1Medium15 marks
A body turns about a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.
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82 questions from the UPSC 2014 examination.
82 questions
A body turns about a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.
Obtain the time-dependent Schrödinger equation for a particle. Hence deduce the time-independent Schrödinger equation.
The mean life time of Lambda (\lambda^0) particle is 2.6 x 10^-10 s. What will be the uncertainty in the determination of its mass in eV ?
In a cubic unit cell, find the angle normals to the plane (111) and (121).
Starting from Maxwell's equation, obtain the wave equation for the electric field E in free space and appropriate wave equation for the electric field E = E_z (x, y, z) z-hat.
Lead in the superconducting state has critical temperature of 6.2 K at zero magnetic field and a critical field of 0.064 MA\text{m}^{-1} at 0 K. Determine the critical field at 4 K.
For initial current conditions I = I0 and dI/dt = 0 at t = 0, show that the time dependent current in the critical damping case for an LCR circuit is given by I = I0(1 + γt/2)e^(-γt/2) where γ = R/L, ω0^2 = 1/LC, ω = √(ω0^2 - γ^2/4) and tan δ = -γ/2ω.
Using Ampere's Law and continuity equation, show that the divergence of the total current density is zero.
Consider a system of free gas particles having f degrees of freedom. Use equipartition theorem to establish the relation f = 2/(Cp/Cv - 1) where Cp and Cv are molar specific heats at constant pressure and constant volume respectively. Obtain the values of Cp/Cv for diatomic and triatomic gases.
Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by: f(E) approx exp((mu - E) / k_B * T), where f(E) is much smaller than unity, mu and k_B * T are the chemical potential and thermal energy of the atom.
In deriving radiation laws, we consider a cubical container of volume V containing a photon gas in equilibrium. Calculate the differential number of allowed normal modes of frequency ω.
How does the energy gap in superconductors differ from the energy gap in insulator ? How does it vary with temperature for superconductors ?
An electron is moving in a one dimensional box of infinite height and width 1 \text{ \AA}. Find the minimum energy of electron.
Consider a rigid body rotating about an axis passing through a fixed point in the body with an angular velocity omega. Determine the kinetic energy of such a rotating body in a coordinate system of principal axis. If the Earth suddenly stops rotating, what will happen to the rotational kinetic energy? Comment in detail.
For initial current conditions I = I_0 and dI/dt = 0 at t = 0, show that the time dependent current in the critical damping case for an LCR circuit is given by I = I_0 (1 + (gamma*t)/2) * e^(-gamma*t / 2), where gamma = R/L, omega_0^2 = 1 / LC, omega = sqrt(omega_0^2 - gamma^2 / 4) and tan(delta) = -gamma / (2*omega).
What is the reciprocal lattice and why is it named so ? Derive the relationships for the primitive translation vectors of the reciprocal lattice in terms of those of the direct lattice.
Show that the energy flow due to a plane electromagnetic wave propagating along z-direction in a dielectric medium is given by z-hat * (k / (omega * mu)) * E_0^2 * cos^2(kz - omega * t), where k and omega are the propagation vector and angular frequency, E_0 is electric field amplitude, mu is the relative permeability of the medium.
Explain the four thermodynamic relations of Maxwell. Using the same, obtain the Clausius-Clapeyron equation dP/dT = L / (T(V_2 - V_1)).
In a Young double slit experiment, the first bright maximum is displaced by y = 2 cm from the central maximum. If the spacing between slits and distance from the screen are 0.1 mm and 1 m respectively, find the wavelength of light.
The density inside a solid sphere of radius a is given by rho = (rho_0 * a) / r, where rho_0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.
Find the deBroglie wave length of (i) a neutron and (ii) an electron moving with a kinetic energy of 500 eV. (1 eV = 1.602 x 10^-19 J)
A mirror is moving through vacuum with a relativistic speed v in the x-direction. A beam of light with frequency omega_i is normally incident (from x = infinity) on the mirror. (i) What is the frequency of the reflected light expressed in terms of omega_i, c and v? (ii) What is the energy of each reflected photon?
State and explain Stefan-Boltzmann Law. Show that log P = log K + 4 log R, where P is the power emitted by black body and R is the resistance of the black body, K is a constant.
Using Maxwell-Boltzmann distribution law prove that there cannot be any negative absolute temperature.
How does holography differ from conventional photography ? What are the requirements for the formation and reading of a hologram?
Show that the group velocity is equal to particle velocity. Also prove that the group velocity of the photons is equal to c, the velocity of light.
