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Wave Optics & Transverse Waves

Malus's Law

Explore how the intensity of polarized light changes as it passes through a rotating analyzer filter. Verify the cosine-square relation with digital meters, phone screens, and camera glare controls.

Malus's Law Verification ⓘ

Vary the angle θ and monitor the intensity drop and light spot brightness.

Simulation active

Malus's Law Telemetry

I = I₀ cos²θ
Incident Intensity (I₀)
100 W/m²
Angle (θ)
60.00°
cos²(θ)
0.25
Transmitted (I)
25 W/m²
Transmission %
25 %

Understanding Malus's Law

Malus's Law dictates how much light intensity passes through a polarizing filter when it is rotated relative to the incoming polarized beam's polarization axis.

Since a polarizing filter only transmits the component of the electric field vector parallel to its transmission axis (i.e. E = E₀ cosθ), and since intensity I is proportional to the square of the electric field amplitude (I ∝ E²), the transmitted intensity varies according to the square of the cosine of the angle:

I = I₀ cos²θ

Polarizer vs Analyzer

The Polarizer converts unpolarized light (like sunlight) into linearly polarized light, reducing its intensity by exactly 50%. The Analyzer is a second polarizer used to control or measure the intensity of the light by rotating its axis relative to the first filter.

Special Angles

Review the three crucial conditions of Malus's Law:

  • θ = 0° (Parallel): cos²(0°) = 1, giving maximum transmission (I = I₀).
  • θ = 45° (Intermediate): cos²(45°) = 0.5, giving exactly half transmission (I = 0.5 I₀).
  • θ = 90° (Crossed): cos²(90°) = 0, giving complete darkness (I = 0).

Applications

Malus's Law is widely used in technology and daily life:

  • LCD Screen Displays: Pixels control light transmission by electronically twisting liquid crystals between crossed polarizers.
  • Photography: Circular polarizers (CPL) block horizontally polarized glare off reflective surfaces to capture richer sky colors.

Step-by-Step Solved Problems

Learn how to apply Malus's Law with these worked mathematical examples.

Example 1 Problem Statement

A beam of plane polarized light of intensity 80 W/m² is incident on a polarizing filter whose transmission axis is at an angle of 30° to the polarization direction of the incident light. Calculate the transmitted light intensity.

View Mathematical Solution Steps
  1. Recall Malus's law: I = I0 cos²θ.
  2. Identify the incident polarized intensity: I0 = 80 W/m².
  3. Identify the angle: θ = 30°.
  4. Calculate cos(30°): cos(30°) = √3 / 2 ≈ 0.866.
  5. Calculate cos²(30°): (0.866)² = 0.75.
  6. Substitute values into the formula: I = 80 × 0.75 = 60 W/m².

Final Derived Answer: Transmitted Intensity I = 60 W/m².

Example 2 Problem Statement

At what orientation angle relative to the polarization direction of incident light will a polarizing filter reduce the light intensity to exactly 25% of its initial value?

View Mathematical Solution Steps
  1. Recall Malus's law: I = I0 cos²θ.
  2. Set the ratio of intensities: I / I0 = 0.25 (or 25%).
  3. Solve for cosθ: cos²θ = 0.25 → cosθ = √0.25 = 0.5.
  4. Find the angle θ: θ = cos⁻¹(0.5) = 60°.

Final Derived Answer: Orientation Angle θ = 60° (or 120°).

Example 3 Problem Statement

Unpolarized light of intensity 160 W/m² passes through a polarizer-analyzer system where the angle between their transmission axes is 45°. Determine the intensity of the light emerging from the analyzer.

View Mathematical Solution Steps
  1. Step 1: Calculate the intensity after passing through the first polarizer. Unpolarized light loses exactly half of its intensity: I1 = Iunpolarized / 2 = 160 / 2 = 80 W/m².
  2. Step 2: Apply Malus's law for the analyzer at 45°: I = I1 cos²(45°).
  3. Recall cos(45°) = 1/√2, so cos²(45°) = 0.5.
  4. Calculate emerging intensity: I = 80 W/m² × 0.5 = 40 W/m².

Final Derived Answer: Emerging Intensity I = 40 W/m².

Self-Check Questions

Question 1

Explain the role of the polarizer and the analyzer in verifying Malus's law.

Show Answer & Explanation

The polarizer takes incoming unpolarized light and restricts its electric field vibrations to a single plane, creating linearly polarized light. The analyzer is a second polarizing filter placed downstream. Rotating the analyzer changes the angle θ between the transmission axes, allowing us to measure how the transmitted intensity varies according to Malus's Law.

Question 2

State what happens to the transmitted intensity of polarized light when the analyzer is rotated to angles of 0°, 45°, and 90°.

Show Answer & Explanation

At 0°, the axes are parallel, cos²(0°) = 1, and the transmission is 100%. At 45°, cos²(45°) = 0.5, and the transmission drops to 50%. At 90°, the axes are orthogonal (crossed), cos²(90°) = 0, and the transmission is 0% (total darkness).

Question 3

How does rotating a polarized sunglass lens over a phone screen demonstrate Malus's law?

Show Answer & Explanation

A phone screen emits plane-polarized light. When you look at the screen through a polarized sunglass lens (which acts as an analyzer) and rotate the lens, the angle θ changes. The light fades and brightens according to I = I₀ cos²θ, going completely black at 90°.

Question 4

Why is the intensity of unpolarized light halved when passing through an ideal polarizing filter?

Show Answer & Explanation

Unpolarized light consists of electromagnetic waves vibrating in all possible directions. When resolved into horizontal and vertical components, the average intensity is equally distributed between the two axes. An ideal polarizer completely blocks one component and transmits the other, reducing the total intensity by exactly 50%.

Question 5

If two polarizers are crossed, why does placing a third polarizer at 45° between them restore some transmitted light?

Show Answer & Explanation

When crossed, the first polarizer polarizes light vertically, and the final analyzer blocks it horizontally. Inserting a 45° middle polarizer resolves the vertical electric field into a 45° component, which is transmitted. When this reaches the horizontal analyzer, it is resolved again, resulting in 25% of the polarized light being transmitted.

Question 6

State the mathematical formula of Malus's law and explain what each variable represents.

Show Answer & Explanation

The formula is I = I₀ cos²θ. I₀ is the intensity of the incident plane-polarized light hitting the analyzer, I is the intensity of the light transmitted through the analyzer, and θ is the angle between the transmission axes of the polarizer and the analyzer.