A beam of unpolarised light of intensity `I_0` is passed through a polaroid and then through another polaroid B which is oriented so that its principal plane makes an angle of `45^circ` relative to that of A. The intensity of the emergent light is

Options

A. `I_(0)//2`
B. `I_(0)//4`
C. `I_(0)//8`
D. `I_(0)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understanding the Initial Conditions
We start with a beam of unpolarized light with an intensity \( I_0 \). When unpolarized light passes through a polaroid, it becomes polarized, and its intensity is reduced.

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

### Step 2: Intensity After the First Polaroid
When unpolarized light passes through the first polaroid (let's call it Polaroid A), the intensity of the light that emerges is given by:

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

This reduction occurs because a polaroid only allows the component of light aligned with its axis to pass through.

**Hint**: The intensity of light after passing through a polaroid is halved for unpolarized light.

### Step 3: Setting Up for the Second Polaroid
Next, the light that has passed through Polaroid A (with intensity \( I_A = \frac{I_0}{2} \)) is then passed through a second polaroid (Polaroid B) that is oriented at an angle of \( 45^\circ \) relative to Polaroid A.

**Hint**: The angle between the two polaroids affects the intensity of the light that passes through the second polaroid.

### Step 4: Applying Malus's Law
According to Malus's Law, when polarized light passes through a second polaroid, the intensity of the light that emerges is given by:

\[ I_B = I_A \cos^2(\theta) \]

where \( \theta \) is the angle between the light's polarization direction (from Polaroid A) and the axis of Polaroid B. Here, \( \theta = 45^\circ \).

**Hint**: Malus's Law states that the intensity of polarized light passing through a polaroid is proportional to the square of the cosine of the angle between the light's polarization direction and the polaroid's axis.

### Step 5: Calculating the Final Intensity
Substituting the values we have:

\[ I_B = \left(\frac{I_0}{2}\right) \cos^2(45^\circ) \]

Since \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \), we have:

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

Thus, the intensity of the emergent light after passing through both polaroids is:

\[ \boxed{\frac{I_0}{4}} \]