Fourier analyse the step-function f(x) = 1 \text{ for } 0 < x < \pi = -1 \text{ for } \pi < x < 2\pi and hence prove that 1 + \frac{1}{9} + \frac{1}{25} + \frac{1}{49} + \dots = \frac{\pi^2}{8}

The equation for displacement of point on a damped oscillator is given by x - 5e^{-0.2t} \sin \frac{\pi}{2}t \text{ metre} Find the velocity of the oscillating point at t = \frac{T}{4} and T_p where T is the time-period of the oscillator.

The speeds v of waves on the surface of a liquid is given by v = \sqrt{\frac{TK}{P} + \frac{g}{k}} where T is the surface tension of the liquid of density P > K = 2\pi / \lambda, \lambda being the wavelength of the wave and g is acceleration due to gravity Find the wavelength and frequency of waves on water which move with minimum speed.

Using Huyghens' construction, discuss the propagation of O and E waves in a biaxial crystal when the optic axis is perpendicular to the plane of incidence and parallel to the boundary surface. Take the incident ray to be oblique.

Write a note on Uses of multiple beam interferometry.

Out of the following three functions only one can be Fourier analysed in the range -\pi \leqslant x \leqslant \pi. Choose the correct function and obtain its Fourier series expansion: (i) f(x) = x^{2} \text{ for } 0 < x < \pi (ii) f(x) = 0 \text{ for } -\pi < x < 0 = \pi \text{ for } 0 < x < \pi (iii) f(x) = 1/x^{2} \text{ for } -\pi < x < \pi

Parallel light is incident normally on diffraction grating having 6,000 lines per cm. Find the angular separation between the maxima for wavelengths 5,890 \text{\AA} and 5,896 \text{\AA} in the second order.

Define and explain the terms resolving power and magnifying power of an optical instrument. For a telescope, on that parameter do these depend? For a given resolving power, what is the optimum magnifying power?

How can one produce (i) left-handed circular motion, and (ii) right-handed circular motion by combining two simple harmonic motions? Justify, your answers.

What is stimulated emission of light? How does it differ from spontaneous emission? Name any device basal on the stimulated emission of light and explain its working.

The angular frequency, \omega, of a wave is related to the wave vector, k, by the relation \omega = \omega_{0} \sin\frac{ka}{2} \text{ for } 0 < k > \frac{\pi}{a} Determine the phase and group velocities of this wave and draw a rough sketch to show their variation with k(0 < k < \pi/a)