An unpolarized beam of light is sent into a stack of four polarizing sheets, oriented so that the angle between the polarizing directions of adjacent sheets is$\mathbf{30}\mathbf{°}$. What fraction of the incident intensity is transmitted by the system?

The incident light is unpolarized.

The stacks of four polarizing sheets are arranged such that the angle between the polarizing directions of adjacent sheets is 30^o

When a polarizing sheet is placed in the path of light, the emerging light is polarized parallel to the polarizing direction of the sheet. If the original light is unpolarized, the emerging light is polarized, having an intensity lesser than the incident light.

Formulae:

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

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

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

It is given that incident light is unpolarized. Thus, the intensity of the transmitted light through the first sheet will be calculated using equation (1) as follows:

$I_1=\frac{1}{2}{I}_0$

Now, this polarized light is incident on the second sheet, which is making an angle of$30°$with the polarizing angle of the first sheet. Thus, the intensity of the light coming out of the second sheet will be given using equation (2) as follows:

$\begin{array}{c}{I}_2=I_{1}co{s}^{2}30°\\ =\frac{1}{2}{I}_0\times co{s}^{2}30°\end{array}$

Similarly, the intensity of the light coming out of the third sheet for the same polarizing angle $30°$will be given using equation (2) as follows:

$\begin{array}{c}{I}_3=I_{2}co{s}^{2}30°\\ =\frac{1}{2}{I}_0\times co{s}^{2}30°\times co{s}^{2}30°\end{array}$

And the intensity of the light coming out of the fourth sheet for the same polarizing angle$30°$will be given using equation (2) as follows:

$\begin{array}{c}{I}_4=I_{3}co{s}^{2}30°\\ =\frac{1}{2}{I}_0\times co{s}^{2}30°\times co{s}^{2}30°\times co{s}^{2}30°\end{array}$

Thus, the fraction of incident intensity transmitted through our sheets will be given as follows:

$\begin{array}{c}\frac{I_4}{I_0}=\frac{1}{2}\times co{s}^{2}30°\times co{s}^{2}30°\times co{s}^{2}30°\\ =\frac{1}{2}\times \frac{3}{4}\times \frac{3}{4}\times \frac{3}{4}\\ =\frac{27}{128}\\ =0.21\end{array}$

Hence, the value of the fraction is.$0.21$