Unpolarised light of intensity I is inciedent on a polarizer and the emerging light strikes a second polarizing filter with its axist at `45^(@)` to that of the first. Determine
a) the intensity of the emerging beam and
b) its state of polarization

Options

A. `(1)/(4)` and parallel to second filter
B. `(1)/(4)` and porpendicular to second filter
C. `(1)/(8)` and parallel to second filter
D. `(1)/(8)` and perpendicular to second filter

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the intensity after the first polarizer
When unpolarized light of intensity \( I \) passes through the first polarizer, the intensity of the light that emerges is given by:

\[ I_1 = \frac{I}{2} \]

This is because a polarizer transmits half of the intensity of unpolarized light.

### Step 2: Apply Malus's Law for the second polarizer
The light that emerges from the first polarizer is now polarized. When this polarized light strikes the second polarizer, which is oriented at an angle of \( 45^\circ \) to the first, we can use Malus's Law to find the intensity of the light that emerges from the second polarizer:

\[ I_2 = I_1 \cdot \cos^2(\theta) \]

where \( \theta = 45^\circ \). Substituting the value of \( I_1 \):

\[ I_2 = \frac{I}{2} \cdot \cos^2(45^\circ) \]

### Step 3: Calculate \( \cos^2(45^\circ) \)
We know that:

\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

Thus,

\[ \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 4: Substitute back to find \( I_2 \)
Now substituting \( \cos^2(45^\circ) \) back into the equation for \( I_2 \):

\[ I_2 = \frac{I}{2} \cdot \frac{1}{2} = \frac{I}{4} \]

### Step 5: State of polarization
The state of polarization after passing through the first polarizer is linear polarization in the direction of the first polarizer's axis. After passing through the second polarizer, the light remains linearly polarized but now in the direction of the second polarizer's axis, which is at \( 45^\circ \) to the first.

### Final Answers
a) The intensity of the emerging beam is \( \frac{I}{4} \).

b) The state of polarization of the light is linear polarization at \( 45^\circ \) to the original direction of the unpolarized light.

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