Viewing ionosphere as a dielectric medium of refractive index \mu = \sqrt{1 - \omega_p^2 / \omega^2}, \omega_p being known as the plasma frequency, determine the group velocity of a radio wave of frequency \omega = \sqrt{2}\omega_p.
A parallel plate capacitor is connected to a battery. What is the electric current while the capacitor is being charged ? What is the displacement current between the plates of the capacitor ? Show that the rate of increase of the electric energy is equal to the surface integral of the Poynting vector over the surface enclosing the volume between the plates of the capacitor.
(i) Write Maxwell's equations in the absence of a medium. (ii) Explain the gauge transformation and gauge invariance. Show that the relation \text{div } \bar{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} is Lorentz-covariant, where \bar{A} and \phi are the vector potential and the scalar potential respectively. (iii) Obtain Maxwell's wave equations satisfied by the scalar and vector potentials in the absence of charge and current.
A plane electromagnetic wave of frequency w_1 travelling in z-direction and polarized in x-direction is incident on a dielectric medium separated by a plane boundary in the x-z plane. Show that the transmission and reflection coefficient are given by : R = \frac{(n_1 - n_2)^2}{(n_1 + n_2)^2} , \quad T = \frac{4 n_1 n_2}{(n_1 + n_2)^2} , where n_1 and n_2 are the refractive indices of the first and the second medium.
An inductance L, capacitance C and resistance R are connected in series to form a circuit with AC source driving with E_{\text{rms}} = 120\text{ V} at f = 60\text{ Hz}. Compute the power factor and average power dissipated in the resistance if R = 200\ \Omega, X_L = 80\ \Omega and X_C = 150\ \Omega.
Explain Kirchoff's circuit laws. Using Kirchoff's laws find currents I_1, I_2 and I_3 in the circuit shown below for R_1 = 100\ \Omega, R_2 = 200\ \Omega, R_3 = 300\ \Omega, E_1 = 3\text{ V}, E_2 = 4\text{ V}.
A sphere of homogeneous linear dielectric material is placed in an otherwise uniform electric field \bar{E}_o. Find the electric field inside the sphere.
Compute the electrostatic energy of a conducting solid sphere of radius R and having total charge Q using the expression for the electrostatic energy in terms of the electric field \bar{E} of the above sphere.
Suppose a cavity of volume V contains blackbody radiation in equilibrium with the walls of the cavity at a temperature T. For a reversible adiabatic change of volume show that : VT^3 = \text{constant}. If the initial temperature is 2000^\circ\text{K} and the volume is increased from 10\text{ cm}^3 to 1250\text{ cm}^3, reversibly and adiabatically, what would be the final temperature of the radiation ?
Prove Stefan's law of radiation from thermodynamic considerations.
Define a black body. How can we realise a black body in practice ? Derive expression for Planck's radiation law.
Show that Wien's law and Stefan-Boltzmann law are limiting cases of Planck's radiation law.
A plane electromagnetic wave is given by E_z = a \cos \omega x \cos \omega t and H_y = - a \sin \omega x \sin \omega t. Evaluate the instantaneous value of the Poynting vector \vec{S} and show that <\vec{S}> = 0.
When the current in an \text{R}-\text{L} circuit is decaying, what fraction of the original energy stored in the inductor has been dissipated after 2\cdot 3 time constant ?
Sea water has resistivity 0\cdot 3\ \Omega\text{m} and its dielectric constant is 81. Calculate the ratio of the amplitudes of the conduction and polarisation current intensities when the applied field is oscillating at 100\text{ MHz}.
What do you understand by the term polarization in dielectrics ? Obtain Clausius-Mossotti formula for linear dielectrics.
Using Poisson equation and spherical co-ordinates, calculate the density of continuous charge distribution that will provide the Yukawa potential \phi = \frac{\exp(-\alpha r)}{r}, where \alpha is a constant.
With reference to ferromagnetic materials, explain the terms hysteresis and hysteresis loops.
The dielectric constant of a medium is 3. The Electric field in the dielectric is 10^6\text{ Vm}^{-1}. What are the electric displacement and polarization ?
Explain the physical significance of the Poynting vector \bar{S}. What is represented by the closed integral \oint \bar{S} \cdot d\bar{a} for a closed surface of area \bar{a} ?
