List of top Questions asked in GATE PH

Two projectile protons \( P_1 \) and \( P_2 \), both with spin up (along the \( +z \)-direction), are scattered from another fixed target proton \( T \) with spin up at rest in the \( xy \)-plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is \( \hat{\lambda} \vec{L} \cdot \vec{S} \), where \( \vec{L} \) is the orbital angular momentum of the system with respect to the target, \( \vec{S} \) is the spin angular momentum of the system, and \( \lambda \) is a negative constant in appropriate units. Which one of the following is correct?


 

The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from \( \pm 15 \, {V} \) supplies. If a \( +1 \, {V} \) signal is applied at the input, the output voltage (rounded off to one decimal place) is:


 

A wheel of mass \( 4M \) and radius \( R \) is made of a thin uniform distribution of mass \( 3M \) at the rim and a point mass \( M \) at the center. The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length \( 2R \), with one end fixed at \( O \), as shown in the figure. The wheel rolls without slipping on horizontal ground with angular speed \( \Omega \). If \( \vec{L} \) is the total angular momentum of the wheel about \( O \), then the magnitude \( \left| \frac{d\vec{L}}{dt} \right| = N(MR^2 \Omega^2) \). The value of \( N \) (in integer) is:


 

The Hamiltonian for a one-dimensional system with mass \( m \), position \( q \), and momentum \( p \) is: \[ H(p, q) = \frac{p^2}{2m} + q^2 A(q) \] where \( A(q) \) is a real function of \( q \). If \[ m \frac{d^2 q}{dt^2} = -5q A(q), \] then \[ \frac{d A(q)}{d q} = n \frac{A(q)}{q}. \] The value of \( n \) (in integer) is:
A system of three non-identical spin-\(\frac{1}{2}\) particles has the Hamiltonian
\[ H = \frac{A \hbar^2}{2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3, \] where \( \vec{S}_1, \vec{S}_2, \vec{S}_3 \) are the spin operators of particles labelled 1, 2, and 3 respectively, and \( A \) is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is:
Potential energy of two diatomic molecules P and Q of the same reduced mass is shown in the figure. According to this diagram, which of the following option(s) is/are correct?
Nuclear radiation emitted from a \( {Co}^{60} \) radioactive source is detected by a photomultiplier tube (PMT) coupled to a scintillator crystal. Which of the following option(s) is/are correct?
Which of the following consideration(s) is/are showing that nuclear beta decay, \( n \to p + e^{-} + \bar{\nu}_e \), has to be a three-body decay?
Which of the following option(s) is/are correct for photons?
Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is \( E_F \). The energy corresponding to the filling of \( \frac{N_A}{2} \) electrons, where \( N_A \) is the Avogadro number, is \( 2^n E_F \). The value of \( n \) is:
The nuclear energy levels of mirror nuclei are similar. Using this empirical fact alone, the nuclear force can be said to be independent of which one of the following properties of the nucleons?
As per the Drude model of metals, the electrical resistance of a metallic wire of length \( L \) and cross-sectional area \( A \) is:
(Consider \( \tau \) as the relaxation time, \( m \) as electron mass, \( n \) as carrier concentration, and \( e \) as electronic charge)
Which one of the following is correct for the phase velocity \( v_p \) and group velocity \( v_g \)? (c is the speed of light in vacuum)

“Why do they pull down and do away with crooked streets, I wonder, which are my delight, and hurt no man living? Every day the wealthier nations are pulling down one or another in their capitals and their great towns: they do not know why they do it; neither do I. It ought to be enough, surely, to drive the great broad ways which commerce needs and which are the life-channels of a modern city, without destroying all history and all the humanity in between: the islands of the past.”
(From Hilaire Belloc’s “The Crooked Streets”)
Based only on the information provided in the above passage, which one of the following statements is true?

As the police officer was found guilty of embezzlement, he was _________ dismissed from the service in accordance with the Service Rules. Select the most appropriate option to complete the above sentence.

An electron with mass \( m \) and charge \( q \) is in the spin up state \[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] at time \( t = 0 \). A constant magnetic field is applied along the \( y \)-axis, \( \mathbf{B} = B_0 \hat{j} \), where \( B_0 \) is a constant. The Hamiltonian of the system is \[ H = -\hbar \omega \sigma_y, \quad \text{where} \quad \omega = \frac{q B_0}{2m} < 0 \quad \text{and} \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}. \]
The minimum time after which the electron will be in the spin down state along the \( x \)-axis, i.e., \[ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \] is:
A system of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \) is confined to a one-dimensional potential well of length \( L \). If the lowest energy of the system is \[ E_{{min}} = \frac{N \pi^2 \hbar^2}{2m L^2}, \] the value of \( N \) (in integer) is:

In the transistor circuit shown in the figure, \( V_{BE} = 0.7 \, {V} \) and \( \beta_{DC} = 400 \). The value of the base current in \( \mu A \) (rounded off to one decimal place) is:


 

Cyclotrons are used to accelerate ions like deuterons (\( d \)) and \( \alpha \) particles. Keeping the magnetic field the same for both, \( d \) and \( \alpha \) are extracted with energies 10 MeV and 20 MeV with extraction radii \( r_d \) and \( r_\alpha \), respectively. Taking the masses \( M_d = 2000 \, {MeV}/c^2 \) and \( M_\alpha = 4000 \, {MeV}/c^2 \), the value of \( \frac{r_\alpha}{r_d} \) (in integer) is:
A linear magnetic material in the form of a cylinder of radius \( R \) and length \( L \) is placed with its axis parallel to the \( z \)-axis. The cylinder has uniform magnetization \( M \hat{k} \). Which of the following option(s) is/are correct?
A point charge \( q \) is placed at the origin, inside a linear dielectric medium of infinite extent, having relative permittivity \( \epsilon_r \). Which of the following option(s) is/are correct?
A neutral conducting sphere of radius \( R \) is placed in a uniform electric field of magnitude \( E_0 \), that points along the \( z \)-axis. The electrostatic potential at any point \( r \) outside the sphere is given by: \[ V(r, \theta) = V_0 - E_0 r \left(1 - \frac{R^3}{r^3} \right) \cos \theta \] where \( V_0 \) is the constant potential of the sphere. Which of the following option(s) is/are correct?
Consider the integral \[ I = \frac{1}{2 \pi i} \oint \frac{1}{(z^4 - 1)(z - \frac{a}{b})(z - \frac{b}{a})} \, dz, \] where \( z \) is a complex variable and \( a, b \) are positive real numbers. The integral is taken over a unit circle with center at the origin. Which of the following option(s) is/are correct?
The energy of a free, relativistic particle of rest mass \( m \) moving along the \( x \)-axis in one dimension, is denoted by \( T \). When moving in a given potential \( V(x) \), its Hamiltonian is \( H = T + V(x) \). In the presence of this potential, its speed is \( v \), conjugate momentum \( p \), and the Lagrangian \( L \). Then, which of the following option(s) is/are correct?

The Lagrangian of a particle of mass \( m \) and charge \( q \) moving in a uniform magnetic field of magnitude \( 2B \) that points in the \( z \)-direction, is given by: \[ L = \frac{m}{2} v^2 + qB(x v_y - y v_x) \] where \( v_x, v_y, v_z \) are the components of its velocity \( v \). If \( p_x, p_y, p_z \) denote the conjugate momenta in the \( x, y, z \)-directions and \( H \) is the Hamiltonian, which of the following option(s) is/are correct?