Two 'crossed' polaroids A and B are placed in the path of a light-beam. In between these, a third polaroid is C is placed whose polarisation axis makes an angle `theta` with a polarisation axis of the polaroid A. If the intensity of light emerging from the polaroid a is `I_(o)` then the intensity of light emerging from polaroid B will be

Options

A. `(I_(0))/(4)sin^(2)2phi`
B. `(I_(0))/(4)sin^(2)phi`
C. `(I_(0))/(4)cos^(2)phi`
D. `(I_(0))/(2)cos^(2)2phi`

Answer

To solve the problem, we will use Malus's Law, which states that when polarized light passes through a polarizer, the intensity of the transmitted light is given by:

\[ I = I_0 \cos^2(\theta) \]

where:
- \( I_0 \) is the intensity of the incident light,
- \( \theta \) is the angle between the light's polarization direction and the axis of the polarizer.

### Step-by-Step Solution:

1. **Identify the Initial Intensity**:
The intensity of light emerging from polaroid A is given as \( I_0 \).

2. **Calculate the Intensity After Polaroid C**:
Polaroid C is placed at an angle \( \theta \) with respect to the polarization axis of A. According to Malus's Law, the intensity of light \( I_1 \) after passing through polaroid C is:
\[ I_1 = I_0 \cos^2(\theta) \]

3. **Calculate the Intensity After Polaroid B**:
Polaroid B is crossed with A, meaning it is at an angle of \( 90^\circ \) to A. The angle between the light emerging from C and the axis of B is \( 90^\circ - \theta \). Again applying Malus's Law:
\[ I_2 = I_1 \cos^2(90^\circ - \theta) \]
Since \( \cos(90^\circ - \theta) = \sin(\theta) \), we can rewrite this as:
\[ I_2 = I_1 \sin^2(\theta) \]

4. **Substituting \( I_1 \) into the Equation**:
Substitute \( I_1 \) from step 2 into the equation for \( I_2 \):
\[ I_2 = (I_0 \cos^2(\theta)) \sin^2(\theta) \]

5. **Final Expression**:
Thus, we have:
\[ I_2 = I_0 \cos^2(\theta) \sin^2(\theta) \]
We can use the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \) to express this in a different form:
\[ I_2 = \frac{I_0}{4} \sin^2(2\theta) \]

### Final Result:
The intensity of light emerging from polaroid B is:
\[ I_2 = \frac{I_0}{4} \sin^2(2\theta) \]