Light, Refraction & Optical Properties

Speed of Light in Media: the Index of Refraction

Light travels at c = 3×10⁸ m/s only in a perfect vacuum. In any material, it slows down. The index of refraction n quantifies how much:

Since n ≥ 1 always (light never goes faster than c in vacuum), and n is dimensionless. The larger n is, the slower light travels in that material.

Table of Common Indices of Refraction

Wavelength and Frequency in a Medium

When light crosses a boundary into a new medium, two of its properties change, and one stays the same. Understanding which is which is half the battle of Chapter 23 exam problems.

What changes, what doesn't

• Frequency: NEVER changes. The source (e.g., a laser) oscillates at a fixed rate. The boundary cannot create or destroy oscillations — every wave crest that arrives at the surface must also leave the surface. f is set by the source alone.
• Speed: DOES change. v = c/n. In a denser medium, the electromagnetic wave interacts more with atoms and slows.
• Wavelength: DOES change. Since v = fλ and f is constant, if v decreases, λ must decrease proportionally.

Worked Example (likely on your exam)

λ₀ = 256 nm in vacuum. Glass has n = 1.6. Find the frequency and wavelength inside the glass.

Snell's Law: Quantifying the Bend

When light hits an interface between two media at an angle, it bends. The exact relationship between the angle coming in and the angle going out is given by Snell's Law:

Critical rule: All angles (θ₁ and θ₂) are measured from the normal — the perpendicular to the surface. NEVER from the surface itself. This is the single most common source of errors on optics exams.

The two cases

• Air → Glass (n₁ < n₂): Since n₂ > n₁, we need sinθ₂ < sinθ₁, so θ₂ < θ₁. The ray bends toward the normal.
• Glass → Air (n₁ > n₂): Since n₂ < n₁, we need sinθ₂ > sinθ₁, so θ₂ > θ₁. The ray bends away from the normal.

How to Read a Refraction Diagram (Critical Exam Skill)

Exam questions frequently show a diagram of a ray crossing a boundary and ask: "which medium has larger n?" or "in which medium is the speed greater?" You do NOT need to know the actual angle values — just compare whether the ray bent toward or away from the normal.

The two key scenarios

Decision tree for exam diagrams

• Draw the normal (mentally). Is the angle in medium 2 smaller than in medium 1? → n₂ > n₁ (medium 2 is denser, slower, shorter λ).
• Is the angle in medium 2 larger than in medium 1? → n₂ < n₁ (medium 2 is less dense, faster, longer λ).
• Are the angles equal? → n₁ = n₂ (same medium on both sides).

Determining Speed Ratio from a Diagram

Once you know how to read a refraction diagram, you can extract quantitative information about the speeds. Snell's Law can be rewritten in terms of wave speeds, which gives a direct connection between angles and speeds.

Reading speed from angle comparison

• If θ₂ > θ₁ (bends away): sinθ₂ > sinθ₁, so v₂ > v₁ — light is faster in medium 2.
• If θ₂ < θ₁ (bends toward): sinθ₂ < sinθ₁, so v₂ < v₁ — light is slower in medium 2.
• The speed RATIO: v₂/v₁ = sinθ₂/sinθ₁ (plug in actual angle values if given).

Determining Wavelength Ratio from a Diagram

Since λ = λ₀/n (or equivalently λ ∝ v, since f is constant), the wavelength relationship follows directly from the angle comparison. The medium with the smaller refraction angle is the medium with the shorter wavelength.

Rules from diagram geometry

• Ray bends toward normal entering medium 2 (θ₂ < θ₁): wavelength is shorter in medium 2 (λ₂ < λ₁).
• Ray bends away from normal entering medium 2 (θ₂ > θ₁): wavelength is longer in medium 2 (λ₂ > λ₁).
• Angles equal: wavelengths are equal.

Total Internal Reflection (TIR)

When light travels from a denser medium to a less dense medium (n₁ > n₂), it bends away from the normal. As the incident angle increases, the refracted angle approaches 90°. At the critical angle θ_c, the refracted ray skims the surface. Beyond θ_c, no refraction occurs at all — all light is reflected back into the denser medium. This is Total Internal Reflection.

Why Air-to-Water TIR is Impossible (Exam Q7 Type)

This is one of the most commonly tested TIR concepts: "Can total internal reflection occur when light travels from air into water?" The answer is NO — and you need to know exactly why.

The proof in three steps

• Step 1 — Check the condition: TIR requires n₁ > n₂ (traveling from denser to less dense). Here n_air = 1.00 and n_water = 1.33. Since 1.00 < 1.33, we are going from LESS dense to MORE dense. The condition is NOT met.
• Step 2 — Attempt the formula: If we try sinθ_c = n₂/n₁, we get sinθ_c = n_water/n_air = 1.33/1.00 = 1.33. This is greater than 1. The sine function can never exceed 1 by definition, so no real angle satisfies this — there is no critical angle.
• Step 3 — Physical interpretation: Going air → water, the ray bends TOWARD the normal (n increases). No matter how large θ₁ gets (even 89.9°), the refracted ray always exists, just barely entering the water at a large angle toward the normal. TIR simply cannot happen.

Total Internal Reflection: Real-World Applications

TIR is not just an exam curiosity — it underlies some of the most important technologies in modern life. Understanding these applications also reinforces the physics, since each example reveals the same principle at work.

