Write down the four Maxwell's equations and explain the contribution of Maxwell in the development of these equations.

(i) Using Maxwell's equations, obtain the relation \frac{1}{c} \frac{\partial}{\partial t} \left( \frac{E^2 + B^2}{2} \right) + \vec{\nabla} \cdot (\vec{E} \times \vec{B}) = 0 (ii) What is Poynting vector ? Deduce Poynting theorem for the flow of energy in an electromagnetic field.

Discuss the reflection and refraction of plane electromagnetic waves at plane dielectric boundaries for normal incidence and also find the reflection and transmission coefficients.

Show that Maxwell's equations of electrodynamics are invariant under Lorentz transformations.

A point charge of q is held above a grounded conducting plane located at z = 0. If the position of the charge is (0, 0, d), obtain an expression for the induced charge density on the plane as a function of the coordinates x and y.

What is the volume density of charge in a region of space where electrostatic potential is given by V = a - b(x^2 + y^2) - c\log(x^2 + y^2), where a, b, c are constants.

An inductor L = 0.1\text{ H} in series with a resistor R = 20\ \Omega is connected across 230\text{ V}, 50\text{ Hz} a.c. voltage. Determine the following : (i) inductive reactance (ii) impedance of the circuit (iii) current in the circuit (iv) voltage across R (v) voltage across L (vi) phase angle between the current and voltage (vii) power factor (viii) power consumed in the circuit

(i) Show that the work done per unit volume of a ferromagnetic substance per one complete cycle of hysteresis is given by W = \oint \vec{H} \cdot d\vec{B} (ii) Further, show that the value of W is equal to the area of B-H loop. Which material is preferred for the core of a transformer and why ?

(i) Derive Planck's law of Black body radiation. (ii) Obtain its limiting forms for (i) very low frequency and (ii) very high frequency.

If the magnetic field \vec{B}, at a point with position vector \vec{r} is uniform, show that the corresponding vector potential \vec{A}(\vec{r}) is given by \vec{A}(\vec{r}) = -\frac{1}{2} \left[ \vec{r} \times \vec{B} \right]

Show that for any electromagnetic wave propagating in free space, the total average energy per unit volume is \frac{1}{2}\epsilon_0 E_{\text{max}}^2; where E_{\text{max}} is the amplitude of the electric field associated with the electromagnetic wave.

Verify whether the electric potential V = 15 x^2 y z - 5 y^3 z satisfies Laplace's equation or not.

Discuss the physical significance of Planck's radiation law in the context of emergence of New Physics. If the blackbody energy density u(\omega) has a functional dependence like u(\omega) \propto x^3 (e^x - 1)^{-1}, where x = \frac{\hbar\omega}{kT}, derive Wien's displacement law and comment on its importance.

The electric potential of a grounded conducting sphere of radius a in a uniform electric field E = E_0\hat{z} is given as \phi(r, \theta) = -E_0 r \left[ 1 - \left( \frac{a}{r} \right)^3 \right] \cos\theta Find the expression for the surface charge density on the sphere.

Using Ampere's law, derive the magnetic field of a toroid (N turns each carrying current I) of inner radius a and outer radius b at a distance r midway between a and b.

Viewing ionosphere as a dielectric medium of refractive index \mu = \sqrt{1 - \omega_p^2 / \omega^2}, \omega_p being known as the plasma frequency, determine the group velocity of a radio wave of frequency \omega = \sqrt{2}\omega_p.

Which of the Maxwell equations imply that there are no magnetic monopoles? Explain how the equations would get modified if magnetic monopoles would exist.

Prove that the work done on the charges by the electromagnetic force is equal to decrease in energy stored in the field, less the energy that flowed out through the surface. Explain what you mean by the Poynting vector.

Calculate the strength of the magnetic field to bring a proton nucleus and a ^{13}\text{C} nucleus to resonate at this frequency. Magnetic moment of \text{proton} = 2\cdot 7927\ \mu_\text{N} and magnetic moment of ^{13}\text{C} = 0\cdot 7022\ \mu_\text{N}. The NMR instrument operates at 30\cdot 256\text{ MHz}.

A plane conducting circular wire loop lies perpendicular to a uniform magnetic field B and its area S(t) is changed as S(t) = S_0 (1 - \alpha t), where 0 < t < \frac{1}{\alpha} (S_0 and \alpha are constants). The wire has resistance per unit length \rho\,\Omega\text{ m}^{-1}. Find the induced current through the wire. If the current in a certain coil varies at a rate of 50\text{ A s}^{-1}, the induced e.m.f. is V = 20\text{ volts}. What is the inductance of the coil?