What are the limitations of Rayleigh-Jeans law in explaining the spectrum of radiations from a blackbody? Explain how these limitations were overcome in Planck's radiation law. Deduce Wien's displacement law from Planck's radiation law.

A current i(t)=(2e^{-t}-e^{-2t})\,\mu\mathrm{A} charges up a 120\,\mathrm{nF} capacitor for a period of 2 seconds. If the final voltage across the capacitor is 15\,\mathrm{V}, what was the initial voltage across it?

When a person carrying something metallic walks through the doorway of a metal detector, it emits a sound. Explain the reason behind it. A 200\Omega resistor and a 15\mu\mathrm{F} capacitor are connected in series to 220 V, 50 Hz a.c. supply. Calculate the current in the circuit and the r.m.s. voltage across the resistor and the capacitor. Is the algebraic sum of these voltages more than the supply voltage? If yes, resolve the paradox.

How large an inductance needs to be connected in series with a 120 V, 60 W lightbulb if it is to operate normally when the combination is connected across a 240 V, 60 Hz supply?

Discuss briefly the features of 'guard rings'. The plates of a capacitor are square-shaped, each of side l. The plates are inclined at an angle \alpha to each other. The smallest distance between the plates is a. Calculate the capacitance when \alpha is small.

Why does a soap bubble expand upon electrification? A sphere of radius R contains a charge +Q and a charge -Q distributed uniformly in the upper and lower hemispheres respectively. Show that the dipole moment of charge distribution is \dfrac{3}{4}QR\hat{k}, where \hat{k} is directed along the polar axis of the spherical coordinate system.

A charge q=2\,\mu\mathrm{C} is placed at a=10\,\mathrm{cm} from an infinite grounded conducting plane sheet. Find the (i) total charge induced on the sheet, (ii) force on the charge q and (iii) total work required to remove the charge slowly to an infinite distance from the plane.

Discuss the principle of 'artificial dielectric'. Where do you find its use?

Write down the electromagnetic wave equations in non-conducting dielectric medium. Hence show that the velocity of wave propagation is given by v=\sqrt{\frac{1}{\varepsilon\mu}}, where the symbols have their usual meanings.

Write down the physical significance of Maxwell's equations and explain the concept of displacement current by using a proper example.

Write down Stefan-Boltzmann law of radiation and derive it from Planck's law of radiation. An aluminium foil of relative emittance 0.1 is placed between two concentric spheres (assumed perfectly black) at temperatures 300\,\mathrm{K} and 200\,\mathrm{K} respectively. Find the temperature of the foil once the steady state is reached.

In the circuit diagram shown below, the voltmeter reads 50 volts when it is connected across the 400\Omega resistance. Calculate what the same voltmeter will read when connected across the 300\Omega resistance.

With the help of a neat diagram, show that the potential due to a dipole at a point is given by V=\frac{1}{4\pi\epsilon_0}\,\frac{p\cos\theta}{r^2}, where p is the dipole moment of the charge distribution, \theta is the angle between the line joining the centre of the dipole to the point of interest and the axis of the dipole.

Briefly explain Planck's law of blackbody radiation. Show that Planck's law reduces to Wien's law and Rayleigh-Jeans law at lower and higher wavelength limits respectively.

Write down Maxwell's equations for linear dielectrics and deduce the equation of continuity.

State and prove Poynting's theorem.

Show that the displacement current between the plates of a parallel-plate capacitor is equal to the conduction current across the conductor.

In free space, the electric field of electromagnetic wave is given as \vec{E}(x,t)=120\cos(\omega t-kx)\hat{y}\,\mathrm{V/m}. Find the average power crossing a circular area of radius one metre in the yz-plane.

An alternating current varying sinusoidally with a frequency of 50 Hz has an r.m.s. value of 40 A. Find the instantaneous value of the current at 0.00125 second after passing through maximum positive value.

The spectral energy curve of the moon shows maxima at 470\,\mathrm{nm} and 14\,\mu\mathrm{m}. What inference can you draw from this data? Also calculate the energy density and radiation pressure in both cases. Given, Wien's constant b = 2.892 \times 10^{-3}\,\mathrm{m\,K}, Stefan's constant \sigma = 5.67 \times 10^{-8}\,\mathrm{J\,m^{-2}\,s^{-1}\,K^{-4}} and speed of light c = 3 \times 10^{8}\,\mathrm{m\,s^{-1}}.