Question

A beam of unpolarized light of intensity $I_0$ is passed through a polaroid $A$ 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

• A. $I_0$
• B. \(\frac{I_0}{2}\)
• C. \(\frac{I_0}{4}\)
• D. \(\frac{I_0}{8}\)

Answer 1

The Correct Option is C

To solve this problem, we need to understand how the intensity of light changes as it passes through polaroids. We begin by considering how unpolarized light becomes polarized and subsequently analyze the effect of passing through a second polaroid.

When unpolarized light of intensity \(I_0\) passes through the first polaroid (Polaroid A), it becomes polarized. The intensity of light after passing through the first polaroid is given by:

\(I_1 = \frac{I_0}{2}\)

This reduction to half is due to Malus's Law, which states that the intensity is halved when unpolarized light passes through a polarizer.

The polarized light of intensity \(I_1 = \frac{I_0}{2}\) then passes through the second polaroid (Polaroid B). According to Malus's Law, the intensity \(I_2\) of the light after passing through the second polaroid, which is oriented at an angle \(\theta\) to the transmission axis of the first polaroid, is given by:

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

Here, \(\theta = 45^\circ\). Therefore,

\(I_2 = \frac{I_0}{2} \cos^2 45^\circ\)

We know \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), so:

\(I_2 = \frac{I_0}{2} \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4}\)

Thus, the intensity of the emergent light after passing through both polaroids is \(\frac{I_0}{4}\), which corresponds to the correct option:

\(\frac{I_0}{4}\)

This logical deduction aligns with the given choices, effectively ruling out the other options. Option \(\frac{I_0}{4}\) is hence confirmed to be correct.