Two identical black bodies A and B are respectively at temperatures T and 2T respectively. The heat energy radiated by A is collected for one minute and is used to heat a mass of water. The temperature of water is found to rise by 0.25 K. If the heat radiated by B were to be collected for one minute and used to heat the same mass of water, what would be the rise in the temperature of water ? Assume, specific heat of water to be temperature independent.
A point charge + 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 coordinates x and y.
A plane electromagnetic wave is travelling in vacuum along a direction which makes an angle of 45^\circ each with the positive x and z-axes. The electric field at time t is given by the expression \bar{E} = E_0 (2 \cos \omega t \hat{j} - \sin \omega t (\hat{i} - \hat{k})) where \hat{i}, \hat{j} and \hat{k} are unit vectors along the x, y and z directions respectively and E_0 = 100 \text{ v/m}. Find the magnetic field vector \bar{B} at time t and obtain the average power transmitted per unit area by the wave.
A plane electromagnetic wave travelling in z-direction and polarized in x-direction is incident normally from left on to an interface between two media at z = 0. The medium to the left (z < 0) has a refractive index n_1, while that to the right is of refractive index n_2. The magnetic permeabilities of both the media are equal and \mu_1 = \mu_2 = \mu_0. Obtain expressions for reflection and transmission coefficients (R and T) and show that R + T = 1.
An infinite ladder network of resistances is connected across the points A and B, as shown in the above figure. Each of the resistance is 1\Omega. Calculate the effective resistance between points A and B.
Derive Poisson equation starting from the Coulomb's law for a set of point charges.
Why does a spinning nucleus process in a magnetic field ? Explain the underlying principle of nuclear magnetic resonance (NMR) spectroscopy. Calculate the radio frequency at which NMR occurs in water kept in a uniform magnetic field of 2.4 T, the magnetic moment of proton being 2.793\\ \mu_N.
Obtain the solution of the Laplace equation in cylindrical coordinates.
Explain the Rayleigh Scattering of light. Show that the energy density of light scattered from an isotropic homogenous medium of a gas is inversely proportional to the fourth power of the wavelength of the incident light.
State Amperes law of magnetostatics. Using this law, find the magnetic field at a point due to an infinitely long filamentary current.
In an a.c. circuit, a resistance (R = 100\ \Omega) and a capacitance (C = 100\ \mu\text{F}) in series are connected to an a.c. source V = 200 \sin(100\ \pi t). Calculate the current through the circuit and the voltages across R and C. Draw a vector diagram representing the magnitudes and phases of the voltages.
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.
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 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 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.
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.
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.
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.
