Malus Law

Light, as an electromagnetic wave, consists of oscillating electric and magnetic fields perpendicular to each other and to the direction of wave propagation. In unpolarized light, the electric field vectors oscillate randomly in all possible planes perpendicular to the direction of propagation.

When such unpolarized light passes through a polarizer, it becomes plane-polarized. A polarizer is an optical device that transmits light waves oscillating in a specific plane (its transmission axis) and blocks or absorbs light waves oscillating in other planes.

The intensity of unpolarized light, $I_{unpol}$, is reduced by half upon passing through an ideal polarizer, resulting in plane-polarized light of intensity $I_0 = I_{unpol}/2$. This $I_0$ now represents the intensity of the plane-polarized light incident on the subsequent optical element, often called an 'analyzer'.

Conceptual Foundation:

Malus's Law specifically deals with the interaction of already plane-polarized light with a second polarizer (the analyzer). The electric field vector of the incident plane-polarized light can be resolved into two components: one parallel to the transmission axis of the analyzer and one perpendicular to it.

Only the component parallel to the transmission axis is transmitted through the analyzer. The intensity of light is proportional to the square of the amplitude of its electric field vector ($I \propto E^2$).

Key Principles and Derivation:

Let the plane-polarized light incident on the analyzer have an electric field amplitude $E_0$. Its intensity is $I_0 \propto E_0^2$. Let the transmission axis of the analyzer make an angle $\theta$ with the plane of polarization (or the direction of the electric field vector) of the incident light. The electric field vector $E_0$ can be resolved into two components:

1
$E_x = E_0 \cos \theta$, parallel to the transmission axis of the analyzer.
2
$E_y = E_0 \sin \theta$, perpendicular to the transmission axis of the analyzer.

The analyzer only transmits the component of the electric field that is parallel to its transmission axis. Therefore, the amplitude of the transmitted electric field is $E = E_0 \cos \theta$. The intensity of the transmitted light, $I$, is proportional to the square of this transmitted amplitude: $I \propto E^2 = (E_0 \cos \theta)^2 = E_0^2 \cos^2 \theta$ Since $I_0 \propto E_0^2$, we can write the relationship as:

It describes how the intensity of plane-polarized light changes as it passes through an analyzer rotated by an angle $\theta$ relative to the incident polarization direction.

Important Scenarios and Interpretations:

Parallel Axes ($\theta = 0^\circ$): — When the transmission axis of the analyzer is parallel to the plane of polarization of the incident light, $\theta = 0^\circ$. Since $\cos 0^\circ = 1$, $I = I_0 (1)^2 = I_0$. The maximum intensity is transmitted.
Crossed Axes ($\theta = 90^\circ$): — When the transmission axis of the analyzer is perpendicular to the plane of polarization of the incident light, $\theta = 90^\circ$. Since $\cos 90^\circ = 0$, $I = I_0 (0)^2 = 0$. No light is transmitted. This condition is known as 'crossed polaroids' and results in complete extinction of light.
Intermediate Angles: — For any angle between $0^\circ$ and $90^\circ$, the intensity will be between $0$ and $I_0$. For example, if $\theta = 45^\circ$, $\cos 45^\circ = 1/\sqrt{2}$, so $\cos^2 45^\circ = 1/2$. Thus, $I = I_0/2$.

Real-World Applications:

Malus's Law is not just a theoretical concept; it underpins numerous practical applications:

