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.

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.

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.

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.

What are geomagnetic storms and why are they potentially disruptive to power grids ?

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+).

In what way Maxwell's concept towards electric field and magnetic field is different from Faraday's concept? Derive magnitude of displacement current in a parallel-plate capacitor and account it for the continuous path for the charges across the capacitor.

Calculate the magnitude of Poynting vector at the surface of the Sun. Given that the power radiated by the Sun = 3{\cdot}8 \times 10^{26}\text{ watts} and the radius of the Sun = 7 \times 10^8\text{ m}.

Write the equation for total scattering cross-section for electromagnetic waves. Under what condition the scattering is known as Rayleigh scattering? Write the importance of Rayleigh scattering in nature.

Each square meter of the Sun's surface radiates energy at the rate of 6{\cdot}3 \times 10^7\text{ J/m}^2\text{ /s} and Stefan constant is 5{\cdot}669 \times 10^{-8}\text{ W/m}^2\text{ /K}^4. Find the temperature of the Sun's surface. Assume that the Stefan's law applies to the radiation.

Define a perfect blackbody. State Kirchhoff's law of radiation. What are the limitations of Kirchhoff's law?

What is a transformer? Is it for a.c. or d.c.? With a suitable diagram and representative symbol, explain the construction and necessary terms used in a transformer.

Show that the energy density in electrostatic field is given as u = \frac{1}{2} \bar{D} \cdot \bar{E} Further, show that the dielectric constant is a symmetric tensor except when the field energy vanishes identically.

What are ferromagnetic materials? By suitable figure, illustrate magnetic domain and magnetization curve for a ferromagnetic material.

Derive Poisson's equation. Under what conditions it takes the form of Laplace's equation? If \phi_1, \phi_2, \cdots, \phi_n are solutions of Laplace's equation, then show that \phi = \lambda_1 \phi_1 + \lambda_2 \phi_2 + \cdots + \lambda_n \phi_n is also a solution. Here \lambda\text{s} are arbitrary constants.

(i) Using the laws of transformation of the electric field, \vec{\text{E}}, and the magnetic field, \vec{\text{B}}, show that (\text{E}^2 - \text{C}^2 \text{B}^2) is relativistically invariant. (ii) Suppose that in one inertial frame \vec{\text{B}} = 0 but \vec{\text{E}} \neq 0 (at some point P). Is it possible to find another inertial frame in which the electric field is zero at P ?

Consider electromagnetic fields in a medium with permittivity \varepsilon and permeability \mu. The fields are space and time dependent. Write down the full system of Maxwell's equations along with the constitutive equations. Then, derive the continuity equation from them. Using this continuity equation, show that the total charge of the universe is conserved.

Write down the formula for Planck's radiation law. Show that it reduces to Wien displacement law for shorter wavelengths and to Rayleigh -- Jeans law for longer wavelengths.

Two cells \text{E}_1 of E.M.F. 2\text{ volts} and internal resistance 1\cdot 5\text{ ohms} and \text{E}_2 of E.M.F. 1\cdot 5\text{ volts} and internal resistance 1\text{ ohm}, are joined in parallel with like poles together (see the figure below). Calculate the current that would pass through a 5\text{-ohm} resistance joined in parallel with the cells.