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Malus Law

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Last updated date: 22nd May 2024
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What is Malus Law?

Malus law deals with the polarization properties of light. It helps us to study the relation of the intensity of light and the polarizer-analyzer. 

The law was derived by Etienne-Louis Malus in 1808. He discovered that natural light could be polarized when reflected by a glass surface. He used a calcite crystal to conduct his experiment.

After conducting the experiment, he observed that two types of polarization occurred in natural light that is s- and p- polarization, which are mutually perpendicular to each other. 

This law is used to relate the intrinsic connection between optics and electromagnetism. This law also demonstrates the transverse nature of electromagnetic waves.

Malus observed that the intensity varied from maximum to minimum when the crystal was rotated. Accordingly, he proposed that the amplitude of reflected ray must be A = A0  cosθ. Malus squared the amplitude relation in order to obtain the intensity. Hence, the intensity equation I(θ) of the reflected polarized light is given by the following equation,


$I = I_0 \cos^2 \phi$


Where,


$I_0 = A_0^2$


This equation is known as Malus’s Law. The below-given figure represents a normalized plot of Malus’s Law.

 

State Malus Law

It states that the intensity of plane-polarized light that passes through an analyzer varies directly with the square of the cosine of the angle between the plane of the polarizer and the transmission axes of the analyzer.

 

Malus Law Proof

Let θ be the angle between the transmission axes of the analyzer and the polarizer. The plane-polarized light that emerges from the polarizer is incident on the analyzer. The intensity of the incident light on the analyzer ‘I0’ is directly proportional to the square of the amplitude of the electric vector ‘E0’.

I ∞ E02

Two rectangular components viz: E0cosθ and E0sinθ can be derived from the electric field E0. Here the electric vector E0cosθ that is parallel to the transmission axis gets transmitted through the analyzer.

The component of the electric vector E0sinθ will be absorbed by the analyzer. The intensity ‘I’ of light transmitted by the analyzer is,

I ∝ (E0  x cosθ)2

Hence,

I / I0 = (E0 x cosθ)2/E02 = cos2θ

I = I0 x cos2θ

Therefore, 

I ∝ cos2θ 

The above equation proves Malus’s law.

 

Malus Law Experiment

Aims and Objective: The aims and objectives of this experiment are to identify the relationship between the intensity of light transmitted through the analyzer and the angle ‘𝜃′ between axes of polarizer and analyzer.

 

Apparatus Used: Requirements of the experiment are:

  • A diode laser

  • Photodetector

  • Polarizer-analyzer pair

  • Output measuring unit (microammeter)

  • A dial fitted with the polarizer, and

  • Optical bench.

 

Malus Law Formula

The transmitted light's intensity is given by the formula:

\[I_{t} = A_{t}^{2} = A_{o}^{2} Cos^{2}\theta  = I_{o} Cos^{2}\theta \]

Where,

It = intensity of light transmitted through the analyzer;

Io = intensity of the incident plane-polarized light l, and

θ = angle between the axis of polarizer and analyzer

 

Theory

Sunlight, candlelight, and the light emitted by an electric bulb is ordinary light and are called unpolarized light. The magnetic and electric fields vibrate in all possible directions perpendicular to each other and also perpendicular to the direction of propagation of the light in unpolarized light.

The unpolarized light is represented in figure- 1(a). The unpolarized light is considered to be composed of two linear and orthogonal polarization states with complete incoherence.

When unpolarized light is incident on a polarizer, the intensity of transmitted light from the polarizer reduces to one-half of the incident light. 

Also, no change in the irradiance of the transmitted light occurs if the polarizer is rotated with respect to the incident light, and its intensity remains the same as half of the incident light.

 

Polarization

Some transparent materials like Nicol and Tourmaline are capable of filtering light, thereby allowing light waves with vibrations in only one place. Such material is called Polaroids.

The filtration of light is possible only because of the structure of the material, which has its cells arranged in a straight line and in a direction parallel to the axis of the polarizer. (Represented in figure 1 (b) and 1( c).)

The phenomenon of filtration of light waves and producing it with vibrations in a single direction is called polarization. Polarization is the property of a material by which it filtrates light and makes it directional.

 

Discussion

  1. The current for different angles of rotation of the analyzer makes a cosine curve of 360° rotation. It indicates the validity of the equation. The experimentally measured current (Iexpt) and (Itheo) that were calculated using equation Itheo = Imax Cos2θ is within the limits of error.

