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. * $heta$: 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: — $heta = 0^circ$ (parallel axes), $I = I_0$.
Minimum Transmission (Extinction): — $heta = 90^circ$ (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 $heta$ 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 $heta = 0^circ$ (axes parallel), and zero intensity (extinction) occurs when $heta = 90^circ$ (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 $ heta$? — 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, $heta = 30^circ$.
3
Maximum and Minimum Transmission:

* Maximum: Occurs when $heta = 0^circ$ (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 $heta = 90^circ$ (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. * $heta$: 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 $ heta$ - Critical Definition:

Always measure $heta$ 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 $heta = |50^circ - 20^circ| = 30^circ$.

4. Special Angles and Intensities:

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