gate questions

Q. Consider the motion of a particle along the x-axis in a potential 𝑉(𝑥) = 𝐹|𝑥|. Its ground state energy 𝐸0 is estimated using the uncertainty principle. Then 𝐸0 is proportional to Ans: Option D Sol:

Q. A solid cylinder of radius 𝑅 has total charge 𝑄 distributed uniformly over its volume. It is rotating about its axis with angular speed 𝜔. The magnitude of the total magnetic moment of the cylinder is Ans: Option (C)

Q. The wave function 𝜓(𝑥) of a particle is as shown below Here 𝐾 is a constant, and 𝑎 > 𝑑. The position uncertainty 〈Δ𝑥〉 of the particle is Ans: Option (B)

Q. The  electric   field  of   an   electromagnetic   wave  is   given   by  𝐸⃗→ = 3 sin(𝑘𝑧 − 𝜔𝑡) 𝑥̂ + 4 cos(𝑘𝑧 − 𝜔𝑡) 𝑦̂. The wave is (C) elliptically polarized in clockwise direction when seen travelling towards the observer (D) elliptically polarized in counter-clockwise direction when seen travelling towards the observer Ans: elliptically polarized in counter-clockwise

Q. A conventional type-I superconductor has a critical temperature of 4.7 K at zero magnetic field and a critical magnetic field of 0.3 Tesla at 0 K. The critical field in Tesla at 2 K (rounded off to three decimal places) is Ans: 0.246 Sol:

Q. At temperature T Kelvin (K), the value of the Fermi function at an energy 0.5 eV above the Fermi energy is 0.01. Then T, to the nearest integer, is___________(𝑘B = 8.62 × 10−5 eV/K) Ans: 1262 Sol:

Q. The Hamiltonian for a quantum harmonic oscillator of mass 𝑚 in three dimensions is where 𝜔 is the angular frequency. The expectation value of 𝑟2 in the first excited state of the oscillator in units of (rounded off to one decimal place) is Ans: 2.5 Sol:

Q. The electric field of an electromagnetic wave in vacuum is given by The wave is reflected from the 𝑧 = 0 surface. If the pressure exerted on the surface is 𝛼𝜖0𝐸20, the value of 𝛼 (rounded off to one decimal place) is                        Ans: 0.8 Sol:

Q. An infinitely long thin cylindrical shell has its axis coinciding with the z-axis. It carries a surface charge density 𝜎0cos𝜙, where 𝜙 is the polar angle and 𝜎0 is a constant. The magnitude of the electric field inside the cylinder is Ans:

Q. A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X? (A) X is charged  (B) Spin of X must be greater than or equal to 2  (C) X is a boson  (D) X must be a baryon Ans: X is a boson