Animated Solution for Physics - Optics: A beam of unpolarised 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

Imagine a beam of unpolarised light traveling towards you. It has an initial intensity of $I_0$. This light is a chaotic mix of electric field vectors vibrating in all possible directions perpendicular to its path of propagation. There is no preferred direction; it is completely random.

Polaroid A

Now, we place our first polaroid, let's call it $A$, in the path of this light. A polaroid acts like a microscopic picket fence. It only allows the component of the electric field that is vibrating parallel to its transmission axis to pass through.

Because the incoming light is unpolarised, the electric field vectors are distributed uniformly across all angles. By symmetry, the average value of ${\cos }^2\theta$ over all possible angles is exactly $\frac{1}{2}$. Therefore, exactly half of the unpolarised light's intensity gets through. The light emerging from polaroid $A$ is now perfectly plane-polarised, and its intensity is:

Polaroid B and Malus's Law

Next, this plane-polarised light encounters a second polaroid, $B$. But here is the twist: polaroid $B$ is tilted! Its transmission axis makes an angle of $45^{\circ }$ relative to the transmission axis of polaroid $A$.

To find the final intensity, we need to invoke Malus's Law. Malus's Law states that when completely plane-polarised light is incident on an analyser, the intensity $I$ of the transmitted light varies directly as the square of the cosine of the angle $\theta$ between the transmission axes of the polariser and the analyser.

Let's set up our master equation. The incident intensity on polaroid $B$ is $I_1$, and our angle $\theta$ is $45^{\circ }$:

Let's plug in the values we have derived. We substitute $I_1=\frac{I_0}{2}$ into our equation:

We know from basic trigonometry that $\cos (45^{\circ })=\frac{1}{\sqrt{2}}$. Squaring this value gives us exactly $\frac{1}{2}$.

Finally, multiplying these fractions, we get our emergent intensity, $I_R$, as $I_0$ divided by $4$. This elegant result perfectly matches option (c).