Show that the ratio of the focal lengths of the two lenses in an achromatic doublet is given by \frac{f_1}{f_2} = - \frac{w_1}{w_2} where w_1 and w_2 are the dispersive powers of the lenses of focal lengths f_1 and f_2 respectively.

A thin converging lens and a thin diverging lens are placed coaxially at a distance of 5 cm. If the focal length of each lens is 10 cm, find for the combination (i) the focal length, (ii) the power, (iii) the positions of the principal points.

Write down the equation of motion for a damped harmonic oscillator assuming the damping force proportional to the velocity of the particle. Obtain the general solution for its displacement as a function of time. Discuss the cases of 'overdamping', 'under-damping' and 'critical damping'.

What is holography ? Describe the experimental set up for Gabor's on-line holographic recording. What are the limitations of Gabors experiments? How these were over come by Leith and Upatneiks?

Two thin convex lenses of focal length 0.2 m and 0.1 m are located 0.1m apart on the axis of symmetry. An object of height 0.1m is placed at a distance of 0.2 m from the first lens. Find by the matrix method, the position and the height of the image.

Two transverse harmonic waves, each of amplitude 5mm, wavelength 1m and speed 3 m/s are travelling in opposite directions along a stretched string fixed at both ends. Obtain an expression for the standing wave Produced. Locate the position at the nodes and antinodes.

An ideal massless spring of force constant k has a mass m attached to one of its ends, the other and being fixed to a rigid support. The spring is horizontal floor, A resistive force -bv, proportional to the velocity v acts on the mass. Assuming the damping to be light, obtain the frequency of oscillation. When m = 0.1\text{ kg} and k = 10\text{ N/m}, it is found that the frequency of oscillation is \sqrt{1/2} times the frequency in the absence of damping. Calculate the value of the constant b.

How does Doppler effect of light of relativistic physics qualitatively differ from its non-relativistic analogue ? Calculate the Doppler shift in the frequency of a photon travelling along the y-axis. with respect to an observer moving along the x-axis with a constant speed v.

Obtain an expression for the intensity of light in the Fraunhofer diffraction pattern due to a circular aperture. What is Airy pattern ? Explain with a neat diagram.

Explain the phenomenon of self focusing of laser beams.

Explain the phenomenon of pulse dispersion in a step index optical fibre.

State Fermat's principle. Apply it to get the laws of reflection from a plane surface.

In an experiment using a Michelson interferometer, explain with the help of suitable ray diagrams.

(i) Why do we need extended source of light,

(ii) Why do we get circular fringes, and

(iii) Shifting of fringes inwards or outwards as we shift the movable mirror.

Derive the expression for resolving power of a diffraction grating with N lines. Calculate the minimum number of lines in the diffraction grating if it has to resolve the yellow lines of sodium (589.0\text{ nm} and 589.6\text{ nm}) in the first order.

Discuss the Fresnel diffraction pattern formed by a straight edge using the Cornu's spiral.

Show that the motion of a classical particle is analogous to the motion of a wave packet, which can be constructed by superposition of a large number of plane waves. Construct such a wave packet and derive an expression for its group velocity.

The phase velocity of the surface wave in a liquid of surface tension T and density \rho is given by u_p = \sqrt{\frac{g\lambda}{2\pi} + \frac{2\pi T}{\lambda\rho}} Show that the group velocity u_g of the surface wave is given by u_g = \frac{g + (12\pi^2 T) / \rho\lambda^2}{2\sqrt{\left(\frac{2\pi g}{\lambda} + \frac{8\pi^3 T}{\rho\lambda^3}\right)}}

A wave is represented by \Psi_1 = 10 \cos(5x - 25t). Find wavelength \lambda, velocity v, frequency f and the direction of propagation. If it interferes with another was given by \psi_2 = 20 \cos(5x + 25t + \pi/3), find the amplitude and the phase of the resultant wave. (All dimensions are in SI system).

How do you know that the light is a transverse wave? What is a quarter wave plate? How is it constructed?

The phase velocity in a material is \sqrt{g/k} where k is the propagation constant. Prove that the group velocity will be half of the phase velocity.