85 questions from the UPSC 2019 examination.
85 questions
(i) How does Reynolds number help in the study of fluid motion ? (ii) In a horizontal pipeline of uniform area of cross-section, the pressure falls by 5 N m−2 between two points separated by a distance of 1 km. Calculate the change in kinetic energy per kg of oil flowing at these points. Density of oil = 800 kg m−3.
(i) What are the fringes of equal thickness and fringes of equal inclination ? (ii) In a Newton's ring arrangement with a source emitting two wavelengths λ1 = 6 × 10−7 m and λ2 = 5.9 × 10−7 m, it is found that the mth dark ring due to one wavelength coincides with the (m + 1)th dark ring due to the other. Find the diameter of the mth dark ring, if the radius of curvature of the lens is 90 cm.
Why are Raman active vibrations and IR vibrations in CO2 molecule complementary to each other?
Why is it not possible to detect the parity violation in weak interaction by observing only the beta decay rate? Justify your answer.
(i) How does Reynolds number help in the study of fluid motion ? (ii) In a horizontal pipeline of uniform area of cross-section, the pressure falls by 5 \text{ Nm}^{-2} between two points separated by a distance of 1 \text{ km}. Calculate the change in kinetic energy per kg of oil flowing at these points. \text{Density of oil} = 800 \text{ kg m}^{-3}.
How do you define density of states? Show that the density of states with wave vector less than k in a three-dimensional cubic box of volume V can be given by D(omega) = (V / (2 pi^2)) k^2 (dk / d omega) in the frequency spectrum between omega and omega + d omega. Here, assume that the number of modes per unit range of k is L / (2 pi), L being the length of each side of the cubic box.
Where do you find the applications of gyroscope ? A top of mass 0.200 kg is made up of a thin disc of radius 0.12 m. It is pierced in the centre and a pin of negligible mass is mounted normal to its plane. The pivot under the disc is 0.03 m long. The top is made to spin with its axis making an angle θ = 20° with the vertical and a precessional angular speed of 2 rad/s. Calculate the angular speed with which it spins.
A silicon semiconductor sample is doped with 6 x 10^16 cm^-3 of aluminium and 7 x 10^15 cm^-3 of phosphorus atoms. Given at T = 300 K, the intrinsic carrier concentration, n_i = 1.5 x 10^10 cm^-3; the band gap, E_g = 1.1 eV; the electron mobility, mu_n = 1250 cm^2 V^-1 s^-1 and the hole mobility, mu_p = 480 cm^2 V^-1 s^-1. Determine in the sample of the following: (i) The type of the semiconductor, n or p (ii) The hole carrier concentration (iii) The electron carrier concentration (iv) The position of the Fermi level in the sample with respect to the bottom of the conduction band (v) The conductivity of the sample
Briefly outline the theory of scattering of electromagnetic radiation by a bound electron and hence derive the conditions for Rayleigh scattering. How can you explain the blue of the sky ?
A gas has only two particles, a and b. With the help of a diagram, show that how these two particles can be arranged in the three quantum series 1, 2, 3 using (i) Maxwell-Boltzmann, (ii) Fermi-Dirac, and (iii) Bose-Einstein statistics.
Define Pauli spin matrices σ_x, σ_y and σ_z. Using these definitions, prove the following: (i) σ_x^2 = σ_y^2 = σ_z^2 = 1 (ii) σ_x σ_y = iσ_z; σ_z σ_x = iσ_y; σ_y σ_z = iσ_x
(i) Explain the origin of the nuclear magnetic moment. Deduce expression for the magnetic dipole moment with the help of the Schmidt single particle model. (ii) For a system consisting of one proton and one neutron (not necessarily a deuteron), write down the various possible states specifying clearly its isospin, spin and orbital quantum numbers.
Find the values of E and H on the surface of a wire carrying a current. By computing the Poynting vector, show that it represents a flow of energy into the wire.
Explain the effect of pressure on the melting and boiling points of a substance using Clapeyron's latent heat equation. Calculate under what pressure, water will boil at 120°C, if the change in specific volume when 1 gram of water is converted into steam is 1676 cm^3. Latent heat of steam = 540 cal/g, 1 atmospheric pressure = 10^6 dynes/cm^2.
