Malus Law Calculator
Free calculate transmitted light intensity through a polarizer using i = i₀cos²(θ). Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Intensity of polarized light before polarizer
Angle between polarization direction and polarizer axis
Results
Enter parameters to calculate
What is Malus's Law?
Malus's Law (named after Étienne-Louis Malus) describes how the intensity of polarized light changes when it passes through a polarizer. It states that the transmitted intensity is proportional to the square of the cosine of the angle between the light's polarization direction and the polarizer's transmission axis.
When polarized light encounters a polarizer, only the component parallel to the polarizer's axis is transmitted. The transmitted intensity follows I = I₀cos²(θ), where θ is the angle between the polarization direction and the polarizer axis.
This law is fundamental in optics, explaining how polarizers work, how sunglasses reduce glare, and how LCD displays control light. It's also used in optical instruments and polarization-based measurements.
Malus's Law Formula
Where:
- • I = Transmitted intensity (W/m²)
- • I₀ = Initial intensity (W/m²)
- • θ = Angle between polarization and polarizer axis (degrees or radians)
Special cases:
θ = 0°: I = I₀ (maximum transmission, parallel)
θ = 90°: I = 0 (complete blocking, perpendicular)
θ = 45°: I = I₀/2 (half intensity)
How to Calculate
-
1
Convert angle to radians (if needed)
For calculations: θ_rad = θ_deg × π/180. Most calculators can work with degrees directly.
-
2
Calculate cos(θ)
Find the cosine of the angle between polarization direction and polarizer axis.
-
3
Square the cosine
Calculate cos²(θ) = (cos(θ))².
-
4
Calculate transmitted intensity
I = I₀ × cos²(θ). Multiply initial intensity by cos²(θ).
Practical Examples
Example 1: Parallel Polarization
I₀ = 100 W/m², θ = 0° (parallel to polarizer axis).
Solution:
I = 100 × cos²(0°) = 100 × 1²
I = 100 W/m² (maximum transmission)
Example 2: 45° Angle
I₀ = 100 W/m², θ = 45°.
Solution:
I = 100 × cos²(45°) = 100 × (√2/2)² = 100 × 0.5
I = 50 W/m² (half intensity)
Example 3: Perpendicular
I₀ = 100 W/m², θ = 90° (perpendicular to axis).
Solution:
I = 100 × cos²(90°) = 100 × 0²
I = 0 W/m² (complete blocking)
Applications
Sunglasses
Polarized sunglasses use Malus's law to block horizontally polarized light (glare from water, roads), reducing eye strain.
LCD Displays
Liquid crystal displays use polarizers and Malus's law to control light transmission, creating images by rotating polarization.
Photography
Polarizing filters reduce reflections, enhance colors, and control light intensity based on Malus's law principles.
Education
Teaching wave optics, understanding polarization, and demonstrating how polarizers affect light intensity.
Frequently Asked Questions
Why is it cos²(θ) and not just cos(θ)?
Intensity is proportional to the square of the electric field amplitude. The transmitted amplitude is E = E₀cos(θ), so intensity I ∝ E² = (E₀cos(θ))² = E₀²cos²(θ) = I₀cos²(θ).
What happens with unpolarized light?
Unpolarized light passing through a polarizer becomes polarized and has intensity I = I₀/2 (half the original), regardless of polarizer orientation. This is because unpolarized light contains all polarization directions equally.
Can intensity ever exceed I₀?
No! Maximum transmission occurs at θ = 0° where I = I₀. At all other angles, I < I₀. Energy is conserved - the "lost" intensity is absorbed or reflected by the polarizer.
What about multiple polarizers?
For two polarizers at angle θ: I = I₀cos²(θ). For three polarizers: I = I₀cos²(θ₁)cos²(θ₂), where θ₁ and θ₂ are angles between consecutive polarizers. Each polarizer reduces intensity according to Malus's law.
How do polarizers work?
Polarizers contain aligned molecules or structures that absorb light with electric field perpendicular to the alignment. Only the parallel component passes through, creating polarized light and reducing intensity based on Malus's law.