Fiber Optics (the most important application)

A fiber optic cable consists of a glass core (n ≈ 1.50) surrounded by a glass cladding (n ≈ 1.46). Since n_core > n_cladding, any light ray that strikes the core-cladding boundary at an angle greater than θ_c undergoes TIR and bounces along the core without escaping.

Other TIR Applications

• Diamond (n=2.42): θ_c = sin⁻¹(1/2.42) = 24.4°. Very small! Most light that enters a diamond totally reflects off the back facets, bouncing around before exiting through the top — creating the characteristic sparkle. Diamonds are cut to maximize TIR.
• Medical endoscopes: Flexible fiber bundles guide images from inside the body using TIR. Each fiber carries one pixel of the image.
• Retroreflectors / prisms: A 45°-45°-90° glass prism reflects light through 90° or 180° via TIR (θ_c for glass-air ≈ 41°, and the ray hits at 45° > 41°).

Dispersion: Why Prisms Make Rainbows

We have been treating the index of refraction n as a single number for a given material. In reality, n depends slightly on the wavelength (color) of the light. Violet light has a slightly larger n than red light in glass. This wavelength-dependence of n is called dispersion.

When white light hits a prism, each wavelength (color) is refracted by a slightly different amount. The result is the visible spectrum — red bends least, violet bends most.

Prism TIR: Exam-Style Calculation

A classic exam problem: a glass prism with faces labeled. Light enters one face along the normal (no refraction on entry), then hits a second face at angle δ. Will TIR occur? How does the answer change if the prism is immersed in water?

Setup: glass prism (n = 1.52) in air

Modified setup: same prism immersed in water (n = 1.33)

Polarization of Light

Light is a transverse electromagnetic wave: the electric field E oscillates perpendicular to the direction of propagation. In unpolarized light, E oscillates in all directions within that perpendicular plane simultaneously. In polarized light, E oscillates in one specific direction only.

Methods of polarization

• Polaroid filters: Transmit only the component of E aligned with the filter axis. Intensity after polarizer: I = I₀/2 (for unpolarized input).
• Malus's Law: If polarized light of intensity I₀ passes through a polarizer at angle θ to the polarization direction: I = I₀ cos²θ.
• Brewster's Law (reflection polarization): At Brewster's angle, reflected light is completely polarized parallel to the surface. tanθ_B = n₂/n₁. At this angle, the reflected and refracted rays are perpendicular to each other (θ_reflected + θ_refracted = 90°).

Exam Problem Type 1: Wavelength and Frequency in Glass

This type of problem is extremely common and straightforward once you know the three-step method. Do NOT skip steps — even if you know the answer, showing the method earns partial credit.

Full Solution

Exam Problem Type 2: Reading Diagrams for n, v, and λ

This is the highest-frequency exam skill in Chapter 23. Practice until you can do these in under 30 seconds. No calculations needed — just compare angles.

The three cases

Practice drill — answer all three questions per diagram

• For Case B: (1) n₂ > n₁ (denser). (2) v₂ < v₁ (slower). (3) λ₂ < λ₁ (shorter wavelength).
• For Case C: (1) n₂ < n₁ (less dense). (2) v₂ > v₁ (faster). (3) λ₂ > λ₁ (longer wavelength).
• For Case A: all three quantities are equal in both media.

What-If Scenarios: Testing Your Conceptual Depth

Exam problems often change one parameter to see if you understand the underlying physics. Work through each scenario carefully — each one targets a different potential misconception.

Scenario 1: What if λ₀ = 500 nm instead of 256 nm?

For glass n=1.60: f = c/λ₀ = 3×10⁸/500×10⁻⁹ = 6.00×10¹⁴ Hz (different frequency from the 256 nm case, but still invariant across the boundary). λ_glass = 500/1.60 = 312.5 nm. The ratio λ_glass/λ₀ = 1/n = 1/1.60 is the same regardless of the starting wavelength. Only the absolute values change.

Scenario 2: What if n = 1.0 (vacuum)?

Scenario 3: What if n = 2.0?

Scenario 4: Light goes from glass (n=1.52) to water (n=1.33)

Now n₁ > n₂ (glass is denser than water). Snell's Law still applies: 1.52 sinθ₁ = 1.33 sinθ₂. Since n₁ > n₂, sinθ₂ > sinθ₁, so θ₂ > θ₁ — ray bends away from normal. TIR is now possible!

Scenario 5: What if the frequency of the incoming light is doubled?

Frequency doubles → wavelength halves (in vacuum: λ₀ = c/f). In glass (n=1.6): the new wavelength is still λ₀/n, just half as large. The speed in glass is still c/n — speed depends only on n, not on f. The index n for most materials barely changes with frequency (dispersion is small), so to a good approximation v_glass is the same regardless of the light's color.

Common Mistakes and Exam Traps: What to Watch For

These are the errors that cost students points on Chapter 23 exams. Read each one, internalize it, and promise yourself you won't make it. Each trap is paired with the correct statement and a quick diagnostic.

Trap 1 — "Frequency changes when entering a medium"

Trap 2 — "Angles are measured from the surface"

Trap 3 — "TIR can happen going from air to water"

Trap 4 — "Wavelength in a medium is λ₀ × n (multiply)"

Chapter 23 Complete Cheat Sheet

Everything you need for Exam 3, Chapter 23 — condensed to one slide. Study this until you can reproduce it from memory.

All Formulas

Common n Values (memorize these)