Three cells are connected in parallel with similar poles connected together with wires having negligible resistance. The emfs of the cells are 2, 1 and 4 volts respectively and the corresponding internal resistances are 4, 3 and 2 ohms. Calculate the current flowing through the 4 V cell.
Explain the various methods of finding the size of the nucleus. How will you determine the nuclear radius from the observation of beta rays resulting from nuclear transition when the initial and final nuclei are mirror nuclei?
Assuming equal masses for up (u) and down (d) quarks, find the ratio (μ_n / μ_p) of the magnetic moments of neutron and proton.
Deduce the Miller indices of the close-packed planes of atoms in the f.c.c. lattice.
Two conducting planes, intersecting at right-angles to each other, are kept at a potential \phi_0. Calculate the potential at a point in space if the total charge on a plane of area \alpha \text{ be } Q.
State and express mathematically the three uncertainty principles of Heisenberg. Highlight the physical significance of these principles in the development of Quantum Mechanics.
Discuss in brief the ultraviolet catastrophe. How did Planck solve this problem ?
Consider a face-centred cubic lattice of side a. Deduce— (i) the primitive translation vectors; (ii) the volume of the primitive cell; (iii) the reciprocal primitive translation vectors; (iv) the volume of the reciprocal lattice.
Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n = (n + 1/2) ħω_0 at energy state n with ω_0 being the natural frequency of vibration of the linear oscillator. Prove that n = 0 energy state has a wave function of typical Gaussian form.
Starting from the expression for the electrostatic potential \phi(\vec{r}) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\vec{r}_0)}{|\vec{r} - \vec{r}_0|} \text{ d}V_0 obtain Poisson's equation \nabla^2 \phi = -\frac{\rho}{\varepsilon_0}. [Symbols have their usual meanings]
Write down Euler's dynamical equations of motion (no derivation) of a rigid body about a fixed point under the action of a torque. Show that the kinetic energy of the torque-free motion is constant.
Prove that when light goes from one point to another via a plane mirror, the path followed by light is the one for which the time of flight is the least.
Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.
Given that the single particle energy separation between 1d_5/2 and 1d_3/2 in ^17O is 5 MeV. Calculate the strength of spin-orbit interaction. It is observed that 1d_5/2 level is lower than 1d_3/2 level.
If the partition function for a perfect gas is given by Z = (V / h^3) (2π mkT)^(3/2) calculate (i) average kinetic energy per molecule and (ii) specific heat of the gas.
Show that the mass and linear momentum of a quantum mechanical particle can be given by m = h / (lambda nu) and p = h / lambda, respectively, where h, lambda and nu are Planck's constant, wavelength and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.
What is axial chromatic aberration ? A convex lens has a focal length of 15.5 × 10−2 m for red colour and 14.45 × 10−2 m for violet colour. If an object is kept at a distance of 40 cm from the lens, calculate the longitudinal chromatic aberration of the lens.
Show that the mass and linear momentum of a quantum mechanical particle can be given by m = h / (λv) and p = h / λ, respectively, where h, λ and v are Planck's constant, wavelength and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.
Define angular momentum of a particle and find out the three components of the angular momentum operator L-hat in Cartesian coordinates. Show that L-hat^2 = - h_bar^2 [r^2 nabla^2 - (partial / partial r) (r^2 (partial / partial r))] Prove that the operator L-hat^2 can also be expressed as L-hat^2 = - h_bar^2 [ (1 / sin theta) (partial / partial theta) (sin theta (partial / partial theta)) + (1 / sin^2 theta) (partial^2 / partial phi^2) ] in spherical polar coordinates (r, theta, phi).
Assuming equal masses for up (u) and down (d) quarks, find the ratio (mu_n / mu_p) of the magnetic moments of neutron and proton.
Why do we prefer to work with a critically damped ballistic galvanometer in a laboratory ? What is external critical damping resistance ?
