Two polaroid `A` and `B` are set in crossed positions. A third polaroid `C` is placed between the two making `/_theta` with the pass axis of first polaroid. Write the expression for intensity of light transmitted from second polaroid. In what orientations will be transmitted intensity be (i) minimum (ii) maximum ?

Explanation

To solve the problem of finding the intensity of light transmitted from the second polaroid (B) when a third polaroid (C) is placed between two crossed polaroids (A and B), we will follow these steps:

### Step 1: Understand the Configuration
- We have two polaroids, A and B, set at 90 degrees to each other (crossed).
- A third polaroid C is placed between A and B, making an angle θ with the pass axis of polaroid A.

### Step 2: Apply Malus' Law for Polaroid C
- According to Malus' Law, the intensity of light transmitted through a polaroid is given by:
\[ I_C = I_0 \cos^2(\theta) \]
where \( I_0 \) is the intensity of the incident light on polaroid C.

### Step 3: Determine the Intensity Incident on Polaroid B
- The intensity of light incident on polaroid B is the intensity transmitted through polaroid C:
\[ I_B = I_C \cos^2(90^\circ - \theta) \]
- Since \( \cos(90^\circ - \theta) = \sin(\theta) \), we can rewrite this as:
\[ I_B = I_C \sin^2(\theta) \]

### Step 4: Substitute for \( I_C \)
- Substitute the expression for \( I_C \) into the equation for \( I_B \):
\[ I_B = (I_0 \cos^2(\theta)) \sin^2(\theta) \]
- Thus, we have:
\[ I_B = I_0 \cos^2(\theta) \sin^2(\theta) \]

### Step 5: Use Trigonometric Identity
- We can use the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \) to simplify \( I_B \):
\[ I_B = \frac{I_0}{4} \sin^2(2\theta) \]

### Final Expression
- Therefore, the expression for the intensity of light transmitted from polaroid B is:
\[ I_B = \frac{I_0}{4} \sin^2(2\theta) \]

### Step 6: Determine Maximum and Minimum Intensity
- **Maximum Intensity**: The maximum intensity occurs when \( \sin^2(2\theta) = 1 \), which happens when \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).
- **Minimum Intensity**: The minimum intensity occurs when \( \sin^2(2\theta) = 0 \), which happens when \( 2\theta = 0^\circ \) or \( \theta = 0^\circ \).

### Summary of Results
- The expression for the intensity of light transmitted from polaroid B is:
\[ I_B = \frac{I_0}{4} \sin^2(2\theta) \]
- The transmitted intensity is:
- **Minimum** when \( \theta = 0^\circ \)
- **Maximum** when \( \theta = 45^\circ \)