# Polarization

> AP Physics 2 · AP Physics 2
> Source: https://www.owlsprep.com/study/ap-physics-2-u6-polarization/

This module covers core polarization concepts including Malus's Law, polarizer stack calculations, polarization by reflection, Brewster's angle, and scattering polarization aligned to AP Physics 2 exam requirements.

**Prerequisites:** [Transverse properties of electromagnetic waves](https://www.owlsprep.com/study/ap-physics-2-u6-electromagnetic-waves/); [Refractive index and Snell's Law](https://www.owlsprep.com/study/ap-physics-2-u6-refraction-snells-law/)

## Learning objectives

- Distinguish between polarized, unpolarized, and partially polarized electromagnetic light waves
- Apply Malus's Law to calculate transmitted intensity through sequential ideal polarizing filters
- Explain mechanisms of polarization including reflection, scattering, and birefringence
- Solve multi-step problems involving Brewster's angle for dielectric interfaces

## Polarized vs Unpolarized Light

Visible light is a transverse electromagnetic wave, where electric and magnetic fields oscillate perpendicular to the direction the wave travels. Polarization describes the orientation of the electric field oscillation for a given light beam.

**Linearly Polarized Light** — Light where the electric field oscillates along exactly one fixed axis for every wavefront in the beam.

*Example:* Light that has passed through a single ideal polarizing filter

> **info**
>
> Nearly all natural light sources including sunlight, incandescent bulbs, and fire produce unpolarized light, where electric field orientations change randomly every few nanoseconds across all possible perpendicular axes.

**Check your understanding**

Test your baseline understanding before proceeding:

1. Unpolarized light has electric fields oscillating in how many distinct axes?

   - 1
   - 2
   - All possible perpendicular axes
   - Zero

   *Why:* Unpolarized light has random, evenly distributed E-field orientations across every axis perpendicular to travel.

## Malus's Law for Ideal Polarizers

An ideal linear polarizer only transmits the component of the incident electric field that is aligned to its transmission axis. The transmitted intensity is proportional to the square of the electric field amplitude, leading directly to Malus's Law for incident polarized light.

$$I = I_0 \cos^2(\theta)$$

> **tip**
>
> For unpolarized incident light, averaging the \cos^2(\theta) term over all possible angles gives a value of 0.5, so the first polarizer always cuts unpolarized light intensity exactly in half, no matter its orientation.

**Worked example:** Unpolarized light of intensity 800 W/m² passes first through a vertical polarizer, then through a second polarizer oriented 60° from vertical. Calculate the final transmitted intensity.

1. Step 1: Process the first polarizer. Since incident light is unpolarized, transmitted intensity is halved:
2. $$I_1 = \frac{800}{2} = 400 \text{ W/m}^2$$
3. Step 2: The light after the first polarizer is vertically polarized. The angle between its polarization axis and the second polarizer is 60°, so apply Malus's Law:
4. $$I_2 = 400 \times \cos^2(60^\circ) = 400 \times (0.5)^2 = 100 \text{ W/m}^2$$
5. Final transmitted intensity is 100 W/m².

> **Exam tip:** AP graders will deduct partial credit if you forget to halve the intensity of unpolarized light before applying Malus's Law for subsequent filters.

## Multi-Polarizer Stack Problems

A common exam trick question involves three or more sequential polarizers, often with the first and last oriented 90° apart (called crossed polarizers). Many students incorrectly assume the final intensity is zero, but an intermediate polarizer reorients the polarization to allow non-zero transmission.

**Worked example:** Unpolarized 1000 W/m² light passes through 3 polarizers: first vertical, second at 45° from vertical, third horizontal. Find the final transmitted intensity.

1. Step 1: First polarizer halves the unpolarized intensity:
2. $$I_1 = 1000 / 2 = 500 \text{ W/m}^2$$
3. Step 2: Angle between first and second polarizer is 45°, apply Malus's Law:
4. $$I_2 = 500 \times \cos^2(45^\circ) = 500 \times 0.5 = 250 \text{ W/m}^2$$
5. Step 3: Angle between second and third polarizer is 45°, apply Malus's Law again:
6. $$I_3 = 250 \times \cos^2(45^\circ) = 125 \text{ W/m}^2$$
7. Final intensity is 125 W/m², not zero.

