A system consists of N number of particles, N » 1. Each particle can have only one of the two energies E1 or E1 + ε (ε > 0). If the system is in equilibrium at a temperature T, the average number of particles with energy E1 is
A mass m, lying on a horizontal, frictionless surface, is connected to one end of a spring. The other end of the spring is connected to a wall, as shown in the figure. At t = 0, the mass is given an impulse.
The time dependence of the displacement and the velocity of the mass (in terms of non- zero constants A and B) are given by
A particle with energy E is incident on a potential given by
The wave function of the particle for E > V0, in the region x > 0 (in terms of positive constants A, B and C) is
The electric field of a light wave is given by The polarization
state of the wave is
Consider the coordinate transformation The relation between the area elements dx'dy' and dxdy is given by dx’dy’ = Jdxdy. The value of J is
The trace of a 2 × 2 matrix is 4 and its determinant is 8. If one of the eigenvalues is 2(1 + i), the other eigenvalue is
Temperature dependence of resistivity of a metal can be described by
A proton from outer space is moving towards earth with velocity 0.99c as measured in earth’s frame. A spaceship, traveling parallel to the proton, measures proton’s velocity to be 0.97c. The approximate velocity of the spaceship, in the earth’s frame, is
A charge φ is at the center of two concentric spheres. The outward electric flux through the inner sphere is φ while that through the outer sphere is 2φ. The amount of charge contained in the region between the two spheres is
At room temperature, the speed of sound in air is 340 m/sec. An organ pipe with both ends open has a length L= 29 cm. An extra hole is created at the position L/2. The lowest frequency of sound produced is
A system comprises of three electrons. There are three single particle energy levels accessible to each of these electrons. The number of possible configurations for this system is
A rigid and thermally isolated tank is divided into two compartments of equal volume V, separated by a thin membrane. One compartment contains one mole of an ideal gas A and the other compartment contains one mole of a different ideal gas B. The two gases are in thermal equilibrium at a temperature T. If the membrane ruptures, the two gases mix. Assume that the gases are chemically inert. The change in the total entropy of the gases on mixing is
A Zener regulator has an input voltage in the range 15V- 20V and a load current in the range of 5 mA- 20 mA. If the Zener voltage is 6.8V, the value of the series resistor should be
The variation of binding energy per nucleon with respect to the mass number of nuclei is shown in the figure.
Consider the following reactions:
A rigid triangular molecule consists of three non- collinear atoms joined by rigid rods. The constant pressure molar specific heat (Cp) of an ideal gas consisting of such molecules is
A satellite moves around the earth in a circular orbit of radius R centered at the earth. A second satellite moves in an elliptic orbit of major axis 8R, with the earth at one of the foci. If the former takes 1 day to complete a revolution, the latter would take
A positively charged particle, with a charge q, enters a region in which there is a uniform electric field E and a uniform magnetic field B both directed parallel to the positive y- axis. At t = 0, the particle is at the origin and has a speed v0 directed along the positive x- axis. The orbit of the particle, projected on the x- z plane, is a circle. Let T be the time taken to complete one revolution of this circle. The y- coordinate of the particle at t = T is given by
Vibrations of diatomic molecules can be represented as those of harmonic oscillators. Two halogen molecules X2 and Y2 have fundamental vibrational frequencies vX = 16.7 × 1012 Hz and vY = 26.8 × 1012 Hz, respectively. The respective force constants are KX = 325 N/m and KY = 446 N/m. The atomic masses of F, Cl and Br are 19.0, 35.5 and 79.9 atomic mass unit respectively.
