Unpolarized light passes through two polaroids, the axis of one is vertical and that of the other is `30^@` to the vertical . What is the orientation and intensity of the transmitted light ?

Explanation

To solve the problem of unpolarized light passing through two polaroids, we will follow these steps:

### Step 1: Understanding the Initial Conditions
We have unpolarized light with an initial intensity \( I_0 \). Unpolarized light means that the light waves vibrate in all directions.

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

### Step 2: First Polaroid
The first polaroid is oriented vertically. When unpolarized light passes through a polaroid, the intensity of the transmitted light is reduced to half of the initial intensity. Therefore, after passing through the first polaroid, the intensity \( I_1 \) is given by:
\[ I_1 = \frac{I_0}{2} \]

**Hint:** The first polaroid will always reduce the intensity of unpolarized light to half.

### Step 3: Second Polaroid
The second polaroid is oriented at an angle of \( 30^\circ \) to the vertical. To find the intensity of the light transmitted through the second polaroid, we use Malus's Law, which states:
\[ I = I_{\text{max}} \cos^2(\theta) \]
where \( I_{\text{max}} \) is the intensity of light entering the polaroid and \( \theta \) is the angle between the light's polarization direction and the axis of the polaroid.

In our case, the intensity entering the second polaroid is \( I_1 = \frac{I_0}{2} \) and the angle \( \theta = 30^\circ \). Thus, we can write:
\[ I_2 = I_1 \cos^2(30^\circ) \]
Substituting the values:
\[ I_2 = \frac{I_0}{2} \cos^2(30^\circ) \]

### Step 4: Calculating \( \cos(30^\circ) \)
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Final Intensity Calculation
Now substituting \( \cos^2(30^\circ) \) back into the equation for \( I_2 \):
\[ I_2 = \frac{I_0}{2} \cdot \frac{3}{4} = \frac{3I_0}{8} \]

### Step 6: Orientation of the Transmitted Light
The light transmitted through the second polaroid will be plane polarized at the angle of the second polaroid, which is \( 30^\circ \) to the vertical.

### Final Answer
The transmitted light has an intensity of \( \frac{3I_0}{8} \) and is plane polarized at an angle of \( 30^\circ \) to the vertical.

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