Malus Law

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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.

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After conducting the experiment, he observed that two types of polarization occurred in natural light that is s- and p- polarization, which is 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 varies 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() = I0cos2 

Where,

 I0 = A02.

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

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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

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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

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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:

It = At2 = Ao2 Cos2θ = Io Cos2θ

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.

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Results

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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 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.

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

FAQ (Frequently Asked Questions)

Q1. What is Analyser in Malus Law?

Ans: 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.

Q2. State the Difference Between Unpolarized Light and Plane-polarized Light?

Ans: 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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Q3. 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?

Ans: An unpolarized light passes through two successive polaroids.

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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.

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Q4. How Do You Measure Light Intensity?

Ans: 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.