PYQ Bank

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.

Show that the interaction energy of two magnetic dipoles $ \bar{m}_1 $ and $ \bar{m}_2 $ separated by a displacement $ \bar{r} $ is given by U = \frac{\mu_0}{4\pi} \cdot \frac{1}{r^3} \left[ \bar{m}_1 \cdot \bar{m}_2 - 3(\bar{m}_1 \cdot \hat{r})(\bar{m}_2 \cdot \hat{r}) \right]. If two magnetic dipoles are held at a fixed distance apart, but allowed to rotate freely, what would be the configuration for stable equilibrium ?

Find the vector potential due to a line segment from x = a to x = b carrying a current I at a point P which is at a distance d from the line segment.

\questiondiagram{
} 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.

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.

State Gauss's Law of electrostatics both in integral form and differential forms. Two charged spheres of radius R each, have their centres a distance d apart such that d < 2R. One of the spheres has a uniform positive charge density \rho per unit volume while the other has opposite charge density -\rho. Show that the electric field in the region of overlap between two spheres is uniform.

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.

Consider a plane electromagnetic wave entering a rarer medium from a denser medium. Show that the Brewster angle is less than the total internal reflection angle.

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.

Explain how Maxwell modified Ampere's law of magnetostatics by introducing the concept of displacement current. How does it resolve the paradox of a charging capacitor ?

State Faraday's Law of induction in terms of emf and magnetic flux. Use Stoke's theorem to express the law in differential form. A uniform, time varying magnetic field $ \bar{B} = \bar{B}_0(1 + \alpha t)\hat{k} $ fills a circular region of radius R lying in the x-y plane with the centre at origin. Here $ \bar{B}_0 $ and $ \alpha $ are appropriately dimensioned constants. Find the induced electric field at a distance r from the centre where r < R.

Find the phase and group velocities for the wave E (z, t) = E_0 \cos (k_1 z - \omega_1 t) + \cos (k_2 z - \omega_2 t) where $ |k_1 - k_2| \ll k_1 k_2 ; |\omega_1 - \omega_2| \ll \omega_1, \omega_2 $.

Let the two waves with parallel electric fields be given by E_1 = 2 \cos \left( \bar{k}_1 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}, E_2 = 5 \cos \left( \bar{k}_2 \cdot \bar{r} - \omega t + \frac{\pi}{3} \right) \text{kV/m}. Find the intensity of each beam $ I_1, I_2 $ and also the interference term $ I_{12} $ at a point where their path difference is zero. Calculate the visibility $ V \left( = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} \right) $ for the interference pattern. [$ \epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2 $, $ \mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 $]

Calculate the Fraunhofer diffraction pattern from a grating of N slits with width e, separated by equal opaque spaces d. Find the condition for principal maxima and the corresponding values of intensity. A parallel beam of Na light is incident normally on a plane grating with 4250 lines per cm. The second order spectral line is observed to be deviated through 30^\circ. Calculate the wavelength of light.

A convex lens of focal length 20 cm and a concave lens of focal length 10 cm are placed 20 cm apart. In between them an object is placed at a distance $ x $ from the convex lens. What is the value of $ x $ in cm so that the images formed by both of them coincide ?

Explain how inversion of population is achieved in a He-Ne laser.

Consider an ensemble of two-level atoms in thermal equilibrium. Show that the ratio of Einstein A and B coefficients is given by \frac{A}{B} = \frac{8\pi h \nu^3}{c^3}. Why is it not possible to achieve inversion of population in a two-level medium ?

A force field is given by \bar{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k}. Is it a conservative field ? If so, what is the scalar potential ?

Derive an expression for the moment of inertia of a rigid body about any axis. What is an "ellipsoid of inertia" ? Explain clearly what you mean by the terms "principal axes" and "principal moments of inertia" ? Find the moment of inertia of a thin rectangular lamin a about an axis passing through the centre of the lamina and perpendicular to its plane. Hence determine the moments of inertia about axes passing through the midpoints of its both sides and perpendicular to its plane.

The length of a moving rod can be defined as the product of its velocity and the time interval between the instants that both the end points of the rod pass a fixed mark in S system. Show that this definition leads to the space contraction.