State and explain Kirchhoff's current law and Kirchhoff's voltage law. Derive these laws from the principles of charge conservation and energy conservation.

(i) Show that the vector potential \vec{\text{A}} at the position defined by the vector \vec{\text{r}} in a uniform electric and magnetic field is \vec{\text{A}} = \frac{1}{2}(\vec{\text{B}} \times \vec{\text{r}}).

(ii) Find out the divergence and curl of the vector potential \vec{\text{A}}.

Determine the position of focal points, principal points and nodal points for a spherical lens of radius 20\text{ cm}. The refractive index of the material of the lens is 3/2. Indicate the positions of all the points in a diagram.

Consider a thick lens of thickness t made of a material of relative refractive index n. Let R_1 and R_2 be the radii of curvature of its two surfaces. Obtain the system matrix of the lens.

What do you understand by pulse dispersion in multimode optical fibres? Deduce an expression for intermodal pulse dispersion in a step-index fibre. Also compare the pulse dispersions of the step-index fibres with the parabolic-index fibres.

Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for the threshold population inversion required for the oscillations of laser.

In a double slit Fraunhofer diffraction experiment, the slit width is 0\cdot 12\text{ mm} and the spacing between the two slits is 0\cdot 48\text{ mm}. The distance of the screen from the slits is 1\cdot 5\text{ m}. If the wavelength of the light used is 600\text{ nm}, determine (i) the missing orders of the interference maxima, and (ii) the distance between the central maxima and the first minima.

For a \text{He}-\text{Ne} laser system, what will be the magnitude of \Delta\omega_D which represents FWHM of the line shape function g(\omega), if resonant frequency \omega_0 = 3 \times 10^{15}\text{ s}^{-1} and temperature T = 300\text{ K}?

A parallel beam of sodium light of wavelength 5893\text{ \AA} is incident on a diffraction grating. The angle between the first order spectra on either side of the normal is 27^{\circ} 42'. Find :

(i) The number of rulings/mm on the grating.

(ii) The greatest number of bright images obtained.

Describe the interference by a plane parallel thin film when illuminated normally and obliquely by a plane wave. Also list some important practical applications of the thin film interference phenomenon.

Within the framework of the Cornu's spiral, describe the intensity distribution at an arbitrary point \text{P} when a plane wave is incident normally on a long narrow slit of width \text{b}. For this case, under what condition will the diffraction pattern be of Fraunhofer type?

Explain the construction and working of a quarter-wave plate. How is it used to produce circularly and elliptically polarized light?

A light beam of wavelength 600\text{ nm} produced by a 20\text{ mW} laser source is incident on a plane mirror. Determine : (i) number of photons per second striking the surface of the mirror. (ii) force exerted by the light beam on the mirror.

A 15\cdot 0\text{ cm} violin string, fixed at both the ends, is vibrating with the largest wavelength. The speed of the wave in this string is 250\text{ m/s}, and the speed of sound in air is 350\text{ m/s}. What is the frequency and wavelength of the emitted sound wave?

m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; A harmonic oscillator is represented by the equation m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0; where m = 0\cdot 25\text{ kg}, \gamma = 0\cdot 07\text{ kg s}^{-1} and k = 85\text{ Nm}^{-1}. Determine (i) the period of oscillation, and (ii) the number of oscillations in which its amplitude will become half of its original value.

A particle of mass \text{m} is suspended from a spring which has mass \text{m} and force constant \text{k}. Show that the oscillation frequency is given by \omega = \frac{\sqrt{3}}{2}\omega_0, where \omega_0 is frequency of oscillation when spring is considered massless.

Consider multiple reflections from a plane parallel film of thickness h and refractive index n_2 and derive an expression for the total reflectivity from the surface of the film.

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Two particles each of rest mass 1\text{ gm} collide head-on with the speed 0\cdot 8\text{c} to stick together at rest. Find the mass of the final composite.

A solid shaft of mass M, length l and radius r is to be replaced by a lighter hollow shaft of the same length l and having the same ratings of \tau/\theta, where \tau is the couple and \theta is the angle of twist. Estimate the percentage reduction in mass of the hollow shaft if the outer radius of the shaft is twice the inner radius. Assume the material of the new shaft is same as that of the replaced shaft.