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Find the energy stored in a system of four charges Q_1 = 1\text{ nC}, Q_2 = 2\text{ nC}, Q_3 = 3\text{ nC} and Q_4 = 4\text{ nC} placed at the cartesian coordinates R_1(1, 1), R_2(2, 1), R_3(1, 4) and R_4(2, 2), respectively. Assume free space.
Write down Stefan-Boltzmann law of radiation and derive it from Planck's law of radiation.
Write down the electromagnetic wave equation in non-conducting dielectric medium. Hence, show that the velocity of wave propagation is given by v = \sqrt{\frac{1}{\mu \epsilon}}, where the symbols have their usual meanings.
A material has \sigma = 10^{-2}\text{ S/m} and \varepsilon = 2\varepsilon_0. At what frequency, the conduction current would be equal to the displacement current?
Show that Continuity equation is embedded in Maxwell's equations.
A spherical shell of radius R, carrying a uniform surface charge \sigma, is set spinning at angular velocity \omega about its axis. Find the vector potential it produces at point \vec{r}.
Assuming that the Sun radiates like a perfect blackbody, at what wavelength does the peak of the solar spectrum occur? Given that the surface temperature of the Sun is about 6000\text{ K}.
A long straight wire of circular cross-section is made of non-magnetic material (\chi_{\text{m}} = 1^\circ to a good degree of approximation). It is of radius a. The wire carries a current I which is uniformly distributed over its cross-section. Compute the energy per unit length stored in the magnetic field contained within the wire.
Consider an L-R-C series circuit with a 300\text{ mH} inductor, a 0\cdot 47\,\mu\text{F} capacitor and a 500\,\Omega resistor. The source has terminal rms voltage \text{V}_{\text{rms}} = 100\text{ V} and variable angular frequency \omega. For what two values of the angular frequency, \omega_1 and \omega_2, is the rms current half the resonance value ? Also calculate the resonance width |\omega_1 - \omega_2|.
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.
Two dipoles 1 and 2 with dipole moments \vec{\text{p}}_1 = -\,5\,\hat{\text{z}}\text{ nC.m} and \vec{\text{p}}_2 = 9\,\hat{\text{z}}\text{ nC.m}, respectively, are located at points (0, 0, -\,2) and (0, 0, 3), respectively. Find the potential at the origin.
A current sheet having \vec{K} = 9.0a_y\,\mathrm{A\,m^{-1}} is located at z = 0. The interface is between the region 1, z < 0, \mu_{r1} = 4, and region 2, z > 0, \mu_{r2} = 3. Given that \vec{H}_2 = 14.5a_x + 8.0a_z\,\mathrm{A\,m^{-1}}. Find \vec{H}_1 and \vec{B}_1.
A metal guitar string with a length of 70\,\mathrm{cm} vibrates at its fundamental frequency of 246.94\,\mathrm{Hz} in a uniform magnetic field of 10\,\mathrm{T} oriented perpendicular to the plane of vibration of the string. Assume a sinusoidal form for the amplitude of the vibrational mode, and a maximum displacement of 3\,\mathrm{mm} at the centre of the string. What is the maximum e.m.f. generated across the length of the guitar string, and at what point in time in the string's motion does that occur? What would be the e.m.f. if the same guitar string vibrates at its second harmonic frequency? Briefly explain.
Consider the R-L-C circuit shown here. Calculate the Q-factor of the circuit. Does the circuit have a resonant frequency? Justify your answer:
A cell of internal resistance 1\,\text{ohm}, 1.5 volt e.m.f. and another cell of internal resistance 2\,\text{ohm}, 2 volt e.m.f. are connected in parallel across the ends of an external resistance of 5\,\text{ohm}. Find the current in each branch of the circuit.
Starting from the Laplace's equation in a cylindrical polar coordinate system and using the method of separation of variables, obtain the differential equations for the solutions of r, \phi and z components of the potential.
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.
A relativistic charged particle moves in the space occupied by uniform and mutually perpendicular electric and magnetic fields \vec{\text{E}} = \text{E}_0\,\hat{\text{x}} and \vec{\text{B}} = \text{B}_0\,\hat{\text{y}}, respectively. The particle moves rectilinearly along the z-direction. Find \vec{\text{E}}' and \vec{\text{B}}' in the reference frame moving translationally with the particle.
In a partially conducting medium, \varepsilon_r = 18.5, \mu_r = 800 and \sigma = 1\ \mathrm{S\,m^{-1}}. Find \alpha, \beta, \eta and the velocity u, for a frequency of 10^9\ \mathrm{Hz}. Determine \vec{H}(z,t). Given, \vec{E}(z,t) = 50e^{-\alpha z}\cos(\omega t - \beta a_z)a_y\ \mathrm{V\,m^{-1}}.
A wire in the form of a hexagon is just enclosed by a circle of radius 10\text{ cm}. If the current in the wire is 1\text{ A}, find the magnetic field at the centre of the hexagon. What would be the direction of the field if the current flows in the anti-clockwise direction ?