  2. The relative intensity of light that comes out from the analyzer is maximum at 0° and 180°. It attains a minimum value of 90° and 270°. The cosine values vary as per the graph between those values.

The curve of light intensity Iexpt versus Cos2θ is a straight line with a unit slope, which indicates the correctness of Malus's law.


Solved Problems:

1. Monochromatic light of intensity $I_0$ passes through a polarizer. If the intensity of light emerging from the polarizer is I, what will be the angle between the transmission axis of the polarizer and the vibration direction of the incident light?

Solution:

Malus' Law is given by $I=I_0 \cos ^2 \theta$, where $I$ is the intensity of light after passing through a polarizer, $I_0$ is the initial intensity, and $\theta$ is the angle between the transmission axis and the vibration direction.

Rearranging the equation for $\theta$ :

$\cos ^2 \theta=\frac{I}{I_0} $

$\theta=\cos ^{-1} \sqrt{\frac{I}{I_0}}$


2. Light of intensity $9 I_0$ passes through two ideal polarizers. If the angle between the transmission axes of the two polarizers is $60^{\circ}$, find the intensity of light emerging from the second polarizer.

Solution:

Using Malus' Law twice, the intensity $I$ after passing through two polarizers is given by: $I=I_0 \cos ^2 \theta_1 \cos ^2 \theta_2$

where $\theta_1$ is the angle between the initial light polarization direction and the first polarizer's transmission axis, and $\theta_2$ is the angle between the two polarizers' transmission axes.

Given $I=9 I_0$ and $\theta_2=60^{\circ}$, you can find $\theta_1$ and then substitute the values into the formula to get the final intensity.


3. A beam of polarized light with intensity $I_0$ passes through two polarizing sheets. If the first sheet is rotated by an angle $\theta_1$ and the second sheet by an angle $\theta_2$, find the intensity of light emerging from the second sheet.

Solution:

The intensity $I$ after passing through two polarizers is given by:

$I=I_0 \cos ^2\left(\theta_1-\theta_2\right)$

This formula is derived by considering the angles between the incident light and the first polarizer $\left(\theta_1\right)$ and between the first polavizer and the second polarizer $\left(\theta_2\right)$.


4. A beam of light is incident on a polarizer with an intensity $I_0$. If the polarizer is rotated by an angle $\alpha$, find the new intensity of the transmitted light.

Solution:

The intensity $I$ after rotating the polarizer is given by:

$I=I_0 \cos ^2 \alpha$

This equation comes directly from Malus' Law, where $\alpha$ is the angle between the original transmission axis and the new transmission axis.


5. Light of intensity $I_0$ passes through a polarizer. If the transmitted light then passes through another polarizer whose transmission axis makes an angle $\theta$ with the original transmission axis, find the intensity of the final transmitted light.

Solution:

The intensity $I$ after passing through two polarizers is given by:

$I=I_0 \cos ^2 \theta$

Here, $\theta$ is the angle between the transmission axes of the two polarizers. Simply plug in the given values to find the final intensity.


Conclusion

Malus Law is a crucial concept in physics, particularly for JEE Main preparation. It explains how light intensity changes when unpolarized light passes through a polaroid. The law states that the intensity of polarized light is proportional to the square of the cosine of the angle between the polarizer and the light's original direction. This mathematical relationship helps analyze and understand the behavior of polarized light, making it valuable in optics. Mastering Malus Law is essential for JEE Main candidates, as it forms the basis for solving problems related to light polarization and optics, contributing to a comprehensive understanding of physics concepts.

FAQs on Malus Law

1. What is Analyser in Malus Law?

When light falls on a polarizer, it gets polarized. When this polarized light falls on another Polaroid (an analyzer), it transmits light as per the orientation of its axis with the polarizer. 

The light’s intensity transmitted through the analyzer is given by Malus' law.

2. State the Difference Between Unpolarized Light and Plane-polarized Light?

The difference between unpolarized light and plane-polarized light is as follows.

  1. The electric and magnetic field of a polarized light oscillates in one direction, but the electric field oscillates in all directions in unpolarized light.

  2. Polarized light is coherent, unpolarized light is incoherent.

  3. The Intensity of polarized light depends on the polarizer, and the intensity of unpolarized light depends on the source.

