Two Polaroids `P_(1)` and `P_(2)` are placed with their axis perpendicular to eachother. Unpolarised light `I_(0)` is nicident on `P_(1)`. A third polaroid `P_(3)` is kept in between `P_(1)` and `P_(2)` such that its axis makes an angle `45^(@)` with that of `P_(1)`. The intensity of transmitted light through `P_(2)` is

Options

A. `(I_(0))/(2)`
B. `(I_(0))/(4)`
C. `(I_(0))/(8)`
D. `(I_(0))/(16)`

Explanation

To solve the problem step by step, we will analyze the effect of each Polaroid on the intensity of light passing through them.

### Step 1: Incident Light on Polaroid P1
When unpolarized light \( I_0 \) strikes the first Polaroid \( P_1 \), the intensity of the transmitted light is reduced by half. This is a property of Polaroids when unpolarized light passes through them.

**Calculation:**
\[ I_1 = \frac{I_0}{2} \]

### Step 2: Light Passing Through Polaroid P3
The light that passes through \( P_1 \) now has an intensity of \( I_1 \). The second Polaroid \( P_3 \) is oriented at an angle of \( 45^\circ \) with respect to \( P_1 \). According to Malus's Law, the intensity of light transmitted through a Polaroid is given by:
\[ I = I_0 \cos^2(\theta) \]
where \( \theta \) is the angle between the light's polarization direction and the axis of the Polaroid.

**Calculation:**
\[ I_2 = I_1 \cos^2(45^\circ) = \frac{I_0}{2} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4} \]

### Step 3: Light Passing Through Polaroid P2
Now, the light that has passed through \( P_3 \) (with intensity \( I_2 \)) is incident on the third Polaroid \( P_2 \), which is oriented at \( 90^\circ \) to \( P_1 \) and thus \( 45^\circ \) to \( P_3 \). Again, we apply Malus's Law.

**Calculation:**
\[ I_3 = I_2 \cos^2(45^\circ) = \frac{I_0}{4} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{4} \cdot \frac{1}{2} = \frac{I_0}{8} \]

### Final Result
The intensity of the transmitted light through \( P_2 \) is:
\[ I_3 = \frac{I_0}{8} \]

### Conclusion
Thus, the final answer is:
\[ \text{The intensity of transmitted light through } P_2 \text{ is } \frac{I_0}{8}. \]