Question

In Fig. 33-42, unpolarized light is sent into a system of three polarizing sheets. The angles${\mathit{\theta }}_{\mathbf{1}}\mathbf{,}{\mathit{\theta }}_{\mathbf{2}}$and${\mathit{\theta }}_{\mathbf{3}}$of the polarizing directions are measured counterclockwise from the positive direction of the yaxis (they are not drawn to scale). Angles${\mathit{\theta }}_{\mathbf{1}}$and ${\mathit{\theta }}_{\mathbf{3}}$are fixed, but angle ${\mathit{\theta }}_{\mathbf{2}}$can be varied. Figure 33-43 gives the intensity of the light emerging from sheet 3 as a function of ${\mathit{\theta }}_{\mathbf{2}}$. (The scale of the intensity axis is not indicated.) What percentage of the light’s initial intensity is transmitted by the system when ${\mathit{\theta }}_{\mathbf{2}}\mathbf{=}\mathbf{30}\mathbf{°}$?

Short answer

Percentage of the light’s initial intensity transmitted by the system when${\theta }_2={30}^0$is$\frac{I_3}{I_0}=9.4\%.$

Step-by-step solution

Step 1: Given

Figure 33-42 is the system of three polarizing sheets.

Figure 33-43 is the graph of l vs$\theta$ .

Determining the concept

Use the concept of change in the intensity of light due to polarization. First, find the angles for the three sheets. Using the equation of intensity for angle, find the fraction of intensity.

Formulae are as follows,

$I=I_0{\cos }^2\theta$

Where, l is the intensity.

(a) Determining the percentage of the light’s initial intensity transmitted by the system when θ2=300.

From the graph, the curve is zero at${\theta }_2=0^0\text{and}{90}^0$. That means sheet 2 is perpendicular to one of the other sheets at${\theta }_2=0^0\text{and}{90}^0$.

For$\theta =0^0$, the first and second sheets are parallel andthethird sheet is perpendicular.

At $\theta ={90}^0,$the second and third sheets are parallel and first sheet is perpendicular.

$$\begin{gathered} {\theta }_1=0^0 \\ {\theta }_3={90}^0 \end{gathered}$$

Given${\theta }_2={30}^0$

The intensity of light fromthefirst sheet as,

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

The intensity of light fromthesecond sheet as,

$$\begin{gathered} I_2=I_1{\cos }^2({\theta }_2-{\theta }_1) \\ {I}_2=I_0{\cos }^2{30}^0 \\ {I}_2=\frac{3}{8}{I}_0 \end{gathered}$$

Similarly, intensity of light from the third sheet is,

$$\begin{gathered} I_3=I_2{\cos }^2({\theta }_3-{\theta }_2) \\ {I}_3=\left(\frac{3}{8}\right)I_0{\cos }^2({90}^0-{30}^0) \\ \left(\frac{3}{8}\right)I_0{\cos }^2\left({60}^0\right) \end{gathered}$$

$I_3=\frac{3}{22}{I}_0$

So the fraction of light as,

$$\begin{gathered} \frac{I_3}{I_0}=\frac{3}{32} \\ \frac{I_3}{I_0}=9.4\% \end{gathered}$$

Therefore, the percentage of the light’s initial intensity transmitted by the system when ${\theta }_2={30}^0$is$\frac{I_3}{I_0}=9.4\%.$