List of top Questions asked in GATE PH- 2021

In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q. The probability that any point picked randomly within the square falls in the shaded area is 

The spin-orbit effect splits the \( ^2p \rightarrow ^2s \) transition (wavelength, \( \lambda = 6521 \, \text{Å} \)) in Lithium into two lines with separation of \( \Delta \lambda = 0.14 \, \text{Å} \). The corresponding positive value of energy difference between the above two lines, in eV, is \( m \times 10^{-5} \). The value of \( m \) (rounded off to the nearest integer) is \(\underline{\hspace{2cm}}\). (Given: Planck's constant, \( h = 4.125 \times 10^{-15} \, \text{eV s} \), speed of light, \( c = 3 \times 10^8 \, \text{m/s} \))

Consider a cross-section of an electromagnet having an air-gap of 5 cm as shown in the figure. It consists of a magnetic material \( \mu = 20000 \mu_0 \) and is driven by a coil having \( NI = 10^4 \), where \( N \) is the number of turns and \( I \) is the current in Ampere. 

Ignoring the fringe fields, the magnitude of the magnetic field \( \vec{B} \) (in Tesla, rounded off to two decimal places) in the air-gap between the magnetic poles is \(\underline{\hspace{2cm}}\).

Consider an atomic gas with number density \( n = 10^{20} \, \text{m}^{-3} \), in the ground state at 300 K. The valence electronic configuration of atoms is \( f^7 \). The paramagnetic susceptibility of the gas \( \chi = m \times 10^{-11} \). The value of \( m \) (rounded off to two decimal places) is \(\underline{\hspace{2cm}}\). 
(Given: Magnetic permeability of free space \( \mu_0 = 4\pi \times 10^{-7} \, \text{H m}^{-1} \), Bohr magneton \( \mu_B = 9.274 \times 10^{-24} \, \text{A m}^2 \), Boltzmann constant \( k_B = 1.3807 \times 10^{-23} \, \text{K}^{-1} \))

A contour integral is defined as \[ I_n = \oint_C \frac{dz}{(z - n)^2 + \pi^2} \] where \( n \) is a positive integer and \( C \) is the closed contour, as shown in the figure, consisting of the line from \( -100 \) to \( 100 \) and the semicircle traversed in the counter-clockwise sense. The value of \( \sum_{n=1}^5 I_n \) (in integer) is \(\underline{\hspace{2cm}}\). 

A uniform block of mass \(M\) slides on a smooth horizontal bar. Another mass \(m\) is connected to it by an inextensible string of length \(l\) of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as \(x\) and \(\theta\), as shown in the figure. 

If \(p_x\) and \(p_\theta\) are the generalized momenta corresponding to \(x\) and \(\theta\), respectively, then the correct option(s) is(are)

As shown in the figure, inverse magnetic susceptibility \( \frac{1}{\chi} \) is plotted as a function of temperature (T) for three different materials in paramagnetic states. 

(Curie temperature of ferromagnetic material = \( T_C \), Néel temperature of antiferromagnetic material = \( T_N \))
Choose the correct statement from the following 
 

Consider the potential \( U(r) \) defined as 
\[ U(r) = -U_0 \frac{e^{-\alpha r}}{r} \] where \( \alpha \) and \( U_0 \) are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with \( U(r) \) for a (wave-vector) momentum transfer \( q \) and \( \alpha \to 0 \), is proportional to 
(Useful integral: \( \int_0^\infty \sin(qr) e^{-\alpha r} \, dr = \frac{q}{\alpha^2 + q^2}  \))

Consider a spherical galaxy of total mass \( M \) and radius \( R \), having a uniform matter distribution. In this idealized situation, the orbital speed \( v \) of a star of mass \( m \ll M \) as a function of the distance \( r \) from the galactic centre is best described by \[ (G \text{ is the universal gravitational constant}) \] 

The free energy of a ferromagnet is given by \[ F = F_0 + a_0 (T - T_C) M^2 + b M^4, \] where \(F_0\), \(a_0\), and \(b\) are positive constants, \(M\) is the magnetization, \(T\) is the temperature, and \(T_C\) is the Curie temperature. The relation between \(M^2\) and \(T\) is best depicted by 

A system of two atoms can be in three quantum states having energies 0, $\epsilon$ and $2\epsilon$. The system is in equilibrium at temperature \( T = (k_B\beta)^{-1} \). Match the following Statistics with the Partition function.

Consider a single one-dimensional harmonic oscillator of angular frequency \( \omega \), in equilibrium at temperature \( T = \left( k_B \beta \right)^{-1 }\). The states of the harmonic oscillator are all non-degenerate having energy \( E_n = \left( n + \frac{1}{2} \right) \hbar \omega \) with equal probability, where \( n \) is the quantum number. The Helmholtz free energy of the oscillator is
 

Consider a state described by \[ \psi(x, t) = \psi_2(x, t) + \psi_4(x, t), \] where} \( \psi_2(x, t) \) and \( \psi_4(x, t) \) are respectively the second and fourth normalized harmonic oscillator wave functions and \( \omega \) is the angular frequency of the harmonic oscillator. The wave function \( \psi(x, t = 0) \) will be orthogonal to \( \psi(x, t) \) at time \( t \) equal to
 

Consider a particle in a one-dimensional infinite potential well with its walls at \( x = 0 \) and \( x = L \). The system is perturbed as shown in the figure. The first order correction to the energy eigenvalue is 

For the given sets of energy levels of nuclei X and Y whose mass numbers are odd and even, respectively, choose the best suited interpretation. 

Consider two concentric conducting spherical shells as shown in the figure. The inner shell has a radius \(a\) and carries a charge \(+Q\). The outer shell has a radius \(b\) and carries a charge \(-Q\). The empty space between them is half-filled by a hemispherical shell of a dielectric having permittivity \(\epsilon_1\). The remaining space between the shells is filled with air having the permittivity \(\epsilon_0\). 

The electric field at a radial distance \(r\) from the center and between the shells \((a < r < b)\) is

Consider a point charge +Q of mass m suspended by a massless, inextensible string of length l in free space (permittivity \( \epsilon_0 \)) as shown in the figure. It is placed at a height \( d \) (\( d > l \)) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane. 

An electromagnetic wave having electric field \( E = 8 \cos(kz - \omega t) \hat{y} \, \text{V cm}^{-1} \) is incident at \(90^\circ\) (normal incidence) on a square slab from vacuum (with refractive index \( n_0 = 1.0 \)) as shown in the figure. The slab is composed of two different materials with refractive indices \( n_1 \) and \( n_2 \). Assume that the permeability of each medium is the same. After passing through the slab for the first time, the electric field amplitude, in V cm\(^{-1}\), of the electromagnetic wave, which emerges from the slab in region 2, is closest to

The above combination of logic gates represents the operation 

Consider the atomic system as shown in the figure, where the Einstein A coefficients for spontaneous emission for the levels are \( A_{2 \to 1} = 2 \times 10^7 \, \text{s}^{-1} \) and \( A_{1 \to 0} = 10^8 \, \text{s}^{-1} \). If \( 10^{14} \, \text{atoms/cm}^3 \) are excited from level 0 to level 2 and a steady state population in level 2 is achieved, then the steady state population at level 1 will be \( x \times 10^{13} \, \text{cm}^{-3} \). The value of \( x \) (in integer) is \(\underline{\hspace{2cm}}\).