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 to achieve either zero or maximum intensity of the emergent light, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have two parallel polaroids A and B. The light incident on polaroid A is unpolarized light.

2. **Intensity after Polaroid A**:
- When unpolarized light passes through a polaroid, the intensity of the transmitted light is reduced to half.
- If the initial intensity of the light is \( I_0 \), then the intensity after passing through polaroid A is:
\[ I_A = \frac{I_0}{2} \]

3. **Introduce Polaroid C**:
- Now, we place a third polaroid C between A and B. The orientation of polaroid C can be adjusted to achieve different intensity outcomes.

4. **Intensity after Polaroid C**:
- Let the angle between the transmission axis of polaroid A and polaroid C be \( \phi \).
- The intensity of light after passing through polaroid C is given by Malus's Law:
\[ I_C = I_A \cdot \cos^2(\phi) = \frac{I_0}{2} \cdot \cos^2(\phi) \]

5. **Intensity after Polaroid B**:
- Let the angle between the transmission axis of polaroid C and polaroid B also be \( \phi \) (since A and B are parallel, the angle between A and B is 0).
- The intensity after passing through polaroid B is:
\[ I_B = I_C \cdot \cos^2(\phi) = \frac{I_0}{2} \cdot \cos^2(\phi) \cdot \cos^2(\phi) = \frac{I_0}{2} \cdot \cos^4(\phi) \]

6. **Condition for Zero Intensity**:
- The intensity \( I_B \) will be zero when \( \cos^2(\phi) = 0 \).
- This occurs when \( \phi = 90^\circ \). Therefore, to achieve zero intensity, polaroid C should be placed at an angle of \( 90^\circ \) to polaroid A.

7. **Condition for Maximum Intensity**:
- The intensity \( I_B \) will be maximum when \( \cos^2(\phi) = 1 \).
- This occurs when \( \phi = 0^\circ \) or \( \phi = 180^\circ \). Therefore, to achieve maximum intensity, polaroid C should be aligned parallel to polaroid A (and B).

### Summary:
- To achieve **zero intensity**: Place polaroid C at \( 90^\circ \) to polaroid A.
- To achieve **maximum intensity**: Place polaroid C parallel to polaroid A (at \( 0^\circ \) or \( 180^\circ \)).