State Rayleigh-Jeans law. Show that the intensity of emissions at a particular wave-length is proportional to the temperature T. Discuss the limitations of this law in describing the intensity distribution of emission spectrum of a blackbody.

What is a magnetic shell ? Define the strength of a magnetic shell. A 2\text{ mm} thick magnetic shell weighing 100\text{ gm} has magnetic moment of 1000\text{ units}. The density of the shell material is 10\text{ gm/cc}. Calculate the intensity of magnetisation and the strength of the shell.

Derive Poisson equation starting from the Coulomb's law for a set of point charges.

A potential in cylindrical coordinates is a function of r and \phi but not of z. Obtain the separated differential equations for R and \Phi, where V = R(r) \Phi(\phi) and solve them.

Write Ampere's circuital law, and obtain a generalised form of this law, for non-stationary case.

Find the magnetic field \vec{B} at the point P due to a short straight length of wire carrying current 'i'. Length of the wire is l. Point P is at a distance r away from the centre of the wire. Angle between \vec{l} and \vec{r} is \theta.

Starting from Maxwell's equation, \nabla . D = \rho, where D is the electric displacement density and \rho is the charge density, derive Poisson's equation. Deduce Laplace's equation for charge-free region from Poisson's equation.

What is molecular polarizability ? Derive Clausius - Mossotti equation relatig the molecular polarizability with the dielectric constant of a dielectric material.

In the case of proton NMR, state the expression for the energy of the dipole in an external magnetic field. How does the NMR originate ? Give the rough range of frequencies of the NMR signals, and the normal magnitude of the applied field.

Using Planck's radiation formula u(\nu) d\nu = \frac{8 \pi h}{c^3} \frac{\nu^3 d\nu}{e^{h\nu/kT} - 1} where the symbols have their usual meaning, find the wavelength of the region where energy density is the greatest. Also calculate the total energy density over all the frequencies

What do you mean by a gauge transformation ? What is its importance ? Show that the Lorentz gauge condition \vec{\nabla} \cdot \vec{A} + \frac{1}{c} \frac{\partial \phi}{\partial t} = 0 is Lorentz invariant. Here \vec{A} and \phi are the vector and scalar potentials.

Starting from Maxwell's equations of electromagnetic field in vacuum obtain the classical wave equations for the four field vectors \vec{E}, \vec{D}, \vec{B} and \vec{H} . Show that the field vectors can be propagated as waves in free space with the velocity of propagation equal to 3 \times 10^8\text{ m/s}, where for free space we have the vacuum permittivity \varepsilon_o = 8.854 \times 10^{-12}\text{ farad/m} & vaccum permeability \mu_o = 1.257 \times 10^{-6}\text{ henry/m}.

Write down the different components of the electromagnetic field tensors F_{\mu\nu} and further prove that Maxwell's equations of electrodynamics are invariant to Lorentz transformations.

7. (a) Write down the macroscopic form of the Maxwell's equations in any isotropic (but inhomogeneous) medium and define the symbols appearing therein. Convert these equations in the integral forms to highlight the laws represented by these equations.

(b) Show that the electric and magnetic field vectors, \vec{E} and \vec{B}, in plane electromagnetic waves are mutually perpendicular in a plane normal to the direction of propagation. How are phases of \vec{E} and \vec{B} related to each other ?

6. (a) Consider two long and straight current carrying wires placed parallel to each other a certain distance apart. Derive an expression for the force per unit length experienced by these wires. Discuss that the attractive (repulsive) nature of this force is related to the directions of flow of currents in the two wires.

(b) Derive approximate expressions for the potential and the radial as well as the azimuthal components of the field due to an electric dipole at points far away from it. Also derive expression and hence describe the effect of a unifrom electric field on a dipole which can rotate freely.

(c) Use the Planck formula for the blackbody radiation u(\omega, T) = \frac{\hbar^2}{\pi^2 c^3} \frac{\omega^3}{\exp(\beta \hbar \omega) - 1} with \beta = \frac{1}{k_B T} to derive Wien's law, Rayleigh-Jeans law and Stefan-Boltzmann law.

(a) Prove the relation \nabla^2 \left( \frac{1}{|\vec{r} - \vec{r}'|} \right) = - 4 \pi \delta \left(|\vec{r} - \vec{r}'|\right) and hence show that \phi(\vec{r}) = \int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} \, d\vec{r}' is a solution of the Poisson equation \nabla^2 \phi(\vec{r}) = - 4 \pi \rho(\vec{r}).

(b) Describe physical significance of the displacement current considering the example of current flow through a capacitor.