Two polarizing sheets, one directly above the other, transmit p% of the initially unpolarized light that is perpendicularly incident on the top sheet. What is the angle between the polarizing directions of the two sheets?

A beam of unpolarized light is sent through two polarizing sheets placed one on top of the other, transmitting p% of the initially unpolarized light.

Light can be polarized by passing it through a polarizing filter or other polarizing material. By using the relation between original intensity and transmitted intensity, we can find the polarizing angle between the polarizing directions of the sheets.

Formulae:

If the incident light is un-polarized, then the intensity of the merging light using one-half rule is given by,$I=0.5I_0\cdot \cdot \cdot \cdot \cdot \cdot \left(1\right)$

If the incident light is already polarized, then the intensity of the emerging light is cosine –squared of the intensity of incident light,$I=I_0{\cos }^2\theta \cdot \cdot \cdot \cdot \cdot \cdot \left(2\right)$

Here,$\theta$is the angle between polarization of the incident light and the polarization axis of the sheet.

Let $I_{0}andI$be the incident and transmitted intensity respectively.

Initially the beam is unpolarized. When it passed through first sheet using equation (1), the transmitted intensityI is half the original intensity. $I_0$ That is given as:

$I=\frac{1}{2}{I}_0$

This is then passed throughthesecond sheet, for which the emergent intensity from sheet 2 from sheet 1 is given using equation (2) as follows:

$\begin{array}{c}{I}_2=I_1{\cos }^2\theta \\ =\frac{1}{2}{I}_0{\cos }^2\theta \cdot \cdot \cdot \cdot \cdot \cdot \left(3\right)\end{array}$

It is given thatthe sheets transmit p% of the initially unpolarized light. Thus, the given emergent intensity from sheet 2 is given as follows:

$\begin{array}{c}I=p\%\text{of}{I}_0\\ =\frac{p}{100}{I}_0\cdot \cdot \cdot \cdot \cdot \cdot \left(4\right)\end{array}$

Using equations (3) and (4), we can get the polarizing angle between the two sheets as follows:

$\begin{array}{c}\frac{1}{2}{I}_0{\cos }^2\theta =\frac{p}{100}{I}_0\\ {\cos }^2\theta =\frac{p}{50}\\ \cos \theta =\sqrt{\frac{p}{50}}\\ \theta ={\text{cos}}^{-1}\sqrt{\frac{p}{50}}\end{array}$

Hence, the angle between the polarizing directions of the sheets is.${\text{cos}}^{-1}\sqrt{\frac{p}{50}}$