When connected in series, L_1, C_1 have the same resonant frequency as L_2, C_2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mentioned.
Explain the working principle of a 3-level laser with a specific example. Comment on why the third level is needed.
Using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i h^-, deduce the commutation relation between the components of angular momentum operator L. [L_x, L_y] = i h^- L_z, [L_y, L_z] = i h^- L_x and [L_z, L_x] = i h^- L_y.
In question 3(a), if the average energy flux of the incident beam is Pi (watts/m^2), what is the average energy flux of the reflected beam?
In deriving radiation laws, we consider a cubical container of volume V containing a photon gas in equilibrium. Calculate the differential number of allowed normal modes of frequency omega.
One kg of water at 20°C is converted into ice at -10°C at constant pressure. Heat capacity of water is 4,200 J/kg.K and that of ice is 2,100 J/kg.K. Heat of fusion of ice at 0°C is 335 x 10^3 J/kg. Calculate the total change in entropy of the system.
One kg of water at 20°C is converted into ice at -10°C at constant pressure. Heat capacity of water is 4,200 J/kg.K and that of ice is 2,100 J/kg.K. Heat of fusion of ice at 0°C is 335 * 10^3 J/kg. Calculate the total change in entropy of the system.
The mean life time of Lambda (λ) particle is 2.6 x 10^-10 s. What will be the uncertainty in the determination of its mass in eV ?
An electron is moving in a one dimensional box of infinite height and width 1 A. Find the minimum energy of electron.
Write down Pauli spin matrices. Express J_x, J_y and J_z in terms of Pauli spin matrices.
A sample of a certain element is placed in a magnetic field of flux density 0.3 tesla. How far apart is the Zeeman component of a spectral line of wavelength 4500 \text{ \AA} ? Given : e/m = 1.76 x 10^11 c/kg, c = 3.0 x 10^8 ms^-1.
What is the physical significance of Einstein's A-coefficient ? Explain, why it is more difficult to achieve Lasing action at X-ray wavelength than at infra-red wavelength.
Explain the four thermodynamic relations of Maxwell. Using the same, obtain the Clausius-Clapeyron equation dP/dT = L / T(V2 - V1).
In a certain engine, a piston undergoes vertical SHM with an amplitude of 10 cm. A washer rests on the top of the piston. As the motor is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston?
Explain the working of SEM and TEM and highlight the major differences in principles. Draw neat schematic diagrams.
What are the magic numbers in nuclei ? List the experimental evidences indicating their existence.
Show that the energy flow due to a plane electromagnetic wave propagating along z-direction in a dielectric medium is given by ẑ k/ωμ E0^2 cos^2(kz - ωt), where k and ω are the propagation vector and angular frequency, E0 is the electric field amplitude, μ is the relative permeability of the medium.
Explain why stable light nuclei have equal number of protons and neutrons whereas heavy nuclei have excess of neutron.
(i) How does liquid drop model explain fission ? (ii) What are the limitations of shell model ?
Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m as the order number and theta as the angle of diffraction for normal incidence, calculate the angular dispersion d(theta)/d(lambda) for an incident light of wavelength lambda.
A sphere of radius R moves with velocity u in an incompressible, non-viscous ideal fluid. Calculate the pressure distribution over the surface of the sphere. Do you think that a force is necessary to keep the sphere in uniform motion?
For a multimode step index optical fibre, the core refractive index is 1.5 and fractional index difference is 0.001. Calculate the pulse broadening for 1 km length of the fibre. Over a length of 2 km of the fibre, calculate the minimum pulse separation that can be transmitted without overlap.
Using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i\hbar, deduce the commutation relation between the components of angular momentum operator L. [L_x, L_y] = i\hbar L_z, [L_y, L_z] = i\hbar L_x and [L_z, L_x] = i\hbar L_y.
It is possible to estimate the nuclear radius from the study of alpha decay ? Explain how.
Starting from Maxwell's equation, obtain the wave equation for the electric field E in free space and appropriate wave equation for the electric field E = Ez(x, y, z) ẑ.
A sample of a certain element is placed in a magnetic field of flux density 0.3 tesla. How far apart is the Zeeman component of a spectral line of wavelength 4500 A ? Given : e/m = 1.76 x 10^11 c/kg, c = 3.0 x 10^8 ms^-1.
Give the basic structure of n-channel depletion type MOSFET. Draw the drain current-drain voltage characteristics both in depletion as well as enhancement modes.
In question 3(a), if the average energy flux of the incident beam is P_i (watts/m^2), what is the average energy flux of the reflected beam?
