Show that the temperature (T) of a plant varies inversely as the square root of its distance (R) from the Sun. (The Sun and Planets are considered to be black-bodies in radiative equilibrium.)

State Planck's formula for black-body spectrum. Show that 'Planck's formula reduceds to Wien's formula at short wavelengths,

In a nuclear explosion, the maximum temperature reached was of the order of 10^8\text{ K}. Estimate the order of wavelength at which the maximum of radiated energy occurs.

Write down Maxwell's equations for an isotropic homogenous dielectric and point out their relations with observational laws. How do the equations lead to the concept of electromagnetic waves?

Write a short note on Uniqueness theorem in electrostatics.

State Coulomb's law and show that the electric field can be derived from a potential function. If the charge distribution is continuous, find the integral formula for determining its field.

If steady voltage is applied to an L-R circuit, show how voltage across the inductance and the current in the circuit changes with time. Explain the term inductive time constant.

A particle executing simple harmonic motion, has an acceleration (\pi^2/3)\text{ cm/sec}^2 when the displacement is 1\text{ cm}. Determine the period.

Give a mathematical analysis of forced vibration and hence explain the phenomenon of amplitude resonance.

State Rayleigh Criterison for limit of resolution. Show that \frac{I_{\text{middle}}}{I_{\text{max}}} = \frac{8}{\pi^2}

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 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}.

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.

State Kepler's laws of planetary motion. Why does a missile need an escape velocity to escape from earth?

Write a short note on The Lorentz transformation.

The half-life of \mu-meason at rest is 2 \times 10^{-8}\text{ sec}. Determine the half life of \mu-meason while travelling with half the speed of light in vacuum.

What do you mean by centre of mass of a system of particles? Derive expressions for the instantaneous position vector and velocity of the centre of mass of such a system of particles.

State Bernoulli's theorem. Find out the velocity of efflux of a liquid from a reservoir in which the pressure is 3.5 \times 10^5\text{ N/m}^2 above the atmospheric pressure. The density of the liquid is 700\text{ kg/m}^3.