Understanding Malus Law in Physics

Malus Law is a fundamental concept in the study of the polarization of light, describing how the intensity of plane-polarized light changes when it passes through a polarizing filter at an angle. This law establishes the quantitative relationship between the transmitted light intensity and the angle between the polarization direction of the incident beam and the axis of the analyzer. It is essential in wave optics and finds usage in optical devices, experiments, and various competitive examinations.

Concept of Polarization and Malus Law

Polarization refers to the orientation of the oscillations of the electric field vector in an electromagnetic wave relative to its direction of propagation. When unpolarized light passes through a polarizer, only the component of the electric field aligned with the transmission axis is allowed to pass, resulting in plane-polarized light. Malus Law mathematically describes the effect of a secondary polarizer, called an analyzer, aligned at an angle with respect to the initial polarization direction.

The understanding of Malus Law is essential to explain phenomena related to the transmission and attenuation of polarized light in media and devices. Students can deepen their grasp of the topic by first reviewing the principles of Polarisation Of Light.

Mathematical Statement and Formula

Malus Law states that the intensity of polarized light transmitted through an analyzer depends on the square of the cosine of the angle between the incident light’s polarization direction and the analyzer's axis. The formula for Malus Law is expressed as:

$I = I_0 \cos^2 \theta$

Here, $I_0$ is the initial intensity of light before the analyzer, $\theta$ is the angle between the polarization direction of the incident light and the axis of the analyzer, and $I$ is the final intensity after passing through the analyzer.

Derivation of Malus Law

Consider a plane-polarized light beam incident on an analyzer at an angle $\theta$ relative to its polarization direction. The electric field vector $E_0$ of the incident light makes an angle $\theta$ with the transmission axis of the analyzer. Its component along the analyzer axis is $E_0 \cos \theta$.

The intensity of light is proportional to the square of the electric field amplitude. Thus, the transmitted intensity is:

$I \propto (E_0 \cos \theta)^2$

$I = I_0 \cos^2 \theta$

This equation confirms the mathematical relationship stated in Malus Law.

Intensity Variation: Special Cases

When the analyzer is parallel to the polarization direction ($\theta = 0^\circ$ or $180^\circ$), $\cos^2 \theta = 1$. Therefore, $I = I_0$, and all intensity is transmitted. When the analyzer is perpendicular ($\theta = 90^\circ$), $\cos^2 90^\circ = 0$, so $I = 0$, meaning no light passes through.

Malus Law Graph: Intensity Versus Angle

The variation of transmitted intensity with the angle $\theta$ is represented graphically as a cosine squared curve. The maximum intensity occurs at $0^\circ$ and $180^\circ$, while zero occurs at $90^\circ$ and $270^\circ$.

Application of Malus Law

Malus Law is widely applied in analyzing the behavior of polarized light in various devices and experimental setups. Its importance extends to technological and scientific fields involving optical filters.

• Verification of polarization in optical experiments
• Design of anti-glare sunglasses and filters
• Measurement of polarization states in laboratories
• Control of light intensity in photographic instruments
• Analysis of stress patterns in transparent materials

To further relate this concept to wave phenomena, refer to the explanations on the Wave-Particle Duality page.

Comparison: Malus Law and Brewster Law

Malus Law for Unpolarized Light

When unpolarized light passes through a polarizer, only the component aligned with the transmission axis is transmitted. The average value of $\cos^2 \theta$ over all possible angles is $\dfrac{1}{2}$, so the transmitted intensity becomes $I = \dfrac{I_0}{2}$ after the first polarizer. Malus Law is then applied to further polarizers for the remaining beam.

Malus Law for Multiple Polarizers

For light passing through more than two polarizers, the intensity after each analyzer is obtained by successive application of Malus Law. For example, with three polarizers, the transmitted intensity is given by $I = \dfrac{I_0}{2} \cos^2\theta_1 \cos^2\theta_2$, where $\theta_1$ and $\theta_2$ are the angles between the transmission axes of consecutive polarizers.

Solved Example: Using Malus Law

A beam of plane-polarized light of intensity $8\,\mathrm{W/m^2}$ is incident on a polarizer oriented at $60^\circ$ to the beam's polarization direction. Calculate the intensity after the polarizer.

Given $I_0 = 8\,\mathrm{W/m^2}$, $\theta = 60^\circ$; using $I = I_0 \cos^2 \theta$:

$\cos 60^\circ = 0.5 \implies \cos^2 60^\circ = 0.25$

$I = 8 \times 0.25 = 2\,\mathrm{W/m^2}$

Critical Points and Common Mistakes

• The angle $\theta$ is always between polarization direction and analyzer axis
• Malus Law does not apply directly to unpolarized light; halve intensity first
• Always use $\cos^2 \theta$ in calculations
• No light is transmitted if polarizer and analyzer are perpendicular

Reviewing the concept of the Huygens Principle can reinforce foundational knowledge in wave optics, supporting the understanding of polarization and related laws.