The dielectric constant of a medium is 3. The Electric field in the dielectric is 10^6\text{ Vm}^{-1}. What are the electric displacement and polarization ?

How do you justify the statement : "A current carrying conductor, although has no net charge experiences a force when placed in a magnetic field."

Explain the term 'mutual inductance' between two coils carrying current. Describe a method of determining mutual inductance between two coils of wire with relevant theory.

Find out the number of photons in a cavity of volume 1\cdot 00\text{ cm}^3 under thermal equilibrium at temperature T = 1000\ ^\circ\text{K}. Assume that \int_0^\infty \frac{x^2 dx}{e^x - 1} = 2\cdot 405.

For the above system, show that the energy per unit cavity volume in the frequency range of \nu and \nu + d\nu can be given by u(\nu)d\nu = \frac{8\pi\nu^2 k_B T}{c^3} d\nu where k_B is the Boltzmann's constant. Discuss the limitations of this formula and how did Planck put forward the correct analysis.

If considered a blackbody as a radiation-filled cavity at a uniform temperature T, demonstrate with an appropriate figure the electromagnetic wave patterns inside the cavity of length L for wavelengths \lambda = L, \frac{3}{2} L and 2L.

Show that the complex propagation constant k of an electromagnetic wave propagating in an isotropic dielectric medium with conductivity \sigma can be given by k = k_r + i k_i with k_r \approx \frac{2\pi}{\lambda_o} n \left[ 1 + \frac{1}{8} \left( \frac{\sigma}{\omega \varepsilon} \right)^2 \right] and k_i \approx \frac{2\pi}{\lambda_o} n \left[ \frac{1}{2} \left( \frac{\sigma}{\omega \varepsilon} \right) \right] where n = \sqrt{\varepsilon / \varepsilon_o} and \lambda_o = \omega / 2\pi c; \varepsilon and c being the permittivity of the dielectric and velocity of light in free space, respectively.

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].

The electric field in a medium is given by \vec{E} = \vec{E}_0 e^{-\alpha z} \sin(kz - \omega t), where \vec{E}_0 is a constant vector with dimensions of the electric field. Prove that \vec{E} cannot have a component along the unit vector, \hat{z}, parallel to the z-axis. Here \alpha is a positive constant.

Prove that the Maxwell's equations in a medium contain the conservation of charge in differential form.

Consider an infinite current sheet with a uniform current density \vec{K} \text{ (Amp/m)}. Show that the magnetic field \vec{H} at a point away from the sheet is \vec{H} = \frac{1}{2} \vec{K} \times \hat{n} where \hat{n} is a unit normal vector directed from the current sheet to the point.

The plane y = 5 carries a current of density 10\\ \hat{z} \text{ (Amp/m)}. Calculate the value of the magnetic field \vec{H} at the point (0, 1, -5).

Consider the Earth as a black body. Radiations from the Sun arrive at the surface of the Earth with an average intensity of S\text{ watts/m}^2. If the reflection coefficient of the Earth's surface is \alpha, determine the temperature of the Earth under equilibrium conditions.

Consider an infinite line charge with charge density \rho\text{ coulomb/meter} located at a distance d\text{ meters} from a grounded conducting plane z = 0. Determine : (i) the magnitude of the potential V for z > 0 and z \le 0. (ii) the surface charge density induced on the conducting plane.

The current density in spherical co-ordinates is given by \vec{J} = \frac{1}{r^3} \left[ 2 \cos \theta \hat{r} + \sin \theta \hat{\theta} \right] \text{A/m}^2 where \hat{r} and \hat{\theta} are unit vectors. Calculate the amount of current passing through a hemisphere of radius 20\text{ cm}.

An inductor of inductance 5\text{ H} is suddenly connected to a 10\text{ V} d.c. power supply through a resistor of 10\\ \Omega. After what time will the current in the circuit be 1/10\text{th} of its steady state value ?

The electromagnetic field inside a device is given by \vec{E} = \hat{y} E_0 \sin(k_x x) e^{-j k_z z}, \quad\text{where } k_x = \frac{n\pi}{a} \vec{H} = E_0 [\hat{x} \frac{-k_z}{\omega\mu} \sin(k_x x) + \hat{z} \frac{j k_x}{\omega\mu} \cos(k_x x)] e^{-j k_z z} Obtain an expression for the z-component of time-averaged Poynting vector \vec{S}.

Consider a perfectly conducting half-space as shown below :

A uniform plane wave given by \begin{align*} \vec{E}^i &= \hat{x} E_0 e^{-j k z} \ \vec{H}^i &= \hat{y} \frac{E_0}{\eta_0} e^{-j k z} \quad \eta_0 = 120\pi\text{ ohm} \end{align*} is incident normally on the boundary. Write down the expressions for the reflected electric and magnetic fields.

For two isotropic media with \mu_1 \neq \mu_2 and \varepsilon_1 \neq \varepsilon_2, find an expression for the Brewster angle \theta_b for parallel polarization.

(ii) The earth may be modeled as a spherical capacitor with a = 6\cdot 5 \times 10^6\text{ m} and b \rightarrow \infty. Determine C, if the medium surrounding the earth is free space.