PYQ Bank

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

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

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]

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.

(i) Derive Planck's law of Black body radiation. (ii) Obtain its limiting forms for (i) very low frequency and (ii) very high frequency.

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.

Show that Maxwell's equations of electrodynamics are invariant under Lorentz transformations.

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.

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}}.

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.

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.

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

(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 ?

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.

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.

Plane polarized light of wavelength 6000\text{ \AA} is incident on a thin quartz plate cut with faces parallel to the optic axis. Given, \mu_0 = 1.544 and \mu_e = 1.553, calculate the following quantities : (i) The ratio of the intensities of the ordinary and the extraordinary light if the planes of vibration of two incident lights make an angle of 30^\circ with the optic axis. (ii) The minimum thickness of the plate for which ordinary and the extraordinary waves will combine to produce plane polarized light.

Explain the principle of producing polarized light by the method of reflection, refraction and double refraction with the help of neat diagrams.

Explain the principle of (i) induced absorption, (ii) spontaneous emission and (iii) stimulated emission. Show that the ratio of Einstein's coefficients is given by \frac{A}{B}=\frac{8\pi h\nu^{3}}{c^{3}}

The refractive indices of core and cladding in a step index optical fiber are 1.52 and 1.48 respectively. The diameter of the core is 30\,\mu\mathrm{m}. If the operating wavelength is 1.3\,\mu\mathrm{m}, calculate the V parameter and the maximum number of modes supported by the fiber.