Experiment 07 · Theory
Malus Law
Malus law describes how the intensity of plane-polarized light changes when it passes through an analyzer — a second polarizer placed after the first — as the angle between their transmission axes is changed. It is the law this experiment sets out to verify, and the whole of it is one cosine, squared.
Polarization of light
Light is an electromagnetic wave in which the electric field vector oscillates perpendicular to the direction the wave travels. In ordinary, unpolarized light that field vibrates randomly in every direction perpendicular to the propagation axis at once. A polarizer transmits only the component of the electric field parallel to its own transmission axis, and what comes out the other side is plane-polarized light: light vibrating in a single plane.
Pass that polarized light through a second polarizer — the analyzer — and how much emerges depends on the angle between the transmission axes of the two.
Statement of the law
If plane-polarized light of intensity I₀, obtained after the first polarizer, falls on an analyzer whose transmission axis makes an angle θ with the polarizer's axis, then the intensity I transmitted through the analyzer is:
Where the square comes from
Let E₀ be the amplitude of the electric field of the polarized light leaving the polarizer, oscillating along that polarizer's transmission axis. When this light falls on the analyzer, only the component of E₀ lying along the analyzer's own axis is transmitted, and resolving the vector gives the transmitted amplitude directly:
Intensity, however, is not amplitude. Intensity goes as the square of the amplitude, I ∝ E², so squaring the line above gives I ∝ E₀² cos²θ, and writing I₀ = kE₀² for the intensity incident on the analyzer leaves I = I₀ cos²θ. That single squaring step is the whole reason the law carries a cos²θ and not a plain cos θ, and it is the point the viva returns to most often.
Four angles worth knowing
| Angle θ | cos²θ | Intensity I | Observation |
|---|---|---|---|
| 0° | 1 | I = I₀ | Maximum intensity — the axes are parallel |
| 45° | 0.5 | I = I₀/2 | Half the intensity gets through |
| 90° | 0 | I = 0 | No light transmitted — the axes are crossed |
| 180° | 1 | I = I₀ | Maximum intensity again |
Read down that table and the shape of the thing is clear: the transmitted intensity is not linear in the angle at all, and it returns to its maximum every 180°. Confirming that cos²θ dependence, rather than assuming it, is what the experiment is for.
How the law is verified
A light source, a convex lens, the polarizer, the analyzer and a photo cell are lined up along the bench at a common height. The polarizer is set once and left alone. The analyzer is turned in steps — 10° at a time in this experiment — from the position of maximum deflection right round to where the meter reads practically nothing, and the microammeter reading is noted at every step.
The current through a photo cell is proportional to the light falling on it, so the meter reading stands in for I. Plot those readings against the calculated cos²θ and, if the law holds, the points fall on a straight line — I ∝ cos²θ. A plot of I against θ itself would be a curve and would prove nothing; it is the choice of cos²θ along the x-axis that turns the law into something you can check with a ruler.