Einstein's molar specific heat capacity of a solid is given by C_V = 3R \left(\frac{\theta_{\text{E}}}{T}\right)^2 \frac{\text{e}^{\theta_{\text{E}}/T}}{(\text{e}^{\theta_{\text{E}}/T} - 1)^2} \text{where } \theta_{\text{E}} = \frac{\hbar \omega}{k_{\text{B}}} Obtain the expressions for the cases : (i) when T \gg \theta_{\text{E}} (ii) when T \ll \theta_{\text{E}} What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature ? The molar specific heat capacity of a solid at constant volume is 2.77 \text{ JK}^{-1} \text{ at } 36.8 \text{ K}. Determine the Debye temperature of the solid.
Where do you find the applications of gyroscope ? A top of mass 0.200 \text{ kg} is made up of a thin disc of radius 0.12 \text{ m}. It is pierced in the centre and a pin of negligible mass is mounted normal to its plane. The pivot under the disc is 0.03 \text{ m} long. The top is made to spin with its axis making an angle \theta = 20^{\circ} with the vertical and a precessional angular speed of 2 \text{ rad/s}. Calculate the angular speed with which it spins.
If the partition function for a perfect gas is given by Z = \frac{V}{h^3} (2\pi mkT)^{3/2} calculate (i) average kinetic energy per molecule and (ii) specific heat of the gas.
Why is population inversion in general not possible in a two-level laser system? Explain it.
What is Carnot's theorem ? Prove that Carnot's reversible engine is the most efficient one and no other engine can be more efficient than Carnot's engine.
What are the conditions for the change in temperature of a van der Waals gas passing through a porous plug ? Prove that the ideal gas passing through the porous plug does not show any change in temperature.
Derive the expression for the average energy of a quantum oscillation of frequency v. Assume Fermi-Dirac distribution and E - E_F > 2, where E_F is the Fermi level.
Derive the expression for the average energy of a quantum oscillation of frequency nu. Assume Fermi-Dirac distribution and E - E_F > 2, where E_F is the Fermi level.
Two \beta-particles A and B emitted by a radioactive source R travel in opposite directions, each with a velocity of 0.9 \text{ c} with respect to the source. Find the velocity of B with respect to A (Here c is the velocity of light).
Sketch the cross-sectional structure of an enhancement mode MOSFET and explain its principles of operation with the help of its output characteristics.
Stable light nuclei have equal number of protons and neutrons, whereas heavy nuclei have excess of neutrons. Explain why.
For a free quantum mechanical particle under the influence of a one-dimensional potential, show that the energy is quantized in discrete fashion. How do these energy values differ from those of a linear harmonic oscillator?
(i) How can one convert a left-handed circularly polarised light into a right-handed one (and vice versa) ? (ii) Calculate the thickness of a quarter-wave plate when the wavelength of light is 589 nm. Given : μo = 1.544 and μe = 1.553.
Define angular momentum of a particle and find out the three components of the angular momentum operator L_hat in Cartesian coordinates. Show that L_hat^2 = -ħ^2 [r^2 ∇^2 - (∂/∂r)(r^2 (∂/∂r))] Prove that the operator L_hat^2 can also be expressed as L_hat^2 = -ħ^2 [ (1 / sin θ) (∂/∂θ) (sin θ (∂/∂θ)) + (1 / sin^2 θ) (∂^2/∂φ^2) ] in spherical polar coordinates (r, θ, φ).
A silicon semiconductor sample is doped with 6 × 10^16 cm^-3 of aluminium and 7 × 10^15 cm^-3 of phosphorus atoms. Given at T = 300 K, the intrinsic carrier concentration, n_i = 1.5 × 10^10 cm^-3; the band gap, E_g = 1.1 eV; the electron mobility, μ_n = 1250 cm^2 V^-1 s^-1 and the hole mobility, μ_p = 480 cm^2 V^-1 s^-1. Determine in the sample of the following: (i) The type of the semiconductor, n or p (ii) The hole carrier concentration (iii) The electron carrier concentration (iv) The position of the Fermi level in the sample with respect to the bottom of the conduction band (v) The conductivity of the sample
What is Franck-Condon principle? Discuss the intensity distribution in the vibrational electronic spectra of a diatomic molecule on the basis of this principle.
