List of top Physics Questions asked in GATE PH

Two 1 kg blocks are connected to two massless springs of spring constants \(8\text{ N.m}^{-1}\) and \(4\text{ N.m}^{-1}\). The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \(\omega_H\) and \(\omega_L\) (\(\omega_H > \omega_L\)). The ratio \(\dfrac{\omega_H}{\omega_L}\) (rounded off to two decimal places) is ______

A capacitor is made of two circular metal plates of radius 1 m, separated by a distance \(d = 1\text{ mm}\). The space between the plates is filled with a dielectric of permittivity \(\epsilon_r = 5\). The capacitor is connected to a voltage \(V = 10\sin(2\pi \times 10^6 \times t)\) volts, where \(t\) is in seconds. The maximum value of the magnetic field between the plates, at a radial distance \(r = 0.5\text{ m}\) from the centre, is \(B \times 10^{-6}\text{ T}\). The value of \(B\) (rounded off to two decimal places) is ______.
(Speed of light in vacuum \(c = 3 \times 10^8\text{ m.s}^{-1}\))

Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an interval of \(20.0\text{ cm}^{-1}\). Its moment of inertia is found to be \(I_0 \times 10^{-47}\text{ kg.m}^2\), the value of \(I_0\) (rounded off to one decimal place) is ______.
(\(h = 6.6 \times 10^{-34}\text{ J.s}\), speed of light in vacuum \(c = 3 \times 10^8\text{ m.s}^{-1}\))

The Hamiltonian of two interacting spin-1/2 particles is \(H = \dfrac{A}{\hbar^2}\vec{S}_1\cdot\vec{S}_2\), where \(\vec{S}_1\) and \(\vec{S}_2\) are the spin angular momenta of particles 1 and 2, respectively. Here, \(A = 10.56\) eV. The energy in eV required to induce an excitation from the ground state to the excited state (rounded off to two decimal places) is ______

Copper has an electron number density of \(8.3 \times 10^{28}\text{ m}^{-3}\). Its Fermi energy in eV (rounded off to one decimal place) is ______.
(\(\hbar = 1.06 \times 10^{-34}\text{ J.s}\), mass of electron \(m_e = 9.10\times 10^{-31}\text{ kg}\), charge of electron \(= 1.60\times 10^{-19}\text{ C}\))

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one's index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed, it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle \(\theta\). The value of \(\theta\) in degrees (rounded off to one decimal place) is ______
(\(g = 9.8\text{ m.s}^{-2}\))

The output signal (current \(I_d\)) of a reversed biased (with a voltage \(V_b\)) photodiode, on which light is incident, is fed to an amplifier (see figure). The output voltage is digitized by a 10 bit Analogue to Digital convertor (ADC) which has a reference voltage of 5 V. The smallest current which can be measured by the circuit in nano-Amperes (rounded off to one decimal place) is ______

Consider the distribution of outcomes generated by \(N\) (\(N \gg 1\)) independent throws of (i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above, which of the following statements is/are true?

An atom has two energy levels with energy difference 2.2 eV between them. A gas of these atoms has \(8 \times 10^{20}\) atoms in the upper state and \(5 \times 10^{20}\) atoms in the lower state. Ignoring spontaneous emission, the maximum possible energy released by this gas of atoms by stimulated emission is \(E\) Joules. The value of \(E\) (rounded off to one decimal place) is ______ \((e = 1.6\times10^{-19}\text{ C})\)

A symmetric rigid body has moment of inertia \(I_1, I_2, I_3\) about its principal axes 1, 2, and 3, respectively, with \(I_1 = I_3 = I_\perp\) and \(I_2 \neq I_\perp\). It is rotating in space with no torque on it so that its angular momentum \(\vec{L}\) is constant. Let \(\omega_1, \omega_2, \omega_3\) be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?

An alpha particle moves towards a fixed nucleus carrying charge \(Ze\), with initial speed \(v_0\) and impact parameter \(b\). Starting from a large distance from the nucleus, its distance of closest approach is \(r_m\) and its speed there is \(v_m\). Then which of the following options is correct?

