Three polarizing sheets are stacked. The first and third are crossed; the one between has its polarizing direction at $\mathbf{45}\mathbf{.0}\mathbf{°}$to the polarizing directions of the other two. What fraction of the intensity of an originally unpolarized beam is transmitted by the stack?

The second sheet has a polarizing direction atto the polarizing directions of the other two sheets.

Light can be polarized by passing it through a polarizing filter or other polarizing material. The intensity of an unpolarized light after passing through a polarized filter is given by the one-half rule or the cosine-square rule. Here, we need to use the concept of polarization using polarizing sheets along with the equations for the intensity of light passing through the polarizing sheet.

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.

For the first sheet, the incident light is unpolarized.

Thus, using equation (1), the intensity of the emergent light can be given as follows:

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

Now, the second sheet has a polarization direction at, Thus, using equation (2), the intensity from the second sheet can be given as follows:

$\begin{array}{c}{I}_2=I_1{\cos }^2\left({45}^0\right)\\ =0.5\times \frac{1}{2}{I}_0\\ =0.25I_0\end{array}$

In a similar way, for the same polarizing direction, the intensity of the emergent light from the third sheet is given as follows:

$\begin{array}{c}{I}_3=I_2{\cos }^{2}45°\\ =0.5\times 0.25I_0\\ =0.125I_0\end{array}$

So, the fraction of light intensity of an originally unpolarized beam is given as:

$\frac{{\text{I}}_{\text{3}}}{{\text{I}}_{\text{0}}}=0.125$

Hence, the value of the fraction is.$0.125$