(i) A reference frame S' moves with respect to rest frame S with a uniform velocity 'v' parallel to x-direction. Show from Lorentz transformation that two events simultaneous (t1 = t2) at different positions (x1 ≠ x2) in S frame are not in general simultaneous in S' frame. (ii) The mean life of π meson is 2 × 10−8 s. Calculate the mean life of a meson moving with a velocity of 0.8 c, where c is the velocity of light.
Find the capacitance of two concentric spherical metal shells having radii a and b.
Show that the cross-section for elastic scattering of a point particle from an infinitely massive sphere of radius R is \frac{R^2}{4}. What is the inference of this result ?
What is Gibbs' phase rule ? Find the values of degrees of freedom when (i) only the liquid \text{CO}_2 is in equilibrium with the gaseous \text{CO}_2. (ii) water is in the vapour-liquid saturation region. (iii) water is in a single-phase region. (iv) water is at the triple point.
Show that the cross-section for elastic scattering of a point particle from an massively sphere of radius R is π R^2 / 4. What is the inference of this result ?
What is Gibbs' phase rule ? Find the values of degrees of freedom when (i) only liquid CO2 is in equilibrium with the gaseous CO2. (ii) water is in the vapour-liquid saturation region. (iii) water is in a single-phase region. (iv) water is at the triple point.
Discuss how population inversion is achieved in Ruby laser. What is 'laser spiking' ? Why does it occur ?
Starting from the expression for the electrostatic potential φ(→r) = 1/(4πεo) ∫ (ρ(→r') / |→r - →r'|) dV_o obtain Poisson's equation ∇^2 φ = -ρ/εo. [Symbols have their usual meanings]
How do you define density of states? Show that the density of states with wave vector less than k in a three-dimensional cubic box of volume V can be given by D(ω) = (V / 2π^2) k^2 (dk / dω) in the frequency spectrum between ω and ω+dω. Here, assume that the number of modes per unit range of k is L / (2π), L being the length of each side of the cubic box.
Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n = - (1 / 2(a/Z)) (Z^2 / 4πε_0) (1 / n^2) where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.
What is axial chromatic aberration ? A convex lens has a focal length of 15.5 \times 10^{-2} \text{ m} for red colour and 14.45 \times 10^{-2} \text{ m} for violet colour. If an object is kept at a distance of 40 \text{ cm} from the lens, calculate the longitudinal chromatic aberration of the lens.
A silicon semiconductor sample at T = 300 K having cross-sectional area of 0.5 μm^2 has a pentavalent donor doping profile given by C(x) = 5 × 10^16 e^(-x/L_n) cm^-3. Given, the mobility of the electrons in the sample is 1250 cm^2 V^-1 s^-1 and the diffusion length of the electrons, L_n, is 4 μm. Calculate the diffusion current in the sample at distance x = 2 μm.
The angles between the tetrahedral bonds of diamond are the same as the angles between the body diagonals of a stack of neighbouring cubes having common edges and not faces. Use vector analysis to find the value of the angle.
Two conducting planes, intersecting at right-angles to each other, are kept at a potential φo. Calculate the potential at a point in space if the total charge on a plane of area α be Q.
(i) A reference frame S' moves with respect to rest frame S with a uniform velocity 'v' parallel to x-direction. Show from Lorentz transformation that two events simultaneous (t_1 = t_2) at different positions (x_1 \neq x_2) in S frame are not in general simultaneous in S' frame. (ii) The mean life of \pi \text{ meson is } 2 \times 10^{-8} \text{ s}. Calculate the mean life of a meson moving with a velocity of 0.8 \text{ c}, where c is the velocity of light.
Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n = (n + 1/2) h_bar omega_0 at energy state n with omega_0 being the natural frequency of vibration of the linear oscillator. Prove that n = 0 energy state has a wave function of typical Gaussian form.
Einstein's molar specific heat capacity of a solid is given by Cv = 3R (θE / T)^2 ((e^(θE / T)) / ((e^(θE / T) - 1)^2) where θE = hω / kB Obtain the expressions for the cases : (i) when T >> θE (ii) when T << θE What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature ? The molar heat capacity of a solid at constant volume is 2.77 JK−1 at 36.8 K. Determine the Debye temperature of the solid.
