Q.The figure shows four pairs of polaroids, seen from front. Each pair is mounted in the path of initially unpolarized light. The axis of polaroids is shown with dotted lines. Through which pair the intensity of light seen is :

Ordinary (unpolarised) light passing through a single polaroid always emerges at exactly half the incident intensity, regardless of the polaroid's orientation.

Sodium light is unpolarised — its electric field vibrates randomly in all directions perpendicular to propagation. When such light of intensity $I_0$ passes through a single polaroid, the polaroid transmits only the component of vibration along its transmission axis. Averaging over all random orientations of the incident field gives a transmitted intensity of exact …

Polaroids work by selectively absorbing one plane of polarisation, which controls the intensity of transmitted light.

A polaroid consists of long chain molecules aligned in a particular direction; it transmits light whose electric vector is parallel to this pass-axis and absorbs the perpendicular component. When two polaroids are used together, rotating one relative to the other varies the transmitted intensity according to Malus's law, $I = I_0\cos^2\theta$. In sunglasses, polaroids cut down glare (reduce the intensity of reflected/scattered light); in 3D movie syst …

Unpolarised light is halved by the first polaroid, then reduced further by $\cos^2\theta$ (Malus' law) through the second polaroid at $30°$, giving $3I/8$.

Working

Step 1 — first polaroid: Unpolarised light of intensity $I$ falling on a polaroid always transmits half its intensity as plane-polarised light:

$I_1 = \dfrac{I}{2}$

Step 2 — second polaroid: This linearly polarised light of intensity $I_1$ now falls on the second polaroid, whose transmission axis is inclined at $\theta = 30°$ to the first. By Malus' law: …

By Malus's law, $I=I_0\cos^2\theta$, transmitted intensity is maximum when the analyzer's pass axis is parallel to the polarizer's.

When unpolarized light passes through a polarizer, it becomes linearly polarized along the polarizer's transmission axis, with intensity $I_0$ (half the incident intensity). If this polarized light then falls on an analyzer whose pass axis makes an angle $\theta$ with the polarizer's axis, Malus's law gives the transmitted intensity as

$I = I_0 \cos^2\theta$ …

Intensity transmitted through a pair of polaroids follows Malus's law, $I=I_0\cos^2\theta$, where $\theta$ is the angle between the two polaroid axes — maximum when the axes are parallel, minimum (zero) when they are perpendicular.

Unpolarized light first becomes polarized (and halved in intensity, $I_0/2$) after passing through the first (upper) polaroid. Passing through the second (lower) polaroid then attenuates it further according to Malus's law:

$I = \frac{I_0}{2}\cos^2\theta$

where $\theta$ is the angle between the transmission axes of the upper and lower polaroids.

In each of the four pairs (a)–(d), the lower polaroid's axis is marked at a fixed angle and the upper polaroid's axis is rotated to a different absolute orientation, which changes $\theta$ (and hence the relative angle between the two axes) from pair to pair. To identify the answer:

• The pair where the two marked axis lines are CLOSEST to being parallel to each other (smallest $\theta$, ideally $\theta \to 0°$) transmits the MAXIMUM intensity ($I \to I_0/2$, its largest possible value).
• The pair where the two marked axis lines are CLOSEST to being perpendicular ($\theta \to 90°$) transmits the MINIMUM intensity ($I \to 0$). …

The intensity of polarised light passing through an analyser is I = I₀ cos²θ.

When plane-polarised light of intensity $I_0$ falls on an analyser (a polaroid), only the component of its electric field along the analyser's transmission axis is passed.

If $\theta$ is the angle between the plane of polarisation of the incident light and the analyser's axis, the transmitted amplitude is $E_0\cos\theta$, and since intensity is proportional to the square of amplitude: …