Unpolarised light of intensity `I_0` passes through five successive polaroid sheets,each of whose transmission axis makes an angle `45^0` With the previous one.The intensity of the finally transmitted light is

Options

A. `(I_@/64)`
B. `(I_@/16)`
C. `(I_@/8)`
D. `(I_@/32)`

Answer

To solve the problem of unpolarised light passing through five successive polaroid sheets, each at an angle of \(45^\circ\) to the previous one, we can follow these steps:

### Step-by-Step Solution:

1. **Initial Intensity**:
The initial intensity of the unpolarised light is given as \(I_0\).

2. **First Polaroid Sheet**:
When unpolarised light passes through the first polaroid sheet, the intensity of the transmitted light is reduced to half:
\[ I_1 = \frac{I_0}{2} \]

3. **Subsequent Polaroid Sheets**:
For each subsequent polaroid sheet, we will apply Malus's Law, which states that the intensity \(I\) after passing through a polaroid is given by:
\[ I = I_0 \cos^2(\theta) \]
where \(\theta\) is the angle between the light's polarisation direction and the transmission axis of the polaroid.

4. **Second Polaroid Sheet**:
The angle between the first and second polaroid is \(45^\circ\):
\[ I_2 = I_1 \cos^2(45^\circ) = \frac{I_0}{2} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4} \]

5. **Third Polaroid Sheet**:
The angle between the second and third polaroid is also \(45^\circ\):
\[ I_3 = I_2 \cos^2(45^\circ) = \frac{I_0}{4} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{4} \cdot \frac{1}{2} = \frac{I_0}{8} \]

6. **Fourth Polaroid Sheet**:
The angle between the third and fourth polaroid is again \(45^\circ\):
\[ I_4 = I_3 \cos^2(45^\circ) = \frac{I_0}{8} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{8} \cdot \frac{1}{2} = \frac{I_0}{16} \]

7. **Fifth Polaroid Sheet**:
The angle between the fourth and fifth polaroid is also \(45^\circ\):
\[ I_5 = I_4 \cos^2(45^\circ) = \frac{I_0}{16} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{16} \cdot \frac{1}{2} = \frac{I_0}{32} \]

8. **Final Intensity**:
Therefore, the intensity of the finally transmitted light after passing through all five polaroid sheets is:
\[ I_f = \frac{I_0}{32} \]

### Final Answer:
The intensity of the finally transmitted light is \(\frac{I_0}{32}\).