Problem — Crossed polarisers

Problem

Two polarising sheets have their transmission axes crossed at $90^\circ$. Unpolarised light enters the first sheet: what comes out of the second? Discuss using Malus’s law.

Solution

After the first sheet $P_1$. Unpolarised light contains all oscillation directions. A polariser transmits half the intensity and makes the light polarised along its own axis (here, vertical): $I_1=\frac{I_\text{nat}}{2}$

At the second sheet $P_2$. The light reaching $P_2$ is vertically polarised, but the axis of $P_2$ is at $90^\circ$ (horizontal). Malus’s law gives the transmitted intensity: $I_2=I_1\cos^2\theta=\frac{I_\text{nat}}{2}\cos^2 90^\circ$

Calculation. Since $\cos 90^\circ=0$: $I_2=\frac{I_\text{nat}}{2}\cdot 0=0$

No light emerges from the crossed polarisers: the vertical component transmitted by $P_1$ has no projection onto the horizontal axis of $P_2$.

$\ev{I_2=0\ \Rightarrow\ \text{total darkness}}$

Collegamenti

Argomenti: Onde elettromagnetiche Concetti: Polarizzazione e legge di Malus · Intensità di un’onda