PYQ Bank

Write a short note on Uniqueness theorem in electrostatics.

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?

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.

State Biot-Savart's law. Derive an expression for the magnetic field at points along the axis of a circular coil.

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.

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.]

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

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.

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

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.

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.

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.

Write a short note on Inertial forces in a rotating frame.

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

Calculate the speed of a satellite orbiting the planet Jupiter at a distance of 108\text{ km}, above its surface (The mass and radius of Jupiter are 1.9 \times 10^{27}\text{ kg} and 69892\text{ km} respectively.)

Using Rutherford's observation that the number of \alpha particles scattered at an angle \phi and falling on unit area of the screen varied as (\text{cosec }(\phi/2))^4, deduce an expression for the probability of scattering between angles \phi and \phi + d\phi.