Calculate (i) the internal energy of the electron gas per unit volume and (ii) the molar specific heat at constant volume for sodium at 100\text{ K} containing one free electron per atom. Given that the density of sodium = 0{\cdot}97 \times 10^3\text{ kg m}^{-3} and the atomic weight of sodium = 23.
Deduce Clausius-Clapeyron equations based on reversible cycle. Show that the specific heat of steam is negative. What is the significance of negative specific heat?
Write down the distribution law obeyed by electron gas and apply the same to derive Richardson-Dushman equation.
There are g cells of energy \varepsilon. Show that the number n of bosons of energy \varepsilon distributed among these cells is given by n = \frac{g}{e^{(\varepsilon - \mu)/kT} - 1} What is \mu and how will you find it?
10\text{ g} of water at 60\text{ }^\circ\text{C} is mixed with 30\text{ g} of water at 20\text{ }^\circ\text{C}. Will the entropy of the system increase or decrease? Calculate the change.
Derive Poisson's equation. Under what conditions it takes the form of Laplace's equation? If \phi_1, \phi_2, \cdots, \phi_n are solutions of Laplace's equation, then show that \phi = \lambda_1 \phi_1 + \lambda_2 \phi_2 + \cdots + \lambda_n \phi_n is also a solution. Here \lambda\text{s} are arbitrary constants.
Show that the energy density in electrostatic field is given as u = \frac{1}{2} \bar{D} \cdot \bar{E} Further, show that the dielectric constant is a symmetric tensor except when the field energy vanishes identically.
Write the equation for total scattering cross-section for electromagnetic waves. Under what condition the scattering is known as Rayleigh scattering? Write the importance of Rayleigh scattering in nature.
Define a perfect blackbody. State Kirchhoff's law of radiation. What are the limitations of Kirchhoff's law?
Each square meter of the Sun's surface radiates energy at the rate of 6{\cdot}3 \times 10^7\text{ J/m}^2\text{ /s} and Stefan constant is 5{\cdot}669 \times 10^{-8}\text{ W/m}^2\text{ /K}^4. Find the temperature of the Sun's surface. Assume that the Stefan's law applies to the radiation.
What are ferromagnetic materials? By suitable figure, illustrate magnetic domain and magnetization curve for a ferromagnetic material.
In what way Maxwell's concept towards electric field and magnetic field is different from Faraday's concept? Derive magnitude of displacement current in a parallel-plate capacitor and account it for the continuous path for the charges across the capacitor.
Calculate the magnitude of Poynting vector at the surface of the Sun. Given that the power radiated by the Sun = 3{\cdot}8 \times 10^{26}\text{ watts} and the radius of the Sun = 7 \times 10^8\text{ m}.
What is a transformer? Is it for a.c. or d.c.? With a suitable diagram and representative symbol, explain the construction and necessary terms used in a transformer.
(i) Obtain an expression for the displacement of the damped harmonic oscillator where the damping force is proportional to the velocity. Discuss the effect of the damping on the displacement and frequency of the oscillator. Discuss the difference physically between overdamped and critically damped by taking example of a pendulum. (ii) A particle of mass 3 moves along the x-axis attracted towards the origin by a force whose magnitude is numerically equal to 12x. The particle is also subjected to damping force whose magnitude is numerically equal to 12 times the instantaneous speed. If it is initially at rest at x = 10, find the position and the velocity of the particle at any time.
Consider the formation of Newton's rings when two closely spaced wavelengths are present, for example, the \text{D}_1 and \text{D}_2 lines of sodium (\lambda_1 = 5890\text{ \AA} and \lambda_2 = 5896\text{ \AA}). What will be the effect of the presence of these two wavelengths as the lens is gradually moved away from the plate? What will happen if the sodium lamp is replaced by a white source? (The radius of curvature of the lens, R = 100\text{ cm})
A glass chamber 25\text{ mm} long filled with air is positioned in front of one of the secondary sources in Young's experiment. The air is removed from the chamber and a test gas is entered in its stead. On comparing the fringe system corresponding to air with that of the test gas, it is found that the entire interference pattern on the screen was displaced by 21 bright bands towards gas-containing chamber. Given that \lambda = 656{\cdot}2816\text{ nm} for which the refractive index of air is n_a = 1{\cdot}000276. Determine the refractive index of the gas (n_g).
Write an expression for the profile (t = 0) of a harmonic wave moving in the +x-direction such that at x = 0, \psi = 10; at x = \lambda/6, \psi = 20 and at x = 5\lambda/12, \psi = 0. (The notations carry usual meanings)
What do you understand by quarter wave plate? How can it be used to convert a linearly polarized wave into a circularly polarized wave?
(i) Describe the wave given by the expression E = \hat{j} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) - \hat{k} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) where \hat{j} and \hat{k} are unit basis vectors in Cartesian coordinates. (ii) Write an expression for linearly polarized harmonic plane wave of scalar amplitude E_0, propagating along a line in the xy-plane at 45^\circ to the x-axis and having the xy-plane as its plane of vibration.