1
LCD Displays: — Liquid Crystal Displays (LCDs) utilize polarization. Each pixel in an LCD contains liquid crystals sandwiched between two crossed polarizers. By applying voltage, the liquid crystals rotate the plane of polarization of light, allowing it to pass through the second polarizer to varying degrees, thereby controlling the brightness of the pixel according to Malus's Law.
2
Polaroid Sunglasses: — These sunglasses reduce glare from reflective surfaces (like water or roads). Light reflected from non-metallic surfaces is partially polarized horizontally. Polaroid sunglasses have vertically oriented transmission axes, effectively blocking the horizontally polarized glare, as per Malus's Law, where $\theta$ approaches $90^\circ$ for the glare component.
3
Photography: — Polarizing filters are used in photography to reduce reflections from water or glass, darken blue skies, and enhance color saturation. By rotating the filter, the photographer can control the amount of polarized light (e.g., reflections) that reaches the camera sensor, based on Malus's Law.
4
Stress Analysis (Photoelasticity): — Transparent materials, when subjected to stress, become optically anisotropic (birefringent), meaning they change the polarization state of light passing through them. By placing a stressed transparent object between crossed polarizers (a polariscope), stress patterns become visible as varying light intensities, which can be analyzed using Malus's Law to determine stress distribution.
5
Optical Instruments: — Many scientific instruments, such as polarimeters (used to measure the rotation of plane-polarized light by optically active substances) and microscopes, incorporate polarizers and analyzers, with their operation governed by Malus's Law.

Common Misconceptions:

$I_0$ is always the original unpolarized intensity: — This is incorrect. In Malus's Law, $I_0$ refers to the intensity of the plane-polarized light incident on the analyzer. If the light initially was unpolarized with intensity $I_{unpol}$, then after passing through the first polarizer, its intensity becomes $I_0 = I_{unpol}/2$. Students often mistakenly use $I_{unpol}$ directly in Malus's Law.
Angle $\theta$ definition: — The angle $\theta$ is specifically between the transmission axis of the analyzer and the plane of polarization of the incident light. It is not necessarily the angle between the two polarizers' transmission axes unless the incident light is polarized along the first polarizer's axis.
Intensity vs. Amplitude: — Students sometimes confuse the linear relationship of amplitude ($E = E_0 \cos \theta$) with the squared relationship of intensity ($I = I_0 \cos^2 \theta$). Remember that intensity is proportional to the square of the amplitude.
Applicability to unpolarized light: — Malus's Law does not directly apply to unpolarized light. Unpolarized light, when passed through a single polarizer, results in plane-polarized light with half the original intensity, regardless of the polarizer's orientation. Malus's Law comes into play only when this polarized light then passes through a second polarizer (analyzer).

NEET-Specific Angle:

For NEET, questions on Malus's Law often involve scenarios with multiple polarizers. A common setup involves an unpolarized light source, followed by a polarizer (P1), then an analyzer (P2), and sometimes a third polarizer (P3) in between. Here's a systematic approach:

1
Unpolarized light through P1: — If unpolarized light of intensity $I_{unpol}$ passes through the first polarizer P1, the transmitted light is plane-polarized with intensity $I_1 = I_{unpol}/2$. The direction of polarization is along P1's transmission axis.
2
Polarized light through P2: — If this light $I_1$ then passes through a second polarizer P2, whose transmission axis makes an angle $\theta_1$ with P1's axis, the intensity transmitted through P2 will be $I_2 = I_1 \cos^2 \theta_1 = (I_{unpol}/2) \cos^2 \theta_1$. The light emerging from P2 is now polarized along P2's transmission axis.
3
Polarized light through P3 (if present): — If a third polarizer P3 is introduced, whose transmission axis makes an angle $\theta_2$ with P2's axis (or $\theta_{total}$ with P1's axis, depending on how the angles are defined), the intensity transmitted through P3 will be $I_3 = I_2 \cos^2 \theta_2$. It's crucial to correctly identify the angle $\theta$ for each step as the angle between the incident polarized light's plane of polarization and the transmission axis of the current polarizer.

NEET questions frequently test understanding of these sequential applications, often asking for the final intensity or the angle required to achieve a certain intensity. Problems might also involve finding the angle for maximum or minimum transmission, or for a specific fraction of the initial unpolarized intensity. Mastery of Malus's Law, combined with a clear understanding of polarization, is essential for these types of problems.