List of top Questions asked in GATE Physics- 2024

Consider the Lagrangian \( L = m\dot{x}\dot{y} - mw_0^2 zxy \). If \( p_x \) and \( p_y \) denote the generalized momenta conjugate to \( x \) and \( y \), respectively, then the canonical equations of motion are _______.

If \( F_1(Q, q) = Qq \) is the generating function of a canonical transformation from \((p, q)\) to \((P, Q)\), then which one of the following relations is correct?

An extrinsic semiconductor shown in the figure carries a current of 2 mA along its length parallel to the \( +x \) axis.

When the majority charge carrier concentration is \( 12.5 \times 10^{13} \, \text{cm}^{-3} \) and the sample is exposed to a constant magnetic field applied along the \( +z \) direction, a Hall voltage of 20 mV is measured with the negative polarity at the \( y = 0 \) plane. Take the electric charge as \( 1.6 \times 10^{-19} \) C. The concentration of minority charge carriers is negligible. Which of the following statements is/are true?

A particle of mass m is moving in the potential
\[ V(x) = \begin{cases} V_0 + \frac{1}{2} m \omega_{0P}^2 x^2 & \quad \text{if } x > 0 \\\infty & \quad \text{if } x \leq 0 \end{cases}\]
Figures P, Q, R, and S show different combinations of the values of \( \omega_0 \) and \( V_0 \).

Let \( E^{(P)}_j \), \( E^{(Q)}_j \), \( E^{(R)}_j \), and \( E^{(S)}_j \) with \( j = 0, 1, 2, \ldots \), be the eigen-energies of the j-th level for the potentials shown in Figures P, Q, R, and S, respectively. Which of the following statements is/are true?

Binding energy and rest mass energy of a two-nucleon bound state are denoted by \( B \) and \( mc^2 \), respectively, where \( c \) is the speed of light. The minimum energy of a photon required to dissociate the bound state is _______.

Consider a three-dimensional system of non-interacting bosons with zero chemical potential. The energy of the system \( \epsilon \propto k^2 \), where \( k \) is the wave vector. The low temperature specific heat of the system at constant volume depends on the temperature as \( C_v = T^{n/2} \). The value of \( n \) is ___ (in integer).

According to the nuclear shell model, the absolute value of the difference in magnetic moments of \({^{15}_{8}O}\) and \({^{15}_{7}N}\), in the units of nuclear magneton (\(\mu\)), is \(\frac{a}{3}\). The magnitude of \(a\) is (in integer).

A particle of mass \( m \) in an infinite potential well of width \( a \) is subjected to a perturbation \( V' = \frac{h^2}{40ma^2} \) as shown in the figure, where \( h \) is Planck's constant.

The first-order energy shift of the fourth energy eigenstate due to this perturbation is \( \frac{h^2}{Nma^2} \). The value of \( N \) is ____ (in integer).

The Lagrangian of a particle of mass \( m \) is given by \( L = \frac{1}{2}m\dot{x}^2 - \lambda x^4 \), where \( \lambda \) is a positive constant. If the particle oscillates with total energy \( E \), then the time period of oscillations \( T \) is given by:
\[ T = a \int_0^{\left(\frac{E}{\lambda}\right)^{\frac{1}{4}}} \frac{dx}{\sqrt{\frac{2}{m}(E - \lambda x^4)}} \]
The value of \( a \) is ___ (in integer).

Consider the operational amplifier circuit shown in figure.

The output voltage V。 is ____ V (in integer).

An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states \( \psi_{n i m} \):\[ \psi = \frac{1}{3} \psi_{100} + \frac{1}{\sqrt{3}} \psi_{210} + \frac{\sqrt{5}}{3} \psi_{320} \]Let \( E_1, E_2, \) and \( E_3 \) represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time \( t_1 \) and the outcome is \( E_2 \). Then total angular momentum is measured at time \( t_2 > t_1 \), and finally, energy is measured again at \( t_3 > t_2 \). The probability of finding the system in a state with energy \( E_2 \) after the final measurement is \( \frac{P}{9} \). The value of \( P \) is ______ (in integer).

The Fourier transform and its inverse transform are respectively defined as:\[ \tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{i \omega x} dx \]and\[ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega) e^{-i \omega x} d\omega \]Consider two functions \( f \) and \( g \). Another function \( f * g \) is defined as:\[ (f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y) g(x - y) dy \]Which of the following relation is/are true?Note: Tilde (~) denotes the Fourier transform.

The typical biasing of a silicon transistor is shown in the figure.

The value of the common-emitter current gain \( \beta \) for the transistor is 100. Ignore reverse saturation current. The output voltage \( V_o \) (in V) is ____ (in integer).

