Two polaroids are crossed. If one of them is rotated through `30^(@)` and unpolarised light of intensity `l_(0)` is incident on the first polaroid, then the intensity of transmitted light will be

Options

A. `(l_(0))/(4)`
B. `(3l_(0))/(4)`
C. `(3l_(0))/(8)`
D. `(l_(0))/(8)`

Explanation

To solve the problem of finding the intensity of transmitted light when two polaroids are crossed and one is rotated through 30 degrees, follow these steps:

### Step 1: Understand the Initial Condition
Initially, we have two polaroids that are crossed, meaning the angle between their transmission axes is 90 degrees. When unpolarized light of intensity \( I_0 \) passes through the first polaroid, the intensity of the light transmitted through the first polaroid can be determined.

**Hint:** Remember that unpolarized light has equal intensity in all directions.

### Step 2: Calculate the Intensity After the First Polaroid
The intensity of light transmitted through the first polaroid is given by Malus's Law, which states that the intensity \( I \) after passing through a polarizer is:

\[ I = \frac{I_0}{2} \]

This is because the unpolarized light is reduced to half its intensity when it passes through the first polaroid.

**Hint:** The factor of \( \frac{1}{2} \) comes from the nature of unpolarized light.

### Step 3: Determine the New Angle Between the Polaroids
After rotating one of the polaroids by \( 30^\circ \), the angle between the two polaroids becomes:

\[ 90^\circ - 30^\circ = 60^\circ \]

**Hint:** Visualize the rotation to understand how the angle changes.

### Step 4: Calculate the Intensity After the Second Polaroid
Now, we need to calculate the intensity of light transmitted through the second polaroid. Using Malus's Law again, the intensity after the second polaroid is given by:

\[ I_2 = I_1 \cos^2(\theta) \]

where \( I_1 = \frac{I_0}{2} \) (the intensity after the first polaroid) and \( \theta = 60^\circ \).

Substituting the values:

\[ I_2 = \left(\frac{I_0}{2}\right) \cos^2(60^\circ) \]

Since \( \cos(60^\circ) = \frac{1}{2} \):

\[ I_2 = \left(\frac{I_0}{2}\right) \left(\frac{1}{2}\right)^2 = \left(\frac{I_0}{2}\right) \left(\frac{1}{4}\right) = \frac{I_0}{8} \]

**Hint:** Remember to square the cosine term when applying Malus's Law.

### Final Answer
The intensity of the transmitted light after passing through both polaroids is:

\[ \boxed{\frac{I_0}{8}} \]