Question

An unpolarized beam of intensity \( I_0 \) is incident on a pair of Nicol’s prism making an angle of 60 degrees with each other. The intensity of the light emerging from the pair is

• A. \( I_0 \)
• B. \( I_0/2 \)
• C. \( I_0/4 \)
• D. \( I_0/8 \)

Hint:

In polarizer problems, always remember that the intensity of light is halved after passing through the first polarizer, and further reduced by the square of the cosine of the angle between the polarizers.

Answer 1

The Correct Option is C

Step 1: Understanding the phenomenon.
The problem involves light passing through two Nicol's prisms. When unpolarized light passes through a polarizer, the intensity of the transmitted light becomes half of the initial intensity. When it passes through a second polarizer, the intensity further reduces depending on the angle between the polarizer axes. In this case, the angle between the prisms is 60 degrees.

Step 2: Intensity after first polarizer.
When unpolarized light passes through the first Nicol’s prism, the intensity reduces to half of the initial intensity: \[ I_1 = \frac{I_0}{2} \]
Step 3: Intensity after the second polarizer.
The second polarizer makes an angle of 60 degrees with the first. The intensity of light emerging from the second polarizer is given by: \[ I_2 = I_1 \cos^2 \theta = \frac{I_0}{2} \cos^2 60^\circ \] Since \( \cos 60^\circ = 0.5 \), we get: \[ I_2 = \frac{I_0}{2} \times \left(0.5\right)^2 = \frac{I_0}{4} \]
Step 4: Conclusion.
Thus, the intensity of the light emerging from the pair is \( \frac{I_0}{4} \). Therefore, the correct answer is (3) \( I_0/4 \).

Answer 2

Unpolarized light of intensity \( I_0 \) first passes through one Nicol prism, and then through a second Nicol prism set at \( 60^\circ \) to the first. Let's work out the intensity emerging at each stage.

Passing unpolarized light through a single polarizer always cuts its intensity in half, regardless of the polarizer's orientation: \[ I_1 = \frac{I_0}{2} \] The second Nicol prism, set at \( 60^\circ \) to the first, transmits a further fraction of this light set by the angle between the two prisms: \[ I = I_1 \times \cos(60^\circ) = \frac{I_0}{2}\times\frac{1}{2} = \frac{I_0}{4} \]

1. \(I_0\): This would require both prisms to transmit their full incoming intensity with no reduction at all, which does not happen even for light aligned with a single polarizer's axis.
2. \(I_0/2\): This is only the intensity right after the first prism; it ignores the further reduction introduced by the second, angled prism.
3. \(I_0/4\): This matches the value obtained above, once the reduction from both prisms is accounted for.
4. \(I_0/8\): This would require an additional reduction beyond what the two-prism arrangement at \(60^\circ\) actually produces here.

Therefore, the correct answer is \(I_0/4\).