Malus's Law

Core Idea

When polarized light of intensity I₀ passes through a polarizer whose transmission axis makes an angle θ with the polarization direction of the incoming light, the transmitted intensity is I = I₀cos²θ. At θ = 0°, full transmission; at θ = 90°, no transmission. For initially unpolarized light passing through a single polarizer, the transmitted intensity is always I₀/2 regardless of orientation, since all directions are equally represented.

How It's Best Learned

Use a photometer behind two polarizers; rotate the second polarizer while recording intensity. Plot I vs. θ and verify the cos²θ dependence. Extrapolate to θ = 90° to confirm complete extinction.

Common Misconceptions

• Malus's law applies to already-polarized incident light; for unpolarized light, the first polarizer always transmits exactly half, independent of its orientation.
• Students apply cosθ instead of cos²θ because they project the electric field amplitude (cosθ) but forget that intensity ∝ E².

Explainer

From your study of polarization, you know that polarized light has its electric field oscillating along a single axis. When such light encounters a polarizer — a filter that only transmits oscillations along one specific direction called the transmission axis — the question is: how much light gets through? The answer depends entirely on the angle θ between the incoming light's polarization direction and the polarizer's transmission axis.

The derivation starts with vector projection. The incoming electric field has amplitude E₀. Only the component of that field along the transmission axis can pass through: E_transmitted = E₀ cos θ. This is a direct application of the trigonometry you studied — projecting a vector onto another direction recovers a factor of cosine of the angle between them. But transmitted intensity is not proportional to amplitude — it is proportional to amplitude squared (intensity scales as the square of the field amplitude). So I_transmitted = I₀ cos²θ. This is Malus's Law, and the squaring step is where students most often err. A cosθ answer confuses amplitude with intensity.

The behavior of cos²θ is worth memorizing through its key values. At θ = 0° (polarizer perfectly aligned with incoming polarization), cos²0° = 1 — full transmission. At θ = 90° (polarizer perpendicular to incoming polarization), cos²90° = 0 — complete extinction, no light passes. At θ = 45°, cos²45° = 0.5 — exactly half the intensity is transmitted. Two polarizers crossed at 90° produce total darkness. Inserting a third polarizer between them at 45° and applying Malus's Law twice in sequence shows that light now passes through the combination — a counterintuitive result that follows directly from the mathematics and can be verified experimentally with three inexpensive polarizing filters.

For initially unpolarized light, no single polarization direction dominates; all orientations of the electric field are equally represented. Averaging cos²θ over all orientations from 0° to 360° gives exactly ½. So a single polarizer always transmits exactly I₀/2 of the incident unpolarized intensity, regardless of how you orient it — the orientation only matters for a second polarizer placed downstream, where Malus's Law then applies with the angle between the two transmission axes.