Explain the physical significance of resolving power of a grating with relevant mathematical expression.

The displacement associated with a three-dimensional plane wave is given by \Psi(x,y,z,t) = a\cos\left[\frac{\sqrt{3}}{2}kx + \frac{1}{2}ky - \omega t\right]. Calculate the angles made by the propagating wave with the x, y and z-axes.

The dispersion relation for deep water waves is given by \omega^2 = gk + ak^3 where g and a are constants. Obtain expressions for phase velocity and group velocity in terms of the wavelength \lambda. \omega and k represent the angular frequency and wave number, respectively.

During an earthquake, a horizontal shelf moves vertically. If its motion can be regarded simple harmonic, calculate the maximum value of amplitude of oscillation so that the books resting on it stay in contact with it always. Take g=9.8\ \mathrm{m\,s^{-2}} and T=0.5\ \mathrm{s}.

In a tungsten filament lamp, thermionic emission takes place at 1.2\times10^3\ \mathrm{K}. Calculate the ratio of spontaneous emission to stimulated emission for non-degenerate energy levels. Interpret your result physically. Take \lambda=550\ \mathrm{nm}, k_B=1.38\times10^{-2}\ \mathrm{J\,K^{-1}}, h=6.67\times10^{-34}\ \mathrm{J\,s} and c=3\times10^8\ \mathrm{m\,s^{-1}}.

Explain why information carrying capacity of an optical fibre can be enhanced by reducing the pulse dispersion. How does one minimize pulse dispersion using a graded index optical fibre?

A pulse of \lambda_0=600\,\mathrm{nm} and \Delta\lambda=10\,\mathrm{nm} propagates through a fibre which has a material dispersion coefficient of 50\,\mathrm{ps} per km per nm at 600\,\mathrm{nm}. Calculate the pulse broadening in traversing a 10\,\mathrm{km} length of the fibre. If the pulse width at the input of the fibre is 12\,\mathrm{ns}, what will be the pulse width at the output of the fibre?

Show that a travelling wave on the string, clamped on both the ends, undergoes a phase change of \pi. Hence obtain the time-independent form of the wave equation representing a standing wave on the string.

Use matrix method to obtain an expression for the focal length of a coaxial combination of two thin lenses having focal lengths f_1 and f_2 separated by distance d.

The motion of a damped mechanical oscillator is represented by m\ddot{x}+\alpha\dot{x}+\beta x=0 where m, \alpha and \beta are constants. The oscillator is critically damped. The system is given an impulse at x=0 and t=0, resulting in an initial velocity v. After how much time the system experiences maximum displacement?

Derive an expression for intermodal dispersion for a multimodal step-index fibre.

Distinguish between Fresnel and Fraunhofer classes of diffraction. Show that the area of each Fresnel half-period zone is same.

Calculate the minimum thickness of a quartz plate which would behave as a quarter-wave plate for wavelength of light, \lambda=6000\,\mathring{\mathrm{A}}. The refractive indices for ordinary and extraordinary rays are \mu_o=1.544 and \mu_e=1.553.

A diffraction grating of width 5 cm with slits of width 10^{-4} cm separated by a distance of 2\times10^{-4} cm is illuminated by light of wavelength 550 nm. What will be the width of the principal maximum in the diffraction pattern? Would there be any missing orders?

The separation between the slits is 0.5\,\text{mm} in Young's double-slit experiment. The interference pattern observed on a screen placed 5\,\text{m} away reveals the location of the first maximum which is 6\,\text{mm} from the centre of the pattern. Calculate the wavelength of light and separation between second and third bright fringes.

Explain the physical significance of group velocity from the concept of phase velocity with relevant expressions.

The Fraunhofer single-slit diffraction intensity is given by I=I_0\frac{\sin^2 x}{x^2} where x=\frac{\pi d y}{\lambda l}, with l as the distance from slit to source, d the slit width, y the detector distance and \lambda the wavelength. What is the value of cumulative intensity \int_{-\infty}^{\infty} I(y)\,dy?

Write down the one-dimensional harmonic oscillator differential equation under damping and its solution for the lightly damped condition, with the meanings of symbols. Determine the dependent energy in the lightly damped condition.

When a thin film of a transparent material is put behind one of the slits in Young's double-slit interference experiment, the zero-order fringe moves to the position previously occupied by the fourth-order bright fringe. The index of refraction of the film is n=1.2 and the wavelength of light, \lambda=5000\,\mathring{\mathrm{A}}. Determine the thickness of the film.

Prove that the group velocity V_g of electromagnetic waves in a dispersive medium with refractive index n(\lambda_0) at wavelength \lambda_0 is given by V_g=\frac{c}{n(\lambda_0)-\lambda_0\dfrac{dn(\lambda_0)}{d\lambda_0}} where c is the free space velocity of light. Find the time taken for the electromagnetic pulse to travel a distance D.