List of top Questions asked in GATE PH

Which of the following option(s) is/are correct for a Type I superconductor?

A two-level quantum system has energy eigenvalues \( E_1 \) and \( E_2 \). A perturbing potential \( H' = \lambda \Delta \sigma_x \) is introduced, where \( \Delta \) is a constant having dimensions of energy, \( \lambda \) is a small dimensionless parameter, and \( \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \).

The magnitudes of the first and the second order corrections to \( E_1 \) due to \( H' \), respectively, are:

Let \( |m\rangle \) and \( |n\rangle \) denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are \( \hat{X} \) and \( \hat{P} \), respectively. The matrix element \( \langle m | \hat{P}\hat{X} | n \rangle \) is non-zero when:

Consider a two-level system with energy states \( +\epsilon \) and \( -\epsilon \). The number of particles at \( +\epsilon \) level is \( N+ \) and the number of particles at \( -\epsilon \) level is \( N- \). The total energy of the system is \( E \) and the total number of particles is \( N = N+ + N- \). In the thermodynamic limit, the inverse of the absolute temperature of the system is: 
(Given: \( \ln N! \approx N \ln N - N \))

An \( \alpha \) particle is scattered from an Au target at rest as shown in the figure. \( D_1 \) and \( D_2 \) are the detectors to detect the scattered \( \alpha \) particle at an angle \( \theta \) and along the beam direction, respectively, as shown. The signals from \( D_1 \) and \( D_2 \) are converted to logic signals and fed to logic gates. When a particle is detected, the signal is 1 and is 0 otherwise. Which one of the following circuits detects the particle scattered at the angle \( \theta \) only?


 

A thin circular ring of radius \( R \) lies in the \( xy \)-plane with its centre coinciding with the origin. The ring carries a uniform line charge density \( \lambda \). The quadrupole contribution to the electrostatic potential at the point \( (0, 0, d) \), where \( d \gg R \), is:

The figure shows a system of two equal masses \( m \) and three massless horizontal springs with spring constants \( k_1 \), \( k_2 \), and \( k_1 \). Ignore gravity. The masses can move only in the horizontal direction, and there is no dissipation. If \( m = 1 \), \( k_1 = 2 \), and \( k_2 = 3 \) (all in appropriate units), the frequencies of the normal modes of the system in the same system of units are:


 

The wire loop shown in the figure carries a steady current \( I \). Each straight section of the loop has length \( d \). A part of the loop lies in the \( xy \)-plane and the other part is tilted at \( 30^\circ \) with respect to the \( xz \)-plane. The magnitude of the magnetic dipole moment of the loop (in appropriate units) is:


 

Consider a monatomic chain of length 30 cm. The phonon density of states is \( 1.2 \times 10^{-4} \) s. Assuming the Debye model, the velocity of sound in m/s (rounded off to one decimal place) is 

Powder X-ray diffraction pattern of a cubic solid with lattice constant \( a \) has the (111) diffraction peak at \( \theta = 30^\circ \). If the lattice expands such that the lattice constant becomes \( 1.25a \), the angle (in degrees) corresponding to the (111) peak changes to \( \sin^{-1} \left( \frac{1}{n} \right) \). The value of \( n \) (rounded off to one decimal place) is _________

The effective magnetic moment (in units of Bohr magneton) for the ground state of an isolated 4𝑓 ion with 6 unpaired electrons in the 4𝑓 shell according to Hund’s rules is (in integer) _____

A linear dielectric sphere of radius \( R \) has a uniform frozen-in polarization along the \( z \)-axis. The center of the sphere initially coincides with the origin, about which the electric dipole moment is \( \vec{p}_1 \). When the sphere is shifted to the point \( (2R, 0, 0) \), the corresponding dipole moment with respect to the origin is \( \vec{p}_2 \). The value of \( \left| \frac{\vec{p}_1}{\vec{p}_2} \right| \) (in integer) is:
One mole of an ideal monatomic gas at absolute temperature \( T \) undergoes free expansion to double its original volume, so that the entropy change is \( \Delta S_1 \). An identical amount of the same gas at absolute temperature \( 2T \) undergoes isothermal expansion to double its original volume, so that the entropy change is \( \Delta S_2 \). The value of \( \frac{\Delta S_1}{\Delta S_2} \) (in integer) is:

A schematic Pressure-Temperature diagram of water is shown in the figure. Which of the following option(s) is/are correct?


 

Which of the following option(s) is/are correct for the ground state of a hydrogen atom?
A paramagnetic material containing paramagnetic ions with total angular momentum \( J = \frac{1}{2} \) is kept at absolute temperature \( T \). The ratio of the magnetic field required for 80% of the ions to be in the lowest energy state to that required for having 60% of the ions to be in the lowest energy state at the same temperature is:
Consider the function \( f(z) = \frac{1}{z^2 (z - 2)^3} \) of a complex variable \( z \).
The residues of the function at \( z = 0 \) and \( z = 2 \), respectively, are:

A logic gate circuit is shown in the figure below. The correct combination for the input \( (P, Q) \) for which the output \( T = 1 \) is:


 

Which one of the following baryons has strangeness quantum number \( S = -1 \)?
The Joule-Thomson expansion of a gas is:
A particle of mass \( m \) is in a potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \) for \( x > 0 \), and \( V(x) = \infty \) for \( x \leq 0 \), where \( \omega \) is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is:
For a two-dimensional hexagonal lattice with lattice constant \( a \), the atomic density is:

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of \( \frac{2\pi}{n} \), is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen? 
 

A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model? 


 

Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?