Two polaroids A and B are parallel. How a third Polaroid C be placed between A and B so that the intensity of the emergent light be (A) zero (B) maximum?

Answer

To solve the problem of how to place a third Polaroid C between two parallel Polaroids A and B, we need to analyze the situation step by step.

### Step-by-Step Solution

1. **Understanding the Setup**:
- We have two parallel Polaroids A and B. Let's denote the intensity of the incoming light as \( I_0 \).
- The light is vibrating in all directions before it hits Polaroid A.

2. **Intensity After Polaroid A**:
- When unpolarized light passes through Polaroid A, the intensity is reduced to half. Therefore, the intensity after Polaroid A is:
\[ I_1 = \frac{I_0}{2} \]

3. **Introducing Polaroid C**:
- Polaroid C is placed at an angle \( \phi \) with respect to the vertical (the direction of polarization of Polaroid A).
- The intensity of light after passing through Polaroid C can be calculated using Malus's Law, which states that:
\[ I_2 = I_1 \cos^2(\phi) = \frac{I_0}{2} \cos^2(\phi) \]

4. **Intensity After Polaroid B**:
- Polaroid B is also parallel to Polaroid A, so the intensity after passing through Polaroid B is:
\[ I_{\text{out}} = I_2 \cdot \frac{1}{2} = \frac{I_0}{2} \cos^2(\phi) \cdot \frac{1}{2} = \frac{I_0}{4} \cos^2(\phi) \]

5. **Condition for Zero Intensity**:
- To achieve zero intensity at the output, we need:
\[ \cos^2(\phi) = 0 \]
- This occurs when \( \phi = 90^\circ \). Thus, Polaroid C should be placed at an angle of \( 90^\circ \) to the direction of light polarization from Polaroid A.

6. **Condition for Maximum Intensity**:
- To maximize the output intensity, we need to maximize \( \cos^2(\phi) \). The maximum value of \( \cos^2(\phi) \) is 1, which occurs when:
\[ \phi = 0^\circ \text{ or } 180^\circ \]
- Therefore, Polaroid C should be placed either parallel to Polaroid A (0°) or anti-parallel (180°) to achieve maximum intensity.

### Summary of Conditions:
- **For Zero Intensity**: Place Polaroid C at \( 90^\circ \) to the polarization direction of Polaroid A.
- **For Maximum Intensity**: Place Polaroid C at \( 0^\circ \) or \( 180^\circ \) to the polarization direction of Polaroid A.