Step 1: Understanding the Concept:
When unpolarized light passes through a polaroid, its intensity is halved.
Subsequent passage through additional polaroids follows Malus's Law: \(I = I_0 \cos^2 \theta\), where \(\theta\) is the angle between the pass axes of successive polaroids.
Step 2: Key Formula or Approach:
1. After \(P_1\): \(I_1 = I_{\text{source}} / 2\).
2. After \(P_2\): \(I_2 = I_1 \cos^2 \theta_{12}\).
3. After \(P_3\): \(I_3 = I_2 \cos^2 \theta_{23}\).
Step 3: Detailed Explanation:
Given source intensity \(I_s = 256 \text{ W/m}^2\).
After \(P_1\): \(I_1 = 256 / 2 = 128 \text{ W/m}^2\).
The axis of \(P_2\) is at \(60^\circ\) to \(P_1\), so \(\theta_{12} = 60^\circ\).
After \(P_2\):
\[ I_2 = 128 \cdot \cos^2(60^\circ) = 128 \cdot (0.5)^2 = 128 \cdot 0.25 = 32 \text{ W/m}^2 \]
The axis of \(P_3\) is at \(90^\circ\) to \(P_1\). Since \(P_2\) was at \(60^\circ\) to \(P_1\), the relative angle between \(P_2\) and \(P_3\) is:
\[ \theta_{23} = |90^\circ - 60^\circ| = 30^\circ \]
After \(P_3\):
\[ I_3 = I_2 \cdot \cos^2(30^\circ) = 32 \cdot (\sqrt{3}/2)^2 \]
\[ I_3 = 32 \cdot (3/4) = 8 \cdot 3 = 24 \text{ W/m}^2 \]
Step 4: Final Answer:
The intensity of light at point 'O' is \(24 \text{ W/m}^2\).