1. Malus’ Law

2. Malus’ Law – Intensity Polarised Light

3. Malus’ Law
This is the law of Malus named after Etienne Malus who published this relationship in 1809.
According to Malus, - when completely plane polarized light is incident on the analyzer, the intensity (I ) of the light transmitted by the analyzer is directly proportional to the square of the cosine of angle between the transmission axes of the analyzer and the polarizer.
i.e   I ∞ cos2θ

4. A 'head-on' view of the analyser will help us to find the intensity of the transmitted beam
Suppose the angle between the transmission axes of the analyzer and the polarizer is θ.
The completely plane polarized light form the polarizer is incident on the analyzer.

5. A little Math
If E0 is the amplitude of the electric vector transmitted by the polarizer, then intensity I0 of the light incident on the analyzer is I0 ∞ E02
(The intensity of a beam, measured in W m-2, is proportional to the square of the amplitude.)

6. The electric field vector E0 can be resolved into two rectangular components i.e E0 cosθ and E0 sinθ.
The analyzer will transmit only the component ( i.e E0 cosθ ) which is parallel to its transmission axis.

7. However, the component E0sinθ will be absorbed by the analyser
However, the component E0sinθ will be absorbed by the analyser. Therefore, the intensity (I )of light transmitted by the analyzer is,
I ∞ ( E0 x cosθ )2
I / I0 = ( E0 x cosθ )2 / E02 = cos2θ
I = I0 x cos2θ
Therefore, I ∞ cos2θ.
This proves law of Malus.

8. When θ = 0° ( or 180° ), I = I0 cos20° = I0 That is the intensity of light transmitted by the analyzer is maximum when the transmission axes of the analyzer and the polarizer are parallel.
When θ = 90°, I = I0 cos290° = 0 That is the intensity of light transmitted by the analyzer is minimum when the transmission axes of the analyzer and polarizer are perpendicular to each other.

9. Example
We will use Malus' law to solve this problem, with I 0 as the intensity of the incident beam and I 0/2 as the intensity of the transmitted beam.
Problem: A sheet of Polaroid is being used to reduce the intensity of a beam of polarised light.
What angle should the transmission axis of the Polaroid make with the plane of polarisation of the beam in order to reduce the intensity of the beam by 50%?