Polarization & Malus's Law
Malus's Law
Light is a wave that can vibrate in any direction sideways as it travels. A polarizer is a filter that only lets through the part of the wave lined up with its own axis, so light dims as the mismatch angle grows — and two polarizers crossed at ninety degrees block the light completely.
The formula
I = I₀ · cos²θ
- I — transmitted intensity (W/m²): the brightness of light that makes it through the polarizer
- I₀ — incoming intensity (W/m²): the brightness of the light before it hits the polarizer
- θ — angle (degrees): the angle between the light's polarization and the polarizer's axis
Worked example
Polarized light with an intensity of 8 watts per square meter hits a polarizer whose axis is turned 60 degrees from the light's direction. How bright is the light that gets through?
- I₀ = 8 W/m²
- θ = 60°
- I = I₀ · cos²θ
- I = 8 W/m² · cos²(60°) = 8 W/m² · 0.25
I = 2 W/m²
Test yourself
The same 8 watt per square meter light now hits a polarizer turned 45 degrees instead. What intensity gets through?
- Correct answer: 4 W/m²
- 8 W/m²
- 2 W/m²
Correct! Cosine squared of forty-five degrees is one half, so 8 W/m² × 0.5 = 4 W/m².
Two polarizers are crossed at 90 degrees, so no light gets through. If you slide a third polarizer in between, tilted 45 degrees from both, what happens?
- Still completely blocked
- Correct answer: Some light now gets through
- All the original light gets through
Right! Each 45 degree stage passes half the light (cosine squared of forty-five degrees is 0.5), so 0.5 times 0.5 equals 0.25 — a quarter of the light sneaks through even though the outer two filters are crossed.
Where you see this
Polarized sunglasses calm the glare off a lake or wet road because reflected light is partly polarized — horizontally, exactly what the lenses are built to block. An LCD screen is millions of tiny dimmers made of polarizers, and 3D-cinema glasses hand each eye its own polarization.
Common mistakes
The dimming law is cos², not cos: at 45 degrees the intensity doesn't drop to some gentle fraction but exactly in half — cos² of 45° is 0.5, so 8 W/m² becomes 4 W/m². The famous surprise is the third filter: two polarizers crossed at 90 degrees block everything, yet sliding one at 45 degrees between them lets light through again, because each stage transmits I = I₀ · cos²θ of what arrives — each cut is a halving, not an annihilation.
How it connects
Interference proved light is a wave; polarization pins down the direction of the vibration — sideways, perpendicular to the travel, which is exactly what a wave needs and a stream of particles lacks. It closes the optics module, with Malus's half-at-45-degrees law running quietly inside every sunglasses lens and LCD pixel you own.