**Check your understanding**

Quick check: What would the final intensity be if you removed the middle 45° polarizer?

1. What is the transmitted intensity for two crossed polarizers with unpolarized incident light?

   - 500 W/m²
   - 250 W/m²
   - 0 W/m²
   - 125 W/m²

   *Why:* With no intermediate polarizer, the second horizontal polarizer is 90° offset from the first vertical polarizer, so cos²(90°) = 0.

## Polarization by Reflection and Brewster's Angle

When light reflects off a smooth dielectric surface like water or glass, the reflected light is partially polarized. At one specific incident angle called Brewster's angle, the reflected light is 100% linearly polarized parallel to the surface (horizontally polarized for a flat lake surface).

$$\tan(\theta_B) = \frac{n_2}{n_1}$$

**Brewster's Angle** — Incident angle where reflected and refracted rays are exactly perpendicular to each other, eliminating all p-polarized light from the reflected beam.

**Exam command terms**

Common AP exam command terms for polarization questions:

- **Explain why polarizing sunglasses reduce road glare** — Requires linking horizontal polarization of reflected road light to the vertical transmission axis of polarizing sunglasses.

- **Calculate Brewster's angle for an air-glass interface** — Requires using the tangent form of the law, not Snell's sine ratio.

## Common pitfalls

- **Wrong:** Applying Malus's Law directly to unpolarized incident light instead of halving intensity first
  - Why it fails: Unpolarized light has evenly distributed E-field orientations, so averaging cos²θ over all angles gives 0.5, not cos² of any single angle
  - Correct: Always reduce unpolarized light intensity by 50% after the first polarizer before applying Malus's Law for subsequent filters
- **Wrong:** Claiming two crossed polarizers will always produce zero transmitted intensity even if there is a third polarizer in between
  - Why it fails: The intermediate polarizer reorients the E-field to a non-90° angle relative to the final polarizer, allowing non-zero transmission
  - Correct: Process each polarizer sequentially, calculating transmitted intensity and new polarization orientation at every step
- **Wrong:** Using Snell's Law sine ratio to solve for Brewster's angle instead of the tangent relation
  - Why it fails: Brewster's angle is derived from the condition that reflected and refracted rays are perpendicular, leading to the unique tangent form of the law
  - Correct: Use \tan(\theta_B) = n2/n1 for Brewster's angle calculations, confirm θ_B + refracted angle = 90°
- **Wrong:** Assuming all reflected light off any surface is fully polarized at all incident angles
  - Why it fails: Full polarization by reflection only occurs exactly at Brewster's angle, light is only partially polarized at all other incident angles
  - Correct: Explicitly state that 100% polarized reflected light only occurs at the specific Brewster's angle for the media pair
- **Wrong:** Claiming scattered blue skylight is randomly unpolarized
  - Why it fails: Rayleigh scattering of sunlight in the atmosphere produces strongly polarized light oriented perpendicular to the line between observer and sun
  - Correct: Note that skylight polarization is the core design principle behind polarizing sunglasses

## Cheatsheet

| Scenario | Formula / Rule |
| --- | --- |
| Unpolarized light after first polarizer | I = I₀ / 2 |
| Polarized light through polarizer at angle θ | I = I₀ cos²θ (Malus's Law) |
| Brewster's Angle (n₁ → n₂) | tan(θ_B) = n₂ / n₁ |
| Two crossed polarizers (no intermediate filter) | Final intensity = 0 |
| 3 polarizers at 0°, 45°, 90° (unpolarized incident) | Final I = I₀ / 8 |

## What's next

Mastering polarization is a critical stepping stone for the remaining physical optics topics on your AP Physics 2 exam. You will regularly combine Malus's Law calculations with wave interference and diffraction problems in multi-part FRQs that test cross-topic mastery. Polarization concepts also appear frequently in lab-based questions, where you may be asked to design an experiment to verify Malus's Law using a light sensor and rotatable polarizer. To reinforce your understanding, move to the linked topics below to practice applying polarization rules alongside related optics content, and work through our dedicated problem set for this sub-topic to lock in your score on this high-frequency exam concept.

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