A solid copper sphere of mass 100\text{ g} kept at 300\text{ K} is suspended inside a closed chamber. The walls of the chamber are kept at 0\text{ K}. Assuming that the sphere acts as a blackbody, calculate the time required for its temperature to drop to half of the initial value. (Given that density of copper is 8\cdot 96 \times 10^3\text{ kg m}^{-3}, its specific heat capacity is 389\text{ J kg}^{-1}\text{ K}^{-1})
Write down Maxwell's equations in a non-conducting medium with constant permeability and susceptibility (\rho = j = 0). Show that \vec{E} and \vec{B} each satisfies the wave equation, and find an expression for the wave velocity. Write down the plane wave solutions for \vec{E} and \vec{B}, and show how \vec{E} and \vec{B} are related.
Consider two point particles of charge q each, separated by a distance d, and travelling at non-relativistic velocity \vec{v}. If the line joining the two charges is perpendicular to \vec{v}, then write an expression for the magnetic force between the two particles, and illustrate the direction of the force on each particle.
The potential at the surface of a sphere (radius R) is given by \text{V}_0(\theta) = \text{k} \cos 3\theta, where k is a constant. Find the potential inside the sphere.
The impedance function of a parallel R, L and C circuit has poles loaded at - 3 \pm 4j\text{ radians/second}. If the value of \text{L} = 1\text{ Henry}, determine the following :
(i) The value of R and C.
(ii) What is the quality factor of this circuit ? Derive an expression for series and parallel circuit.
Show that in an A.C. circuit with inductor only, current lags behind emf in phase by \dfrac{\pi}{2}. If the current through 0\cdot 5\text{ Henry} inductor varies sinusoidally with an amplitude of 10\text{ amp} and a frequency 50\text{ Hz}, calculate the potential difference across the terminals of the inductor.
In the circuit as shown in figure :
where V = 10\text{ V}, R = 10\ \Omega, L = 1\text{ H}, C = 10\ \mu\text{F}, and v_c(0) = 0, find i(0+), \dfrac{di}{dt}(0+) and \dfrac{d^2i}{dt^2}(0+).
Write Maxwell's equations in differential and algebraic forms. Give the physical significance of each. Show that the equation of continuity is contained in Maxwell's equations.
Consider the two branch parallel circuit shown in the diagram. Determine the resonant frequency of the circuit.
Calculate the skin depth of electromagnetic waves of 1\,\mathrm{MHz} incident on a good conductor having \sigma = 5.8 \times 10^{7}\,\mathrm{S\,m^{-1}}. Assume that inside the conductor \mu = \mu_0 = 4\pi \times 10^{-7}\,\mathrm{H\,m^{-1}}.
The spectral composition of solar radiation is similar to that of a black body radiator whose maximum emission corresponds to the wavelength 0.48\,\mu\mathrm{m}. Find the mass lost by the Sun every second due to radiation. Evaluate the time interval during which the mass of the Sun reduces by 1 per cent. Given: Stefan Boltzmann constant = 5.669 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}, radius of the Sun = 6.957 \times 10^{8}\ \mathrm{m}, surface temperature of the Sun = 5772\ \mathrm{K} and mass of the Sun is 1.9885 \times 10^{30}\ \mathrm{kg}.
A region 1, z<0, has a dielectric material with \varepsilon_r=3.2 and a region 2, z>0 has a dielectric material with \varepsilon_r=2.0. Let the displacement vector in the region 1 be, \vec{D}_1=-30a_x+50a_y+70a_z\,\mathrm{nC\,m^{-2}}. Assume the interface charge density is zero. Find in the region 2, the \vec{D}_2 and \vec{P}_2, where \vec{P}_2 is the electric polarization vector in the region 2.
Write the Laplace's equation in Cartesian, spherical polar and cylindrical coordinates in one-dimensional space. Using a suitable Laplace's equation out of these three, find the capacitance of parallel plate capacitor.
Justify the importance of method of images. Using this method, evaluate induced charge density on the surface of a grounded conducting sphere when a point charge is placed near it.
Define gauge transformation. Under what condition can the gauge transformation be regarded as restricted gauge transformation ? Further, generate the symmetrical forms of Maxwell's equations in terms of scalar and vector potentials under Lorentz gauge.
Given that the electric potential of a system of charges is V=\dfrac{12}{r^2}+\dfrac{1}{r^3} volt. Calculate the electric field vector at the Cartesian point (4,2,3)\,\mathrm{m}.
Given an infinite line charge of charge density 2\,\mathrm{nC\,m^{-1}} parallel to the y-axis and passing through the point (3,0,4)\,\mathrm{m} and an infinite sheet of charge of charge density 4\,\mathrm{nC\,m^{-2}} parallel to the x-y plane and passing through the point (0,0,6)\,\mathrm{m}. Calculate the electric field intensity at the point (10,10,10)\,\mathrm{m}. Assume free space.
Define a black body and evaluate Planck's radiation law. Show that at high temperature, this law resembles Rayleigh – Jeans law.
In the given circuit, L=2.0\,\mu\mathrm{H}, R=1.0\,\Omega, R_0=2.0\,\Omega and E=3.0\,\mathrm{V}. Find the amount of heat generated in the coil after the switch S is disconnected. The internal resistance of the source is negligible.
A rod of length l is perpendicular to a uniform magnetic field \mathrm{B}. The rod revolves at an angular speed \omega about an axis passing through one end of the rod and parallel to the magnetic field \mathrm{B}. Find the voltage induced across the rod's ends.
