An unpolarized light is travelling along Z axis through three polarizing sheets. The polarzing direction of the first and third sheet are respectively parallel to X axis and Y axis whereas that of the second one is at `60^(@)` to the Y axis. Then, the fraction of the initial light intensity that emerges from the system in about,

Options

A. zero
B. `(3I_(0))/(32)`
C. `(I_(0))/(32)`
D. `(9I_(0))/(32)`

Explanation

To solve the problem of finding the fraction of the initial light intensity that emerges from the system of three polarizing sheets, we can follow these steps:

### Step 1: Understand the Initial Conditions
We start with unpolarized light of intensity \( I_0 \) traveling along the Z-axis.

### Step 2: First Polarizing Sheet
The first polarizing sheet is aligned with the X-axis. When unpolarized light passes through a polarizing sheet, the intensity of the transmitted light is given by:
\[ I_1 = \frac{I_0}{2} \]
This is because half of the intensity of unpolarized light is transmitted through the first polarizer.

### Step 3: Second Polarizing Sheet
The second polarizing sheet is at an angle of \( 60^\circ \) to the Y-axis. Since the first sheet is aligned with the X-axis, the angle between the first and second sheets is \( 30^\circ \) (since \( 90^\circ - 60^\circ = 30^\circ \)). The intensity after passing through the second polarizing sheet is given by:
\[ I_2 = I_1 \cdot \cos^2(30^\circ) \]
Substituting \( I_1 \):
\[ I_2 = \frac{I_0}{2} \cdot \cos^2(30^\circ) \]
Knowing that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[ I_2 = \frac{I_0}{2} \cdot \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{I_0}{2} \cdot \frac{3}{4} = \frac{3I_0}{8} \]

### Step 4: Third Polarizing Sheet
The third polarizing sheet is aligned with the Y-axis. The angle between the second sheet (which is at \( 60^\circ \) to the Y-axis) and the third sheet (which is at \( 90^\circ \) to the Y-axis) is \( 30^\circ \). The intensity after passing through the third polarizing sheet is given by:
\[ I_3 = I_2 \cdot \cos^2(30^\circ) \]
Substituting \( I_2 \):
\[ I_3 = \frac{3I_0}{8} \cdot \cos^2(30^\circ) = \frac{3I_0}{8} \cdot \frac{3}{4} = \frac{9I_0}{32} \]

### Step 5: Calculate the Fraction of Intensity
The fraction of the initial light intensity that emerges from the system is:
\[ \text{Fraction} = \frac{I_3}{I_0} = \frac{9I_0/32}{I_0} = \frac{9}{32} \]

### Final Answer
Thus, the fraction of the initial light intensity that emerges from the system is \( \frac{9}{32} \).

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