Explain how the nuclear spin I depends on the mass number A and atomic number Z of atoms.
Obtain an expression for the normal Zeeman shift. Illustrate the Zeeman splitting of spectral lines of H atom and the allowed transitions for the l = 1 and l = 2 states.
Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression?
If the debye temperature of copper is 348 K, then determine the Debye temperature of gold. Take the densities of gold and copper as 1.93 \times 10^4 \text{ kg}/\text{m}^3 and 0.89 \times 10^4 \text{ kg}/\text{m}^3 respectively.
Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by f(E) ≃ exp((μ - E)/kBT), where f(E) is much smaller than unity, μ and kBT are the chemical potential and thermal energy of the atom.
When connected in series, L1, C1 have the same resonant frequency as L2, C2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mentioned.
Lead in the superconducting state has critical temperature of 6.2 K at zero magnetic field and a critical field of 0.064 MAm^-1 at 0 K. Determine the critical field at 4 K.
Discuss the vibrational spectra of a diatomic molecule treating it as an anharmonic oscillator.
Consider a rigid body rotating about an axis passing through a fixed point in the body with an angular velocity ω. Determine the kinetic energy of such a rotating body in a coordinate system of principal axis. If the Earth suddenly stops rotating, what will happen to the rotational kinetic energy ? Comment in detail.
The density inside a solid sphere of radius a is given by p = ρ0 a / r, where ρ0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.
If x^ and p^ are the position and momentum operators, prove the commutation relation [p^2, x^] = -2i h^- p^.
(i) What are salient features of nuclear forces ? (ii) Discuss Yukawa's theory of nuclear forces.
If I' and I be the Moments of Inertia of a body about an axis passing through an arbitrary origin and about a parallel axis through the centre of mass respectively, show that I' = MR^2 + I, where R is the position vector of the centre of mass with respect to the arbitrary origin and M is the mass of the body.
Consider a system of free gas particles having f degrees of freedom. Use equipartition theorem to establish the relation f = 2 / ((C_p / C_v) - 1), where C_p and C_v are molar specific heats at constant pressure and constant volume respectively. Obtain the values of C_p / C_v for diatomic and triatomic gases.
A mirror is moving through vacuum with a relativistic speed v in the x-direction. A beam of light with frequency ωi is normally incident (from x = ∞) on the mirror. (i) What is the frequency of the reflected light expressed in terms of ωi, c and v? (ii) What is the energy of each reflected photon?
Solve the Schrödinger equation for a particle of mass m confined in a one-dimensional potential well of the form : V = 0, when 0 <= x <= L = infinity, when x < 0, x > L Obtain the discrete energy values and the normalized eigen functions.
(i) Obtain an expression for the resonance condition in NMR. (ii) Explain the relaxation processes in NMR spectroscopy.
Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m as the order number and 0 as the angle of diffraction for normal incidence, calculate the angular dispersion dθ/dλ for an incident light of wavelength λ.
A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time period of the motion.
Solve the Schrödinger equation for a particle of mass m confined in a one-dimensional potential well of the form : V = 0, when 0 \le x \le L, = \infty, when x < 0, x > L. Obtain the discrete energy values and the normalized eigen functions.
If \hat{x} and \hat{p} are the position and momentum operators, prove the commutation relation [p^2, \hat{x}] = -2i\hbar p.
Define Enthalpy and show that it remains constant in a throttling process.
In a cubic unit cell, find the angle between normals to the plane (111) and (121).
An electric field of 100 V/m is applied to a sample of n-type semiconductor whose Hall coefficient is -0.0125 \text{ m}^3/\text{coulomb}. Determine the current density in the sample assuming \mu_x = 0.36 \text{ m}^2 \text{ v}^{-1}\text{s}^{-1}.
The velocity of sound in f.c.c. gold and f.c.c. copper is 2100 m/s and 3800 m/s respectively. If the Debye temperature of copper is 348 K, then determine the Debye temperature of gold. Take the densities of gold and copper as 1.93 x 10^4 kg/m^3 and 0.89 x 10^4 kg/m^3 respectively.
Given that the spacing between vibrational levels of CO molecules is 8.45 x 10^-2 eV of energy. Find the force constant of the molecule.
How are elementary particles classified on the basis of their participation in fundamental interaction ?
An electric field of 100 V/m is applied to a sample of n-type semiconductor whose Hall coefficient is -0.0125 m^3/coulomb. Determine the current density in the sample assuming mu_x = 0.36 m^2 v^-1s^-1.