List of top Questions asked in GATE PH

The Coulomb energy component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is \(\underline{\hspace{2cm}}\).
\text{(Given: } \frac{e^2}{4 \pi \epsilon_0} = 1.44 \, \text{MeV fm}).}

A function \( f(t) \) is defined only for \( t \geq 0 \). The Laplace transform of \( f(t) \) is \[ \mathcal{L}(f; s) = \int_0^\infty e^{-st} f(t) \, dt \] whereas the Fourier transform of \( f(t) \) is \[ \tilde{f}(\omega) = \int_0^\infty f(t) e^{-i\omega t} \, dt. \] The correct statement(s) is(are)

P and Q are two Hermitian matrices and there exists a matrix \( R \), which diagonalizes both of them, such that \( RPR^{-1} = S_1 \) and \( RQR^{-1} = S_2 \), where \( S_1 \) and \( S_2 \) are diagonal matrices. The correct statement(s) is(are)

Consider a system of three distinguishable particles, each having spin \( S = \frac{1}{2} \) such that \( S_z = \pm \frac{1}{2} \) with corresponding magnetic moments \( \mu_z = \pm \mu \). When the system is placed in an external magnetic field \( H \) pointing along the z-axis, the total energy of the system is \( \mu H \). Let \( x \) be the state where the first spin has \( S_z = \frac{1}{2} \). The probability of having the state \( x \) and the mean magnetic moment (in the +z direction) of the system in state \( x \) are

In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobility of the hole to electron is

Consider a spin \( S = \hbar/2 \) particle in the state \( | \phi \rangle = \frac{1}{\sqrt{3}} | 2 \rangle + i | 2 \rangle \). The probability that a measurement finds the state with \( S_x = + \hbar/2 \) is

The donor concentration in a sample of n-type silicon is increased by a factor of 100. Assuming the sample to be non-degenerate, the shift in the Fermi level (in meV) at 300 K (rounded off to the nearest integer) is \(\underline{\hspace{2cm}}\).
\text{(Given: } k_B T = 25 \, \text{meV at 300 K).}

Two observers \( O \) and \( O' \) observe two events \( P \) and \( Q \). The observers have a constant relative speed of 0.5c. In the units, where the speed of light, \( c \), is taken as unity, the observer \( O \) obtained the following coordinates:
\text{Event P: } x = 5, y = 3, z = 5, t = 3
\text{Event Q: } x = 5, y = 1, z = 3, t = 5
\text{The length of the space-time interval between these two events, as measured by \( O' \), is \( L \). The value of \( |L| \) (in integer) is \(\underline{\hspace{2cm}}\).}

A hoop of mass \( M \) and radius \( R \) rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass \( m \) slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is \(\underline{\hspace{2cm}}\). \includegraphics[width=0.5\linewidth]{image20.png}

A material is placed in a magnetic field intensity \(H\). As a result, bound current density \(J_b\) is induced and magnetization of the material is \(M\). The magnetic flux density is \(B\). Choose the correct option(s) valid at the surface of the material.

Assume that \(^{13}N (Z = 7)\) undergoes first forbidden \( \beta^+ \) decay from its ground state with spin-parity \( J_i^\pi \), to a final state \( J_f^\pi \). The possible values for \( J_i^\pi \) and \( J_f^\pi \), respectively are

In an experiment, it is seen that an electric-dipole (E1) transition can connect an initial nuclear state of spin-parity \( J_i^\pi = 2^+ \) to a final state \( J_f^\pi \). All possible values of \( J_f^\pi \) are

Choose the correct statement from the following.

Which of the following sentences are grammatically correct?
(i) Arun and Aparna are here.
(ii) Arun and Aparna is here.
(iii) Arun's families is here.
(iv) Arun's family is here.