The halogen molecules X2 and Y2 are
A hollow, conducting spherical shell of inner radius R1 and outer radius R2 encloses a charge q inside, which is located at a distance d (< R1) from the centre of the spheres. The potential at the centre of the shell is
Doppler effect can be used to measure the speed of blood through vessels. Sound of frequency 1.0522 MHz is sent through the vessels along the direction of blood flow. The reflected sound generates a beat signal of frequency 100 Hz. The speed of sound in blood is 1545 m/sec. The speed of blood through the vessel, in m/sec, is
Which of the following circuits represent the Boolean expression
A conducting wire is in the shape of a regular hexagon, which is inscribed inside an imaginary circle of radius R, as shown. A current I flows through the wire. The magnitude of the magnetic field at the center of the circle is
An observer is located on a horizontal, circular turntable which rotates about a vertical axis passing through its center, with a uniform angular speed of 2 rad/sec. A mass of 10 grams is sliding without friction on the turntable. At an instant when the mass is at a distance of 8 cm from the axis, it is observed to move towards the center with a speed of 6 cm/sec. The net force on the mass, as seen by the observer at that instant, is
Miller indices of a plane in cubic structure that contains all the directions ,  and  are
Seven uniform disks, each of mass .. and radius .., are inscribed inside a regular hexagon, as shown. The moment of inertia of this system of seven disks, about an axis passing through the central disk and perpendicular to the plane of the disks, is
A nucleus has a size of 10–15 m. Consider an electron bound within a nucleus. The estimated energy of this electron is of the order of
Consider a vector field Let C be the circle x2 + y2 = 4 on the plane z = 2, oriented counter- clockwise. The value of the contour integral
Consider the equation with the boundary condition y(1) = 1. Out of the following, the range of x in which y is real and finite, is
The Fourier series for an arbitrary periodic function with period 2.. is given by For the particular periodic function shown in the figure, the value of a0 is
The phase of the complex number (1 + i)i in the polar representation is
For an electromagnetic wave traveling in free space, the electric field is given by Which of the following statements are true?
In an ideal Op- Amp circuit shown below, R1 = 3kW, R2 = 1 kW and Vi = 0.5 sin wt (in Volt). Which of the following statements are true?
A particle of mass m is moving in x- y plane. At any given time t, its position vector is given by where A, B and w are constants with A ≠ B. Which of the following statements are true?
A rod is hanging vertically from a pivot. A particle, traveling in horizontal direction, collides with the rod as shown in the figure. For the rod- particle system, consider the linear momentum and the angular momentum about the pivot. Which of the following statements are NOT true?
A particle is moving in a two- dimensional potential well V(x,y) = 0, 0 ≤ x ≤ L, 0 ≤ y ≤ 2L = ∞, elsewhere.
Which of the following statements about the ground state energy E1 and ground state eigenfunction φ0 are true?
Consider the circuit, consisting of an AC function generator V(t) = V0 sin 2φvt with V0 = 5V, an inductor L = 8.0 mH, resistor R = 5 W and a capacitor C = 100 mF. Which of the following statements are true if we vary the frequency?
Muons are elementary particles produced in the upper atmosphere. They have a life time of 2.2 μs. Consider muons which are traveling vertically towards the earths surface at a speed of 0.998c. For an observer on earth, the height of the atmosphere above the surface of the earth is 10.4 km. Which of the following statements are true?
As shown in the P-V diagram, AB and CD are two isotherms at temperatures T1 and T2, respectively (T1 > T2). AC and BD are two reversible adiabats. In this Carnot cycle, which of the following statements are true?
The following figure shows a double slit Fraunhofer diffraction pattern produced by two slits, each of width a, separated by a distance b, a < b. Which of the following statements are correct?
A unit cube made of a dielectric material has a polarization units. The edges of the cube are parallel to the Cartesian axes. Which of the following statements are true?
The power radiated by sun is 3.8 × 1026 W and its radius is 7 × 105 km. The magnitude of the Poynting vector (in W/cm2) at the surface of the sun is ___________.
A particle is in a state which is a superposition of the ground state .... and the first excited state φ1 of a one- dimensional quantum harmonic oscillator. The state is given by The expectation value of the energy of the particle in this state (in units of hw, w being the frequency of the oscillator.. is ___________.
In an experiment on charging of an initially uncharged capacitor, an RC circuit is made with the resistance R = 10 kW and the capacitor C = 1000 µF along with a voltage source of 6V. The magnitude of the displacement current through the capacitor (in mA), 5 seconds after the charging has started, is ___________.