The canonical partition function of an ideal gas is given by \( Q(T,V,N) = \frac{1}{N!} \left( \frac{V}{\lambda(T)^3} \right)^N \), where \( T \), \( V \), \( N \), and \( \lambda(T) \) denote temperature, volume, number of particles, and thermal de Broglie wavelength, respectively. Let \( k_B \) be the Boltzmann constant and \( \mu \) be the chemical potential. Using \( \ln(N!) = N \ln(N) - N \), if the number density \( \left( \frac{N}{V} \right) \) is \( 2.5 \times 10^{25} \) m\(^{-3} \) at temperature \( T \), then \( e^{\mu/(k_B T)} / (\lambda(T))^3 \times 10^{-25} \) is ___ m\(^{-3} \) (rounded off to one decimal place).

The temperature \( T \) dependence of magnetic susceptibility \( \chi \) (Column I) of certain magnetic materials (Column II) are given below. Which of the following options is/are correct?

Column I	Column II
(1)	(P) Diamagnetic
(2)	(Q) Paramagnetic
(3)	(R) Ferromagnetic
(4)	(S) Antiferromagnetic

The curves \( P \) and \( Q \) schematically show the variation of X-ray intensity with wavelength at two different accelerating voltages for a given target material. In the figure, \( \lambda_1 = 0.25 \) Å, \( \lambda_2 = 0.5 \) Å, \( \lambda_3 = 1.0 \) Å, and \( \lambda_4 = 2.25 \) Å. Take Planck's constant as \( 6.6 \times 10^{-34} \) Js, the speed of light as \( 3 \times 10^8 \) m/s, and the elementary charge as \( 1.6 \times 10^{-19} \) C.

Which of the following statement is/are true?

An oscillating electric dipole of moment \(\overrightarrow{d}(t) = d_0 \cos(\omega t) \hat{z}\) is placed at the origin as shown in the figure. Consider a point \(P(r, \theta, \phi)\) at a very large distance from the dipole. Here, \(r\), \(\theta\), and \(\phi\) are spherical polar coordinates. Which of the following statements is/are true for the intensity of radiation?

Apart from the acoustic modes, 9 optical modes are identified from the measurements of phonon dispersions of a solid with the chemical formula \( A_nB_m \), where \( A \) and \( B \) denote the atomic species, and \( n \) and \( m \) are integers. Which of the following combinations of \( n \) and \( m \) is/are possible?

A material behaves as a superconductor below a critical temperature \( T_c \) and as a normal conductor above \( T_c \). A magnetic field \(\overrightarrow{B} = B\hat{z}\) is applied when \( T > T_c \). The material is then cooled below \( T_c \) in the presence of \( \vec{B} \). Which of the following figures represents the correct configuration of magnetic field lines?

Consider a vector field \( \vec{F} = (2xz + 3y^2)\hat{y} + 4yz^2\hat{z} \). The closed path (\( \Gamma \): \( A \rightarrow B \rightarrow C \rightarrow D \rightarrow A \)) in the \( z = 0 \) plane is shown in the figure.

\( \oint_\Gamma \vec{F} \cdot d\vec{l} \) denotes the line integral of \( \vec{F} \) along the closed path \( \Gamma \). Which of the following options is/are true?

Given that \( A^\alpha \) and \( B_\beta \) (\(\alpha, \beta = 1, 2, 3, \ldots, n\)) are contravariant and covariant vectors, respectively, and by convention, any repeated indices are summed over. Which of the following expressions is/are tensors?

Two point charges of charge \( +q \) each are placed a distance \( 2d \) apart. A grounded solid conducting sphere of radius \( a \) is placed midway between them. Assume \( a^2 \ll d^2 \). Which of the following statements is/are true?

The non-relativistic Hamiltonian for a single electron atom is
\[ H_0 = \frac{p^2}{2m} - V(r) \]
where \( V(r) \) is the Coulomb potential and \( m \) is the mass of the electron.Considering the spin-orbit interaction term:
\[ H' = \frac{1}{2m^2c^2} \frac{1}{r} \frac{dV}{dr} \overrightarrow{L} \cdot \overrightarrow{S} \]
added to \( H_0 \), which of the following statements is/are true?

Decays of mesons and baryons can be categorized as weak, strong and electromagnetic decays depending upon the interactions involved in the processes. Which of the following option is/are true?

In a parallel plate capacitor, the plate at \( x = 0 \) is grounded and the plate at \( x = d \) is maintained at a potential \( V_0 \). The space between the two plates is filled with a linear dielectric of permittivity \( \epsilon = \epsilon_0 \left(1 + \frac{x}{d}\right) \), where \( \epsilon_0 \) is the permittivity of free space. Neglecting the edge effects, the electric field (\(\vec{E}\)) inside the capacitor is .