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Malus law is an important law when it comes to learning and understanding the polarization properties of light. The law helps us to study the light intensity relation of the polarizer-analyzer. Malus law is named after Étienne-Louis Malus. He discovered that natural incident light could be polarized when it was reflected by a glass surface. He used calcite crystal for his experiment. The year of the discovery of Malus Law was 1808.

Formulation of the Law

After observing the results, he further put forth a concept that natural light consisted of the S- and P-polarization. He discovered that these were perpendicular to each other. Today, this law is used to define the intrinsic connection between optics and electromagnetism. It is used in the application and demonstration of the transverse nature of electromagnetic waves.

To summarize, according to Malus Law, when a completely plane-polarized light is incident on the analyzer, the intensity of the light transmitted by the analyzer is directly proportional to the square of the cosine of the angle between the transmission axes of the analyzer and the polarizer.

Malus Law Formula

The law helps us quantitatively verify the nature of polarized light. Let us understand the expression for Malus’ law.

Point A – The Unpolarized light is incident on an ideal polarizer. Here, the intensity of the transmitted light is exactly half that of the incident polarized light no matter how the polarizing axis is oriented.

Point B – An ideal polarizing filter passes 100% of incident unpolarized light, that is polarized in the direction of filter’s (Polarizer) Polarizing axis.

From point (A) and point (B) we can assume I = Io cos2 ϕ

The average value of I (< I >):

This satisfies point (B) mentioned above.

To show point (A), let us consider ϕ = 0

That implies cos2ϕ = 1

I = I0

To determine the direction of polarization we need one polarizer which is known as analyzer oriented making an angle (p) with the polarizer.

Polaraization of Light

What happens, when the linearly polarized light emerging from a polarizer passes through a second polarizer (analyzer) in general the polarizing axis of the second polarizer (analyzer) makes an angle (d) with the polarizing axis of the first polarizer.

The intensity of an electromagnetic wave is proportional to the square of the amplitude of the wave. The ratio transmitted to incident amplitude is cos ϕ, so the ratio transmitted to incident intensity is cos2ϕ.

Solved Question and Answer

1. Which wave can be polarized?


Following Malus Law, the Lightwave gets polarised. The phenomenon of polarization takes place only in the transverse nature of waves. So sound waves cannot be polarised.

2. What is the difference between unpolarized light and plane-polarized light?


The orientation of electric vectors is in all possible direction


The orientation of electric field vectors will be in all possible directions

orientation of electric field vectors is perpendicular to direction of prorogation of light


The orientation of Electric field vectors will be in a direction on a plane perpendicular to the direction of prorogation of light.

3. An Unpolarised light with intensity (I) is passing through a polarizer. What happens to the intensity of incident light? Explain using Malus Law.


An unpolarized light of intensity (I) passes through a polariser, outcoming light intensity becomes half of its initial value.

An unpolarised light of intensity

An unpolarized light of intensity

4. How an Unpolarised light of intensity (I0) get plane-polarized when passing through a polaroid?

(i) An Unpolarised light of intensity (I0) passes through two successive polaroids (P1 and P2) and corresponding intensities of light coming out from them is (I1 and I2) clearly distinguish the difference between (I1) and (I2).

(ii) What is the necessary condition to get maximum intensity after passing through two successive polaroids?


Following Malus’ law, an unpolarized light passes through a polaroid. The electric field vector of unpolarised light gets polarised along a direction. This is a plane perpendicular to the direction of propagation of light. The intensity of the plane polarised light becomes half its incident light intensity. This plane polarised light passes through another polaroid called analyser. The outcoming light, intensity as a function of cosine square angle between the plane of polariser and analyzer.

Where θ – is the angle between the plane of the polarizer and analyser

The angle (θ) between the plane of transmission of polariser and analyser must be zero (or) the polarizer and analyser must be parallel to each other

Unpolarised light of intensity (I0) getting plane polarized

Unpolarised light of intensity (I0) getting plane polarized

Intensity coming out from polaroid P1 and P2 (I1 and I2)

Following Malus law, when (I0) intensity of unpolarised light passes through a polaroid (P1) its intensity becomes (I1), an unpolarized light have electric field vector in all possible directions, when they pass through a polaroid they get polarised along a direction

Then the same unpolarised light of intensity (I0) after passing through a polaroid its intensity becomes (I1) and this (I1) passes through second polaroid (Analyser) then its intensity becomes (I2).

It is a function of the square of the cosine angle (θ) between the arms of polarization of polaroid and analyser.