Wave Optics Chapter-Wise Test 5

For crossed polaroids, intensity after the second polaroid is \( I_0 \cos^2 \theta \), and after the third (at \( 90^\circ - \theta \)) is \( I = I_0 \cos^2 \theta \sin^2 \theta = \frac{I_0}{4} \sin^2 2\theta \).

Maximum occurs at \( \theta = 45^\circ \), so \( I = \frac{I_0}{4} \).

Unpolarised light has its intensity reduced to half after passing through a polaroid, so \( I = \frac{I_0}{2} \).

Unpolarized light has electric vectors oscillating in all directions in the plane perpendicular to the propagation direction, randomly changing over time.

When the angle of incidence exceeds the critical angle, no refraction occurs, and the ray undergoes total internal reflection.

Minima occur at \( \sin \theta = \frac{n\lambda}{a} \). For the second minimum, \( n = 2 \).

\( \lambda = 6.4 \times 10^{-7} \, \text{m} \), \( a = 8.0 \times 10^{-6} \, \text{m} \).

\( \sin \theta = \frac{2 \times 6.4 \times 10^{-7}}{8.0 \times 10^{-6}} = 0.16 \), \( \theta = \sin^{-1}(0.16) \approx 9.2^\circ \).

Phase difference \( \phi = \frac{2\pi}{\lambda} \Delta \).

For \( \Delta = \frac{3\lambda}{4} \), \( \phi = \frac{2\pi}{\lambda} \cdot \frac{3\lambda}{4} = \frac{3\pi}{2} \).

Secondary maxima occur at \( \theta \approx \frac{(n + \frac{1}{2})\lambda}{a} \). For the second secondary maximum, \( n = 2 \), \( \theta \approx \frac{5\lambda}{2a} \).

Total internal reflection occurs when the angle of incidence exceeds the critical angle, and light travels from a denser to a rarer medium.

The central maximum results from constructive interference of all secondary wavelets in phase, while secondary maxima involve partial cancellations, reducing their width and intensity.

Wavelength in a medium \( \lambda_m = \frac{\lambda_{\text{air}}}{n} \).

Given \( \lambda_{\text{air}} = 660 \, \text{nm} \), \( n = 1.65 \), \( \lambda_m = \frac{660}{1.65} = 400 \, \text{nm} \).