ↓ Light out

Advantages over mirrors:

• No coating needed
• 100% reflection (mirrors are ~95%)
• No degradation over time

Why it works: glass has $n = 1.5$, so its critical angle with air is $\sin^{-1}(1/1.5) = 41.8°$. Light striking the long face of the prism at 45° exceeds that, so it is totally reflected.

Master Check: Total Internal Reflection ✓

Test your understanding of TIR!

Before You Move On — Two things students get wrong about TIR.

Real-World Check — An engineer wants stronger light confinement in a fiber optic cable.

🧊 Refraction Through Several Surfaces

Part 8 of 8 — The Grand Finale

You've learned reflection, refraction, Snell's law, and TIR one boundary at a time. Real light often crosses several boundaries in a row — through a window, a glass block, or layers of different materials.

In this part you'll learn to:

• Follow a ray through a block of glass and predict how it leaves
• Use the fact that $n\sin\theta$ carries across parallel boundaries
• Measure an index of refraction from lab data with a straight-line graph

Light Through a Glass Block 🧊

Straight In, Straight Out

If light hits a surface along the normal ($\theta_1 = 0°$), then $\sin\theta_2 = 0$ too: the light slows down but does not change direction.

A Block With Parallel Faces

Problem: Light in air strikes a flat glass block ($n = 1.50$) at 40° from the normal. Follow it through.

Entering (air → glass): $\sin\theta_2 = \frac{(1.00)\sin 40°}{1.50} = \frac{0.643}{1.50} = 0.429 \quad \Rightarrow \quad \theta_2 = 25.4°$

The ray bends toward the normal.

Leaving (glass → air): the faces are parallel, so the ray reaches the bottom face at the same 25.4° from the normal: $\sin\theta_3 = \frac{(1.50)\sin 25.4°}{1.00} = 0.643 \quad \Rightarrow \quad \theta_3 = 40°$

The ray bends away from the normal by exactly the same amount.

What This Shows

• The ray leaves parallel to the direction it came in
• It is shifted sideways — the thicker the block, the bigger the shift
• This is why objects seen through a thick window look slightly displaced but not tilted

Layers of Material

Across parallel boundaries, Snell's law chains together: $n_1\sin\theta_1 = n_2\sin\theta_2 = n_3\sin\theta_3 = \cdots$

So the quantity $n\sin\theta$ is the same in every layer. Example: light in water ($n = 1.33$) at 30° passes through a flat glass plate ($n = 1.50$) into air:

• In water: $n\sin\theta = (1.33)(0.500) = 0.665$
• In glass: $\sin\theta = 0.665/1.50 = 0.443$, so $\theta = 26.3°$
• In air: $\sin\theta = 0.665$, so $\theta = 41.7°$ — the same angle as if the glass weren't there

For parallel layers, the final angle depends only on the first and last materials.

Check Your Understanding — Several surfaces in a row.

Measuring an Index of Refraction 🔬

The Lab

Shine a laser from air into a block of clear plastic. For several angles of incidence $\theta_1$, measure the angle of refraction $\theta_2$.

Make It a Straight Line

Snell's law with air ($n_1 = 1.00$) says $\sin\theta_1 = n\sin\theta_2$. So if you plot

• vertical axis: $\sin\theta_1$
• horizontal axis: $\sin\theta_2$

the points fall on a straight line through the origin whose slope is $n$.

Sample Data (glass)

Slope $= \dfrac{0.866 - 0.342}{0.578 - 0.228} = 1.50$, so $n = 1.50$.

What Else the Slope Tells You

• Speed in the material: $v = \dfrac{c}{n} = \dfrac{3.0 \times 10^8}{1.50} = 2.0 \times 10^8$ m/s
• Wavelength in the material: $\lambda = \dfrac{\lambda_{\text{air}}}{n}$ — the frequency doesn't change
• Critical angle with air: $\theta_c = \sin^{-1}(1/n) = 41.8°$

Why a Glass Block Never Traps Light From Air

Light entering from air can reach at most $\theta_2 = 41.8°$ inside (when $\theta_1$ approaches 90°). With parallel faces, it meets the far face at that same angle — never beyond the critical angle. So light that enters one face of a parallel-sided block from air always gets back out through the opposite face. TIR needs a surface the light meets at a steeper angle, as in a reflecting prism or the wall of a fiber.

Master Check: Reflection & Refraction Complete ✓

Final quiz covering the entire topic!

Summary of Key Concepts 📚

You've completed the full Reflection & Refraction topic! Here's what you've mastered:

Sign Convention (AP: "real is positive")

• Object distance $s_o$: positive for real objects
• Image distance $s_i$: positive for real images, negative for virtual images
• Focal length $f$: positive for converging (concave mirror, convex lens), negative for diverging
• Heights: above the axis positive, below negative

Law of Reflection

$\theta_i = \theta_r$

• Angles measured from normal
• Always obeys this law (smooth surfaces)

Index of Refraction

$n = \frac{c}{v}$

• Always $n \geq 1$ (light slower in materials)
• Higher $n$ = slower light = denser medium

Snell's Law

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

• Entering denser medium ($n_2 > n_1$): bend toward normal
• Entering less dense ($n_2 < n_1$): bend away from normal
• Along the normal: no bending

Total Internal Reflection

$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$ (only when $n_1 > n_2$)

• If $\theta_1 > \theta_c$: 100% reflection
• Powers fiber optics, creates sparkle in diamonds

Several Surfaces

• $n\sin\theta$ is the same in every layer of parallel boundaries
• A parallel-sided block shifts a ray sideways without changing its direction
• A graph of $\sin\theta_1$ vs. $\sin\theta_2$ has slope $n_2/n_1$

What's Next?

Now that you understand reflection and refraction, you're ready for:

• Mirrors (curved surfaces that reflect)
• Lenses (curved surfaces that refract)
• Wave optics (interference, diffraction)

Final Real-World Check — A beam travels from air into a glass block, then from the glass into water. In the water it reaches a water–air surface that is tilted relative to the glass faces, strikes it at an angle beyond the critical angle, and does not get out.