In the given circuit, VCC = 10V and b = 100 for the n- p- n transistor. The collector voltage VC (in Volts) is ___________.
Unpolarized light is incident on a calcite plate at an angle of incidence 50°, as shown in the figure. Take n0 = 1.6584 and ne = 1.4864 for calcite. The angular separation (in degrees) between the two emerging rays within the plate is ___________.
In the hydrogen atom spectrum, the ratio of the longest wavelength in the Lyman series (final state n = 1) to that in the Balmer series (final state n = 2) is ___________.
A rod is moving with a speed of 0.8c in a direction at 60° to its own length. The percentage contraction in the length of the rod is ___________.
X- rays of wavelength 0.24 nm are Compton scattered and the scattered beam is observed at an angle of 60o relative to the incident beam. The Compton wavelength of the electron is 0.00243 nm. The kinetic energy of scattered electrons in eV is ___________.
A diode at room temperature (kT = 0.025 eV) with a current of 1µA has a forward bias voltage VF = 0.4 V. For VF = 0.5 V, the value of the diode current (in µA) is ___________.
GaAs has a diamond structure. The number of Ga- As bonds per atom which have to be broken to fracture the crystal in the (001) plane is ___________.
In the thermodynamic cycle shown in the figure, one mole of a monatomic ideal gas is taken through a cycle. AB is a reversible isothermal expansion at a temperature of 800 K in which the volume of the gas is doubled. BC is an isobaric contraction to the original volume in which the temperature is reduced to 300 K. CA is a constant volume process in which the pressure and temperature return to their initial values. The net amount of heat (in Joules) absorbed by the gas in one complete cycle is ___________.
In a region of space, a time dependent magnetic field B(t) = 0.4 t Tesla points vertically upwards. Consider a horizontal, circular loop of radius 2 cm in this region. The magnitude of the electric field (in mV/m) induced in the loop is ___________.
A plane electromagnetic wave of frequency 5 × 1014 Hz and amplitude 103 V/m, traveling in a homogeneous dielectric medium of dielectric constant 1.69, is incident normally at the interface with a second dielectric medium of dielectric constant 2.25. The ratio of the amplitude of the transmitted wave to that of the incident wave is ___________.
For the arrangement given in the following figure, the coherent light sources A, B and C have individual intensities of 2 mW/m2, 2 mW/m2 and 5 mW/m2, respectively at point P. The wavelength of each of the sources is 600 nm. The r esultant intensity at point P ( in mW /m2) is _______________.
One gram of ice at 0oC is melted and heated to water at 39oC. Assume that the specific heat remains constant over the entire process. The latent heat of fusion of ice is 80 Calories/gm. The entropy change in the process (in Calories per degree) is _______________.
A uniform disk of mass m and radius R rolls, without slipping, down a fixed plane inclined at an angle 30° to the horizontal. The linear acceleration of the disk (in m/sec2) is _______________.
A nozzle is in the shape of a truncated cone, as shown in the figure. The area at the wide end is 25 cm2 and the narrow end has an area of 1 cm2. Water enters the wider end at a rate of 500 gm/sec. The height of the nozzle is 50 cm and it is kept vertical with the wider end at the bottom. The magnitude of the pressure difference in kPa (1 kPa = 103 N/m2) between the two ends of the nozzle is _______________.
A block of mass 2 kg is at rest on a horizontal table. The coefficient of friction between the block and the table is 0.1. A horizontal force 3 N is applied to the block. The speed of the block (in m/ s) after it has moved a distance 10 m is _______________.
A homogeneous semi- circular plate of radius R = 3 m is shown in the figure. The distance of the center of mass of the plate (in meter) from the point O is _______________.
Consider a 20µm diameter p-n junction fabricated in silicon. The donor density is 1016 per cm3. The charge developed on the n- side is 1.6 × 10–13 C. Then the width (in µm) of the depletion region on the n- side of the p- n junction is _______________.