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.
(i) Considering an isotropic, linear, non-conducting, non-magnetic and inhomogeneous dielectric medium with \vec{D}=\epsilon\vec{E}=\epsilon_0n^2(x,y,z)\vec{E}, show that the electromagnetic wave equation for the field \vec{E} is given by \nabla^2\vec{E}+\vec{\nabla}\left(\frac{1}{n^2}\vec{\nabla}n^2\cdot\vec{E}\right)-\mu_0\varepsilon_0n^2\frac{\partial^2\vec{E}}{\partial t^2}=0.
(ii) Write down the scalar equation for E_x from the above equation.
(iii) Interpret physically the situation if we move from homogeneous to an inhomogeneous medium.
(iv) Obtain the similar vector equation for the magnetic field \vec{H} in inhomogeneous medium.
With reference to ferromagnetic materials, explain the terms hysteresis and hysteresis loops.
For a uniform wire of length L and radius a having a potential difference V between the ends and a current I along it, calculate the energy per unit time delivered to the wire by Poynting vector.
The electric field of a plane e.m. wave travelling along the z-axis is \vec{E}=(E_{0x}\hat{x}+E_{0y}\hat{y})\sin(\omega t-kz+\phi). Determine the magnetic field.
A series LCR circuit has resonant frequency \omega_0 and a large quality factor Q. Write down in terms of R, \omega, \omega_0 and Q, its (i) impedance at resonance, (ii) impedance at half-power points and (iii) the approximate forms of its impedance at low and high frequencies.
Consider the equation for a series RLC circuit and compare this to the parallel resonant circuit shown below:
Calculate the value of R_p if a series RLC circuit and the parallel RLC circuit are to have same equations for the potential of capacitance while they both have the same L, C and Q with Q being the total charge.
Consider the following coupled inductor - capacitor circuit:
Calculate the ratio of the frequencies of the anti-symmetric and symmetric modes \omega_a/\omega_s. \left(\text{Given } k=\frac{1}{LC},\ k'=\frac{1}{LC_1}\right)
In the circuit diagram shown below, calculate the current passing through the milliammeter.
ABCD is a rectangle in which charges of +10^{-11}\,\mathrm{C}, -2\times10^{-11}\,\mathrm{C} and 10^{-11}\,\mathrm{C} are placed at corners B, C and D, respectively.
Calculate the potential at the corner A and the work done in carrying a charge of 2 coulombs to A.
The dispersion relation for deep water waves is given by \omega^2 = gk + ak^3 where g and a are constants. Obtain expressions for phase velocity and group velocity in terms of the wavelength \lambda. \omega and k represent the angular frequency and wave number, respectively.
Write down the equation of motion of a weakly damped harmonic oscillator driven by a harmonic force. Obtain an expression for the maximum amplitude of oscillation under steady-state conditions.
An oscillator of mass 0\cdot 01\text{ kg} draws maximum power at a frequency of 96\text{ Hz} with half power points at 93\text{ Hz} and 99\text{ Hz}. If the amplitude of driving force is 3\cdot 88\text{ N}, calculate (i) quality factor, (ii) the damping factors and amplitude of the oscillator at resonance.
A block of mass m is attached to one end of a combination of two massless springs connected in series. The other end of the combination is fixed to a rigid support. The spring constants of the two springs are k_1 and k_2, respectively. The block is free to move on a frictionless horizontal surface. If the block is pulled a little and then released, calculate the frequency of oscillation of the block.
A plano-convex lens of radius 1\cdot 0\text{ m} is placed on an optically flat glass plate. The space between the lens and the glass plate is filled with a liquid. The diameter of the 5^{\text{th}} ring changes to 3\cdot 0 \times 10^{-3}\text{ m}. Calculate the refractive index of the liquid when the ring is bright. The wavelength of light used is 589\text{ nm}.
If a thin sheet of glass of thickness t and refractive index \mu is placed in the path of one of the interfering waves, show that the distance through which a fringe is displaced is a function of t.
In a double slit interference experiment, show that the fringe shape is a hyperbola.
Consider a zone plate with radii r_n = 0\cdot 11 \sqrt{n}\text{ cm} illuminated by a monochromatic light of wavelength 589\text{ nm}. Calculate the positions of first two foci.