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3. Unpolarized Light Passes through Two Successive Polaroids (P1 and P2) the Polaroid P1 Makes Angle θ with the Axis of the Polaroid P2. Find Out the Intensity of the Final Coming Light? And if θ is Varied from 0 to 27, Plot the Intensity Variation Graph?

An unpolarized light passes through two successive polaroids.


We know that when polarized light passes through a polaroid, the intensity of light becomes one-half of the original. If the light is further passed through the polaroids, then its intensity is given by Malus' law

I = 1/2cos2θ

Where,

θ = angle between the axis of the polaroids. Intensity varies from 0 to 2π.

4. How Do You Measure Light Intensity?

To measure the intensity of light first, you have to measure the distance between the source of light and the point. Convert the distance into S.I. units.


For example, if the point of application is at a distance of 56 cm from the source of light, the unit should be 0.56 m.

5. What is the Importance of Polarization in the Concept of Malus Law?

A polarizer signifies a kind of filter that lets the waves of light that are of specific polarization conduct through the filter while blocking the waves of the lights that are of other polarizations. A polarizer is an optical filter that has the capacity of identifying the polarization of the light waves and, depending on that, allows the light wave with a specific polarization to pass through it. It successfully does so by blocking The other light waves that are of specific polarization. 


It is especially efficient to filter the light waves that are of undefined polarization or mixed polarization and transform the beam of light into a beam of light that is of well-defined polarization. Since the law of Malus helps students to establish the relationship between the intensity of plane polarised light and analyzer and a polariser, understanding the concept of the polarizer and how the optical filter helps the light waves of a particular polarization pass through the filter is extremely important.

6. What are the Practical Applications of Polarizer?

To understand the concept of how a camera lens works it is very important that you have a good understanding of the concept of a polariser. A polarizer works as an optical filter that lets the light waves of a particular polarization pass through the filter and at the same time it blocks the other light waves there not of the same polarization. 


The concept of the polarizer and the usages can be better understood with the help of a camera and the work process of the camera lens. In case you want to click a picture at a place that is very bright and you want to reduce the exposure or the brightness of the ambiance, then a polarizer can help you in doing that. The proper functioning of a polarizer can reduce the quantity of light that enters the lens of the camera and hence it can help you to reduce the brightness of the ambiance as well. 


But there are also some limitations with using the polarizer. The polarizers will not help you to click a brilliant shot if you are clicking a picture where there is bright sunlight available. Depending on the layers of the glass of your camera lens, using polarizers can create unnecessary flares. Also, you cannot click a good shot in a dark place or at a place where sunlight is not available. It is because the work of the polarizer is to filter particular waves of lights with a specific polarization while blocking the other light waves with other polarizations. If there is no significant amount of light available at a place, then there is no point in filtering a particular light wave as well.

7. What is the Basic Difference between Polarizer and Analyser and their Mechanism?

Polarizers are specific devices that have the ability to convert white light whereas analyzers are sort of devices that are used to determine the characteristics of a particular light. Polarizers can convert white lights into plane polarised lights but analyzers can only determine if a particular light is plane-polarized or not. Since polarizers can convert the white light, that is why they can convert the light beam into a plane-polarized beam of light as well. Analyzers can only detect the characteristics of a particular type of light and hence they can only be used in detection. 


There are also some differences in the usages of polarizers and analyzers and the positions in which they have to be placed. Polarizers are generally placed under the specimen whereas analyzers are always placed above the specimen. Polarizers have the ability to rotate completely and hence they can easily rotate at 360 degrees angle. 


The analyzers cannot rotate as they can only move according to the path of the light. Polarizers are used to convert the light waves that are coming from a particular light source. Analyzers are used to determine the characteristics of the light and to determine whether the light is polarized or not. Analyzers are used to determine the polarization of the light in order to determine the characteristics of the specimen and to conclude if the specimen has the optical properties of refractive index. Analyzers are used to do so because the refractive index of a particular specimen depends on the polarisation of lights.

8. What are the study materials that I should consult in order to understand the concept of Malus law?

The detailed study guide is now available on the website of Vedantu with the in-depth elaboration of various concepts of physics. The Malus law is especially important for the students of class 12 as it will help them to have the foundational knowledge of the characteristics of light waves and the mechanism of an optical system. All the course materials are formatted in PDF versions.