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.
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.
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.
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.
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
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.
Show that Maxwell's equations of electrodynamics are invariant under Lorentz transformations.
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}}.
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.
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.
Write down Maxwell's equations for linear dielectrics and deduce the equation of continuity.
State and prove Poynting's theorem.
(i) Derive Planck's law of Black body radiation. (ii) Obtain its limiting forms for (i) very low frequency and (ii) very high frequency.
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.
(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 ?
Prove that the group velocity V_g of electromagnetic waves in a dispersive medium is given by V_g = \frac{c}{n + \omega\ dn/d\omega} where c is the velocity of light in vacuum and n is the refractive index of the medium for the angular frequency \omega of the waves.
State Huygen's principle. Using this principle and suitable diagram, show that this principle could lead to Snell's law of refraction.
Discuss the principles of Fresnel's half period zone and explain how these are used in the construction of a zone plate. Show how the zone plate has several foci.
A diffraction grating has 5000\text{ lines per cm}. For illumination at normal incidence, determine the dispersive power of the grating in the second order spectrum in the range of wavelengths around 500\text{ n.m.}
