WAV-07

Malus's law

I = I0 cos² θ after an ideal polariser.

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PolarisationMalus 1808

Governing equation

I=I_0\cos^2\theta

$\theta \in \left[0, 90\right]$

where

I_0: Incident intensity (W/m²)
\theta: Angle to axis (°)
I: Transmitted intensity (W/m²)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

I=I_0\cos^2\theta

$\theta \in \left[0, 90\right]$

$\theta = 0$ $I = 1$

Along the curve

②

Domain of the variable — blue

\theta_{\min} = 0,\quad \theta_{\max} = 90

$\theta \in \left[0, 90\right]$

$\theta = 0$ $\theta = 0$

Along the curve

③

Response curve — green

I=I_0\cos^2\theta

$\theta \in \left[0, 90\right]$

$\theta = 0$ $I = 1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}\theta}

$\theta \in \left[0, 90\right]$

$\theta = 0.566$ $I^{\prime} = -3.45\times 10^{-4}$

Along the curve

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-07 — Malus's law) is the form associated with Malus 1808. Working symbols: $I_0$ , $\theta$ $\rightarrow$ $I$ . The transmitted amplitude is the projection of E onto the polariser axis, so intensity — the square — follows cos² θ.

Purpose

Purpose: compute

I

from

I_0

\theta

in Waves & optics via

I=I_0\cos^2\theta

I = I0 cos² θ after an ideal polariser. Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I_0 = 1.000\,\mathrm{W/m^{2}}

\theta = 35.000\,\mathrm{^{\circ}}

, the governing relation

I=I_0\cos^2\theta

yields

I = 0.6710\,\mathrm{W/m^{2}}

. Linearly polarised input, lossless polariser. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Incident intensity I_01 W/m²Angle to axis \theta35 °

Outputs

Transmitted intensity I0.6710 W/m²

Reading speed

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Narration of this film

Linearly polarised input, lossless polariser.

The transmitted amplitude is the projection of E onto the polariser axis, so intensity — the square — follows cos² θ.

Reading speed

Watch on YouTube