The wire AB shown in the figure below has mass m, resistance r and it can slide on the smooth horizontal parallel rails separated by a distance l. A uniform magnetic field B exists in the rectangular region and a resistance R connects the rails outside the field region. At t = 0, the wire AB is pushed towards right with a speed v_0.

Find out the current in the loop at an instant when the speed of the wire is v. Find out the velocity v as a function of x. (Assume that the resistance of the rails is negligible)

Explain Ampere's circuital law of magnetic field. What modifications Maxwell made in this equation to derive Maxwell's fourth equation?

An infinitely long coaxial cylinder structure has an inner conductor of radius a and an outer conductor of radius b. The inner conductor is attached to V_0, while the outer one is grounded. Calculate the field \vec{E} in between a and b.

If the electrostatic potential in spherical polar coordinate is \phi(r) = \phi_0 e^{-r/r_0} where \phi_0 and r_0 are constants, what will be the charge density at a distance r = r_0?

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?

What is Lenz's law? Is Lenz's law the same as Faraday's law? What happens if Lenz's law is reversed? Explain.

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}.

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 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.

The spectral energy density curve of the moon shows maxima at 14\,\mu\text{m}. Estimate the temperature of the moon and the energy density of the lunar radiation.

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.

Consider a medium with an electric current density \vec{\text{J}}, permittivity \varepsilon and conductivity \sigma, which is described by Ohm's law \vec{\text{E}} = \sigma \vec{\text{J}}. Write down Maxwell's equations in such a medium. Using these equations, derive the wave equation for the electric field \vec{\text{E}}. Show that plane waves of the form \vec{\text{E}} = \vec{\text{E}}_0 \text{ e}^{\text{i}(\omega\text{t} - \text{kx})}, where \vec{\text{E}}_0 is a constant vector, are solutions of the wave equation but with complex k. Determine the explicit expression for k in terms of \varepsilon, permeability \mu_0, \sigma and the angular frequency \omega.

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 ?

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})

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.

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.