At what temperature are the rates of spontaneous and stimulated emission equal? (Assume \lambda = 500\text{ nm})

A plane-polarised light is incident perpendicularly on a quartz plate cut with faces parallel to optic axis. Find the thickness of the quartz plate which introduces phase difference of 60^\circ between e- and o-rays.

What are the characteristics of stimulated emission? Show that in the optical region, stimulated emission is negligible compared to spontaneous emission.

Obtain an expression for the resolving power of a plane transmission grating. Deduce the missing orders for a double-slit Fraunhofer pattern, if the slit widths are 0\cdot 16\text{ mm} and 0\cdot 8\text{ mm} apart.

Explain the phenomenon of interference in thin films. Why is the contrast better in brightness of fringes obtained from the interference of reflected light rays compared to the transmitted light rays?

Let E_A = E_1 \sin \omega t \quad \text{and} \quad E_B = E_2 \sin (\omega t + \delta) By using analytical method, obtain an expression to explain interference. Also show that intensity varies along the screen in accordance with the law of cosine square in interference pattern.

For stationary waves on a string whose ends are fixed, show that the energy density is maximum at antinodes and minimum at nodes.

A convex lens of focal length 20 cm and a concave lens of focal length 10 cm are placed 20 cm apart. In between them an object is placed at a distance $ x $ from the convex lens. What is the value of $ x $ in cm so that the images formed by both of them coincide ?

Explain how inversion of population is achieved in a He-Ne laser.

Let the two waves with parallel electric fields be given by E_1 = 2 \cos \left( \bar{k}_1 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}, E_2 = 5 \cos \left( \bar{k}_2 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}. Find the intensity of each beam $ I_1, I_2 $ and also the interference term $ I_{12} $ at a point where their path difference is zero. Calculate the visibility $ V \left( = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} \right) $ for the interference pattern. [$ \epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2 $, $ \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 $]

Consider an ensemble of two-level atoms in thermal equilibrium. Show that the ratio of Einstein A and B coefficients is given by \frac{A}{B} = \frac{8\pi h \nu^3}{c^3}. Why is it not possible to achieve inversion of population in a two-level medium ?

Find the phase and group velocities for the wave E (z, t) = E_0 \cos (k_1 z - \omega_1 t) + \cos (k_2 z - \omega_2 t) where $ |k_1 - k_2| \ll k_1 k_2 ; |\omega_1 - \omega_2| \ll \omega_1, \omega_2 $.

Calculate the Fraunhofer diffraction pattern from a grating of N slits with width e, separated by equal opaque spaces d. Find the condition for principal maxima and the corresponding values of intensity. A parallel beam of Na light is incident normally on a plane grating with 4250 lines per cm. The second order spectral line is observed to be deviated through 30^\circ. Calculate the wavelength of light.

What are chromatic and spherical aberrations Explain. Calculate the longitudinal aberration for rays at a height of 0.01\text{ m} from the principal axis and incident on the cowed surface (radius of curvature = 0.1\text{ m}) of a thin piano-convex lens made of glass (refractive index = 1.5).

Describe the working of a Fabry-Perot interferometer. Determine the intensity of the fringes of the transmitted light. Why the fringes obtained in the Fabry-Perot interferometer are comparatively sharper than those obtained from the Michelson interferometer ?

The radius of the first zone in a zone plate is 2\text{ mm}. What will be the position of the first image of a point source of light of wavelength \lambda = 500\text{ nm} placed at a distance of 5\text{ m} from the zone plate ?

Establish the relationship between the phase velocity V_p and the group velocity V_g of waves. Under what physical conditions V_g < V_p, V_g = V and V_g > V_p can be possible ?

Define resolving power and dispersive power of a grating. Two spectral lines of wavelengths 500\text{ nm} and 500.5\text{ nm} are seen clearly resolved in second order spectrum of a grating. If the grating has 250 lines per cm, what should be the minimum width of the grating?

Consider superposition of two plane polarized electromagnetic waves: E_y = \hat{y} a \cos(kx - \omega t) \text{ and} E_z = \hat{z} b \cos(kx - \omega t + \phi) Discuss the conditions for the resultant wave to be left circularly and right circularly polarized adopting the convention as seen by an observer travelling with the wave.