Show that the particle velocity in case of a plane progressive wave is given by \frac{\partial y}{\partial t} = -V \frac{\partial y}{\partial x} (where V is the wave velocity). Hence obtain the differential equation of wave motion.
Give an outline of Fresnel's explanation of optical rotation. How does optical rotation due to a material vary with \lambda? For an optically active material the difference between the refractive indices for right-handed and left-handed vibrations (\mu_R - \mu_L) for \lambda = 4500\text{ \AA} is 12 \times 10^{-5}. Estimate the optical rotation caused by 1\text{mm} thick plate in light of 1 = 4500\text{ \AA}. [Assume (m_R - m_L) as independent of \mu.]
A ruby laser produces a beam of light of wavelength 0.3\text{ \AA} with a circular cross-section 1\text{ cm} in diameter. Calculate the diameter of this beam at a distance of 1000\text{ kilometers}.
A diffraction grating with 3 \times 10^4 lines is used in the second order in the range of wavelength 6000\text{\AA}. Find the smallest \Delta\lambda it can resolve.
Distinguish between Fresnel and Fraunhofer classes of diffraction of light. Discuss the theory of a plane grating and hence find an expression for the angular dispersion of a plane grating.
Give a mathematical analysis of forced vibration and hence explain the phenomenon of amplitude resonance.
The phase velocity of surface waves of wavelength \lambda is v_p = \left( \frac{2 \pi T}{\lambda \rho} + \frac{g \lambda^{1/2}}{2 \pi} \right) where T is the surface tension and \rho the density of the liquid and g is acceleration due to gravity. Find the group velocity and express it in terms of the phase velocity. For which wavelength is the phase velocity a minimum ?
Write a note on He-Ne laser.
Deduce the possible thickness of a quarter wave plate of quartz which is to be used for sodium light of wavelength 5890 Å, (\mu_o = 1.558, \mu_e = 1.486.)
Define geometrical resolving power of an optical instrument. Derive the expression for the resolving power of a microscope and comment on the relation between resolving power and magnifying power, if any exists.
Deduce an expression for the diffraction waves at single slit and discuss how it is related with a Fourier transform.
Write a note on Purity of spectral lines and coherence length.
Explain the laws of refraction of light on the basis of Huygens principle.
A soap film of refractive index 1.33 is illuminated with light of different wavelength at an angle of 45^\circ. There is complete destructive interference for \lambda = 5890\text{\AA}. Find thickness of the film.
A particle of mass 10 g lies in a potential given by V(x) = 32 x^2 + 0.2 where x is in metres and V(x) in joules. Write down the equation of motion and solve it. What is the frequency of oscillation?
Write down the differential equation for a damped simple harmonic oscillator. Solve it and discuss the characteristics of dead-beat motion.
For a certain wave, system the angular velocity \omega and the wave vector k are related as follows: \omega = \omega_0 \left| \sin \frac{k a}{2} \right| \quad \text{for } -\frac{\pi}{a} < k < \frac{\pi}{a} Determine and plot the phase velocity and the group velocity for this system.
Two transverse sine waves each of amplitude 4\text{ mm} wavelength 2\text{m} and time period 1\text{ s} and in-phase at x=0, t=0 are travelling along the x-axis in opposite directions. Obtain the equation of the resultant wave and comment on its nature. Calculate the maximum displacement at x=2.3\text{ m}. Also locate the antinodes and nodes.
Write a note on Ruby laser.
Nine Kilograms of mercury is poured into a glass U-tube of uniform internal diameter of 1.2\text{ cm}. It oscillates freely about its equilibrium position. Calculate the period of oscillation.