In what way is holography different from conventional photography ? Discuss the salient features of a hologram. What are the requirements for the formation and reading of a hologram ?
(i) Find the moments of inertia of rigid diatomic molecule about different axes of symmetry through the centre of mass. (ii) A proton is 1837 times heavier than an electron. Find the centre of mass of hydrogen atom.
What is Lamb shift? Discuss its significance in determining the fine structure of H_α Balmer line in hydrogen atom.
A 5 cm^2 Ge solar cell with a dark reverse saturation current of 2 nA has solar radiation incident upon it, producing 4 × 10^17 electron-hole pairs per second. The electron and hole diffusion lengths are given to be 5 μm and 2 μm, respectively. Calculate for the cell of the following: (i) The short-circuit current (ii) The open-circuit voltage
Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n = - (1 / (2 (a / Z))) (1 / (4 pi epsilon_0 n^2)) where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.
(i) What is a central force ? Give two examples of the central force. (ii) Show that the angular momentum (→L) of the particle in a central force field is a constant of motion.
(i) What are the fringes of equal thickness and fringes of equal inclination ? (ii) In a Newton's ring arrangement with a source emitting two wavelengths \lambda_1 = 6 \times 10^{-7} \text{ m} \text{ and } \lambda_2 = 5.9 \times 10^{-7} \text{ m}, it is found that the m^{\text{th}} dark ring due to one wavelength coincides with the (m+1)^{\text{th}} dark ring due to the other. Find the diameter of the m^{\text{th}} dark ring, if the radius of curvature of the lens is 90 \text{ cm}.
Three cells are connected in parallel with similar poles connected together with wires having negligible resistance. The emfs of the cells are 2, 1 and 4 volts respectively and the corresponding internal resistances are 4, 3 and 2 ohms. Calculate the current flowing through the 4 \text{ V} cell.
(i) How can one convert a left-handed circularly polarised light into a right-handed one (and vice versa) ? (ii) Calculate the thickness of a quarter-wave plate when the wavelength of light is 589 \text{ nm}. \text{Given : } \mu_{\text{O}} = 1.544 \text{ and } \mu_{\text{E}} = 1.553.
What is Lamb shift? Discuss its significance in determining the fine structure of H_alpha Balmer line in hydrogen atom.
A 5 cm^2 Ge solar cell with a dark reverse saturation current of 2 nA has solar radiation incident upon it, producing 4 x 10^17 electron-hole pairs per second. The electron and hole diffusion lengths are given to be 5 mu m and 2 mu m, respectively. Calculate for the cell of the following: (i) The short-circuit current (ii) The open-circuit voltage
Explain the effect of pressure on the melting and boiling points of a substance using Clapeyron's latent heat equation. Calculate under what pressure, water will boil at 120^{\circ}\text{C}, if the change in specific volume when 1 \text{ gram of water is converted into steam is } 1676 \text{ cm}^3. \text{Latent heat of steam} = 540 \text{ cal/g}, 1 \text{ atmospheric pressure} = 10^6 \text{ dynes/cm}^2.
Two β-particles A and B emitted by a radioactive source R travel in opposite directions, each with a velocity of 0.9 c with respect to the source. Find the velocity of B with respect to A (Here c is the velocity of light).
Estimate the size of hydrogen atom and the ground state energy from the uncertainty principle.
(i) What is a central force ? Give two examples of the central force. (ii) Show that the angular momentum (\vec{L}) of the particle in a central force field is a constant of motion.
Define Pauli spin matrices sigma_x, sigma_y and sigma_z. Using these definitions, prove the following: (i) sigma_x^2 = sigma_y^2 = sigma_z^2 = 1 (ii) sigma_x sigma_y = i sigma_z; sigma_z sigma_x = i sigma_y; sigma_y sigma_z = i sigma_x
A silicon semiconductor sample at T = 300 K having cross-sectional area of 0.5 mu m^2 has a pentavalent donor doping profile given by C(x) = 5 x 10^16 e^(-x/L_n) cm^-3. Given, the mobility of the electrons in the sample is 1250 cm^2 V^-1 s^-1 and the diffusion length of the electrons, L_n, is 4 mu m. Calculate the diffusion current in the sample at distance x = 2 mu m.