\( \left(k = \dfrac{1}{4\pi\epsilon_0} \text{ and } r_0 = k\dfrac{Ze^2}{mv_0^2}\right) \)

The dispersion (\(E(k)\)) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?

Consider a particle of mass \(m = 9.0 \times 10^{-5}\) kg and charge \(q = 3.0 \times 10^{-4}\) C in a uniform electromagnetic field \(\vec{E} = 2\,\hat{x}\) V.m\(^{-1}\), \(\vec{B} = 3\,\hat{z}\) V.m\(^{-2}\).s. The particle is released from the coordinates \((0, 5\text{ m}, 0)\) at time \(t = 0\). Starting from initial speed zero, it comes back to the \(y\)-axis for the first time at time \(t\). The value of \(t\) in seconds (rounded off to two decimal places) is ______

An infinitely large thin sheet in the \(xy\)-plane carries uniform positive charge density and is moving with constant velocity \(\vec{v}\) in the \(+x\) direction (see figure below). The direction of the corresponding Poynting vector is

A positive point charge is fixed at the origin. At some distance from it on the \(x\) axis, a point dipole is kept pointing in the \(+y\) direction. The force on the dipole is

Which of the following operators is/are self-adjoint?

Two frames \(S\) (solid lines) and \(S'\) (dashed lines) with common origin are shown in the figure below. Frame \(S\) is inertial while \(S'\) is rotating about the common \(z\)-axis. There is a point mass fixed at \(P\) on the \(x\)-axis of the \(S\) frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the \(S'\) frame is \(F_{cen}\) and \(F_{cor}\), respectively. Which of the following options is correct for these forces?

Consider operators \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\) for three observables of a quantum system satisfying \([\hat{A},\hat{B}] = 0\), \([\hat{B},\hat{C}] = 0\), and \([\hat{A},\hat{C}] \neq 0\), with uncertainties \(\Delta A, \Delta B, \Delta C\), respectively. From the options given below, which is/are implied by the commutation relations among \(\hat{A}\), \(\hat{B}\), and \(\hat{C}\)?

Consider two operators \(\hat{A}\) and \(\hat{B}\) which are related as \(\hat{A} = \exp(i\theta \hat{B})\). If \(\theta\) is a non-zero real number, which of the following statements is/are true?

In the circuit shown, \(V(t) = 2\sin(2000\pi t)\) Volts, where \(t\) is in seconds. The source \(V(t)\) drives the non-inverting (+) input of an opamp directly, the inverting (-) input is tied to ground (0 V), and the opamp is powered from +15 V and -15 V rails with no feedback network between its output and either input. Take the opamp to be ideal. Which of the following options is correct?

The energy \(E\) and degeneracy \(d\) of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency \(\omega\) are

Two identical particles with a fixed total energy \(E = 2\hbar\omega\) are in thermal equilibrium in a one-dimensional harmonic oscillator potential \(\frac{1}{2}m\omega^2x^2\). Let the entropy of the particles be denoted by \(S_F\) if they are fermions with spin \(\frac{1}{2}\) (\(S_z = \pm\frac{\hbar}{2}\)) and by \(S_B\) if they are bosons with spin 0. Then, which of the following options is correct? (\(k_B\) is the Boltzmann constant)

A gas of \(N\) classical particles that can occupy energy levels \(\epsilon_1\) and \(\epsilon_2 = \epsilon_1+\Delta\) is in equilibrium with a reservoir at temperature \(T\). From the schematics shown below, choose the correct dependence of the internal energy \(U\) on \(T\).

The Lagrangian \(L_0 = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2q^2\) with the generalized coordinate \(q\) is transformed to \(L = L_0 + \alpha \frac{df(q)}{dt}\). Consider the following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of motion does not change.
Which of the following options is correct for the above statements?

Considering the circuit and the associated signals measured at different pins (numbered as 1, 2, 3, 4, 5) shown in the figure, the correct option is