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.
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.
The equation of a progressive wave moving on a string is y = 5\sin \pi(0.01x - 2t). In this equation, y and x are in centimetres and t is in seconds. Calculate amplitude, frequency and velocity of the wave. If two particles at any instant are situated 200\,\mathrm{cm} apart, what will be the phase difference between these particles?
In Michelson interferometer, 100 fringes cross the field of view when the movable mirror is displaced through 0.029\,\text{mm}. Calculate the wavelength of the light source used.
Obtain the conditions for constructive interference and destructive interference in a thin film due to reflected light.
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.
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}}
Explain the principle of producing polarized light by the method of reflection, refraction and double refraction with the help of neat diagrams.
Can D_1 and D_2 lines of sodium light (\lambda_{D_1}=5890\mathring{\mathrm{A}}\ \text{and}\ \lambda_{D_2}=5896\mathring{\mathrm{A}}) be resolved in second-order spectrum if the number of lines in the given grating is 450? Explain.
Obtain an expression for the resolving power of a grating explaining the Rayleigh's criterion of resolution.
Calculate the percentage contraction in the length of a rod in a frame of reference, moving with velocity 0.8c in a direction
(i) parallel to its length and
(ii) at an angle of 30^\circ with its length. What is the orientation of the rod in the moving frame of reference in case (ii)?
Given a proton for which \beta=0.995 measured in the laboratory. What are the corresponding relativistic energy and momentum? Take, m_p=1\cdot67\times10^{-24}\,\mathrm{g}.
(i) The distance between the centres of the carbon and oxygen atoms in the carbon monoxide (CO) gas molecule is 1.130\times10^{-10}\ \mathrm{m}. Locate the centre of mass of the molecule relative to the carbon atom.
(ii) Find the centre of mass of a homogeneous semicircular plate of radius a.
A student is working in a physics laboratory, which is at temperature 27^{\circ}\mathrm{C}, on a sonometer to study formation of stationary waves. The cross-sectional area of the sonometer wire is 0.85\times10^{-6}\ \mathrm{m^2} and a tension of 20\ \mathrm{N} is applied on it. If the rigid supports are 1.2\ \mathrm{m} apart and the temperature of the wire drops by 7^{\circ}\mathrm{C}, calculate the (i) final tension and (ii) fundamental frequency of vibration of the wire. Take, coefficient of linear expansion and isothermal Young's modulus as 1.5 \times 10^{-5}\ \mathrm{K}^{-1} and 2.0 \times 10^{11}\ \mathrm{N\,m}^{-2} respectively.
A body moving in an inverse square attractive field traverses on elliptical orbit with eccentricity e and period \gamma. Find the time taken by the body to traverse the half of the orbit that is nearer the centre of force. Explain briefly why a comet spends only 18\% of its time on the half of its orbit that is nearer the sun.
A horizontal pipe of non-uniform bore has water flowing through it such that the velocity of flow is 40\ \mathrm{cm/s} at a point where the pressure is 2\ \mathrm{cm} of mercury column. What is the pressure at a point where the velocity of flow is 60\ \mathrm{cm/s}? (Take, g = 980\ \mathrm{cm/s^2} and density of water = 1\ \mathrm{g/c.c.})
Four solid spheres A, B, C, and D each of mass m and radius a, are placed with their centres on the four corners of square of side b as shown in the figure below:
Calculate the moment of inertia of the system about one side of the square, Also, calculate the moment of inertia of the system about a diagonal of the square.
Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio \sigma are related by the equation Y = 2n(1 + \sigma).
Derive the expression for Coriolis force and show that this force is perpendicular to the velocity and to the axis of rotation. What is the nature of this force?
Describe an operational amplifier based integrator. Using operational amplifier integrators, design a circuit to solve the following differential equation: \frac{d^2v}{dt^2}+2\frac{dv}{dt}+3v=0
