8.5 Malus's Law

8.5 Malus's Law

Malus's law states that the intensity $I$ of plane polarized light after passing through an analyzer is directly proportional to the square of the cosine of the angle $\theta$ between the transmission axis of the analyzer and polarizer. That is:

where $I_{0} =$ Intensity of the incident polarized light.

Actually, Malus's law gives a mathematical relation between the intensity of the light incident on the first polaroid (i.e., polarizer) and the intensity of light obtained after passing it through the second polaroid (i.e., analyzer). This is shown in Fig. 8.7(a). An analyzer is also a polarizer that is placed after a polarizer. Rotation of the analyzer affects the intensity of the polarized light. It is used to further reduce the intensity of light and also adjust it by adjusting the angle of the analyzer with respect to the polarizer.

Certain transparent crystalline materials, like tourmaline, calcite crystals, etc., are capable of confining vibrations of light waves in only one plane. Such materials are called polaroids which have high directionality in crystal structure. Light can also be polarized by natural phenomena like reflection, refraction and scattering.

If a piece of polaroid is rotated in front of a polarized ray of light, it causes a variation in the intensity of the light that gets through. The reason that causes the variation of intensity is the angle between the initial polarizer and the axis of second polarizer.

When the incident polarized light of amplitude $A_{0}$ strikes the analyzer at an angle $\theta$, it is resolved into two components $A_{0}\cos\theta$ and $A_{0}\sin\theta$ as shown in Fig. 8.7(b). The component $A_{0}\sin\theta$ is absorbed in the analyzer. Since, only $A_{0}\cos\theta$ passes through the analyzer, the amplitude $A$ of the transmitted light is therefore:

Since intensity $I$ is proportional to the square of the amplitude $A$, it can be expressed as:

$I \propto A^{2}$ or $I = kA^{2}$

where $k = 1$, we can write: $I = A^{2}$

or $I = A_{0}^{2}\cos^{2}\theta = I_{0}\cos^{2}\theta$ ($\because I_{0} = A_{0}^{2}$)

Here $I_{0}$ is the intensity of the incident polarized light.

There are two extreme conditions of $\theta$ followed by the above equation given as:

(i) If $\theta = 0^{\circ}$, then $I = I_{0}$. This means the intensity transmitted through the analyzer is equal to the initial light intensity that passes through the polarizer.

(ii) If $\theta = 90^{\circ}$, then $I = 0$. This means the light is extinguished completely, i.e., no light is allowed to pass through the analyzer.