Malus Law — Revision Notes

⚡ 30-Second Revision

Malus's Law: — $I = I_0 \cos^2 \theta$

* $I$: Transmitted intensity. * $I_0$: Intensity of plane-polarized light incident on analyzer. * $\theta$: Angle between plane of polarization of incident light and analyzer's transmission axis.

Unpolarized light through 1st polarizer: — $I_{polarized} = I_{unpolarized}/2$.
Maximum Transmission: — $\theta = 0^\circ$ (parallel axes), $I = I_0$.
Minimum Transmission (Extinction): — $\theta = 90^\circ$ (crossed axes), $I = 0$.
Key values: — $\cos 0^\circ = 1$, $\cos 30^\circ = \sqrt{3}/2$, $\cos 45^\circ = 1/\sqrt{2}$, $\cos 60^\circ = 1/2$, $\cos 90^\circ = 0$.

2-Minute Revision

Malus's Law is your go-to formula for calculating the intensity of plane-polarized light after it passes through a second polarizer, called an analyzer. Remember the core formula: $I = I_0 \cos^2 \theta$.

Here, $I_0$ is crucial – it's the intensity of the already polarized light hitting the analyzer, not the initial unpolarized light. If you start with unpolarized light of intensity $I_{unpol}$, after the first polarizer, its intensity becomes $I_{unpol}/2$.

This halved intensity is your $I_0$ for the subsequent application of Malus's Law. The angle $\theta$ is the angle between the direction of polarization of the light incident on the analyzer and the analyzer's transmission axis.

Maximum intensity ($I_0$) is transmitted when $\theta = 0^\circ$ (axes parallel), and zero intensity (extinction) occurs when $\theta = 90^\circ$ (axes crossed). For problems with multiple polarizers, apply the law sequentially, carefully determining the incident intensity and the correct angle for each step.

Don't forget to square the cosine term!

5-Minute Revision

Let's solidify Malus's Law. It's the quantitative description of how a polarizer (acting as an analyzer) affects the intensity of incident plane-polarized light. The formula is $I = I_0 \cos^2 \theta$.

Key Points to Master:

1
What is $I_0$? — This is the intensity of the plane-polarized light that is incident on the analyzer. It's a common mistake to use the initial unpolarized intensity here. If you start with unpolarized light of intensity $I_{unpol}$, after passing through the first polarizer, its intensity becomes $I_0 = I_{unpol}/2$. This $I_0$ is then used in Malus's Law.
2
What is $\theta$? — This is the angle between the plane of polarization of the incident light and the transmission axis of the analyzer. Always measure this angle carefully. If light is polarized vertically and the analyzer is at $30^\circ$ to the vertical, $\theta = 30^\circ$.
3
Maximum and Minimum Transmission:

* Maximum: Occurs when $\theta = 0^\circ$ (or $180^\circ$), meaning the analyzer's axis is parallel to the incident polarization. $\cos^2 0^\circ = 1$, so $I = I_0$. * Minimum (Zero): Occurs when $\theta = 90^\circ$ (or $270^\circ$), meaning the analyzer's axis is perpendicular ('crossed') to the incident polarization. $\cos^2 90^\circ = 0$, so $I = 0$.

1
Multiple Polarizers: — For systems with three or more polarizers, apply Malus's Law step-by-step. The output intensity and polarization direction from one polarizer become the input for the next. For example, if P1 is at $0^\circ$, P2 at $30^\circ$, P3 at $70^\circ$:

* $I_1 = I_{unpol}/2$ (polarized at $0^\circ$). * $I_2 = I_1 \cos^2(30^\circ - 0^\circ)$ (polarized at $30^\circ$). * $I_3 = I_2 \cos^2(70^\circ - 30^\circ)$ (polarized at $70^\circ$).

Worked Example: Unpolarized light of $160\,\text{W/m}^2$ passes through a polarizer P1, then an analyzer P2 whose axis is $60^\circ$ to P1. Find the final intensity.

1
After P1: $I_1 = 160/2 = 80\,\text{W/m}^2$. (Light is polarized along P1's axis).
2
After P2: $I_2 = I_1 \cos^2 60^\circ = 80 \times (1/2)^2 = 80 \times (1/4) = 20\,\text{W/m}^2$.

Mastering these points ensures you can tackle most NEET problems on Malus's Law.

Prelims Revision Notes

Malus's Law: Quick Recall for NEET

1. The Fundamental Formula:

$I = I_0 \cos^2 \theta$

* $I$: Intensity of light transmitted through the analyzer. * $I_0$: Intensity of plane-polarized light incident on the analyzer. * $\theta$: Angle between the plane of polarization of the incident light and the transmission axis of the analyzer.

2. Initial Unpolarized Light:

If unpolarized light of intensity $I_{unpol}$ passes through the first polarizer, the transmitted light is plane-polarized with intensity $I_{polarized} = I_{unpol}/2$.
This $I_{polarized}$ then becomes the $I_0$ for any subsequent application of Malus's Law.

3. Angle $\theta$ - Critical Definition:

Always measure $\theta$ as the angle between the current direction of polarization of the light and the transmission axis of the polarizer it is currently passing through.
Example: If light is polarized at $20^\circ$ to the vertical, and the next polarizer's axis is at $50^\circ$ to the vertical, then $\theta = |50^\circ - 20^circ| = 30^\circ$.

4. Special Angles and Intensities:

Parallel Axes ($\theta = 0^\circ$): — $\cos^2 0^\circ = 1 \implies I = I_0$ (Maximum transmission).
Crossed Axes ($\theta = 90^\circ$): — $\cos^2 90^\circ = 0 \implies I = 0$ (Complete extinction).
$\theta = 45^\circ$: — $\cos^2 45^\circ = (1/\sqrt{2})^2 = 1/2 \implies I = I_0/2$.
$\theta = 30^\circ$: — $\cos^2 30^\circ = (\sqrt{3}/2)^2 = 3/4 \implies I = 3I_0/4$.
$\theta = 60^\circ$: — $\cos^2 60^\circ = (1/2)^2 = 1/4 \implies I = I_0/4$.

5. Multiple Polarizers Strategy:

Apply the intensity calculation sequentially for each polarizer.
The output of one polarizer (intensity and polarization direction) becomes the input for the next.
Common Scenario: — Unpolarized light $\xrightarrow{P1} I_{unpol}/2 \text{ (polarized along P1 axis)} \xrightarrow{P2 \text{ at } \theta_1 \text{ to P1}} (I_{unpol}/2)\cos^2 \theta_1 \text{ (polarized along P2 axis)} \xrightarrow{P3 \text{ at } \theta_2 \text{ to P2}} (I_{unpol}/2)\cos^2 \theta_1 \cos^2 \theta_2$.

6. Common Traps:

Using $I_{unpol}$ directly in Malus's Law instead of $I_{unpol}/2$.
Incorrectly identifying the angle $\theta$ between the incident polarization and the analyzer's axis.
Forgetting to square the cosine term.

Vyyuha Quick Recall

To remember Malus's Law: Intensity Is Often Cos-squared Theta. (I = I₀ cos² θ)