Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Paper: Physics
Time Allowed: 3 Hours
PAPER-I (Subjective) 80 Marks
PART-II
Attempt ONLY FOUR questions from PART-II, ALL Questions carry EQUAL marks.
Q. No. 2.
(a) A particle of unit mass moves in the potential
V(x) = ax² + b/x²
where a and b are positive constants. Find the angular frequency of small oscillations.
(b) A hollow spherical shell carries a charge density
ρ = k/r
in the region a ≤ r ≤ b.
Find the electric field in the following regions:
(i) r < a
(ii) a < r < b
(iii) r > b
(c) A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. Determine its angle of projection.
Q. No. 3.
(a) Assume that a star has uniform density. Show that the gravitational pressure P is proportional to V^(-3/4), where V is the volume.
(b) Derive the expressions for electric potential and electric field associated with a point charge located near an infinite grounded conducting plane.
(c) Determine the equations of motion of the masses attached to the string of an Atwood machine using the Lagrangian method.
Q. No. 4.
(a) Q cm³ of water flows per second through a horizontal tube of uniform bore of radius r and length L. Another tube of half the length but radius 2r is connected in parallel to the same pressure head. Find the total quantity of water flowing per second through the two tubes.
(b) A linear quadrupole consists of a charge −2q at the origin and charges +q at the two points (±a, 0, 0). Show that for distances much greater than a (r ≫ a), the potential is approximately
V = (qa² / 4πϵ₀r³) (3cos²θ − 1)
(c) Two soap bubbles of radii r₁ and r₂ coalesce to form a larger bubble of radius r. Show that
r = √(r₁² + r₂²)
Q. No. 5.
(a) Explain the wave function. Derive the wave equation and explain phase velocity and group velocity.
(b) Two semi-infinite grounded metal plates, parallel to each other and to the xz-plane, are located at y = 0 and y = a. Their left ends at x = 0 are closed by an insulated strip of width a maintained at a constant potential V₀(x). Find the potential distribution in the slot.
(c) A two-level system has energies 0 and E. The zero-energy level is non-degenerate, while the level with energy E is triply degenerate. Find the mean energy of a classical particle in this system at temperature T.
Q. No. 6.
(a) Explain the particle in a finite potential well with all possible cases and solutions. Compare it with the infinite potential well.
(b) The potential V(θ, φ) is specified on the surface of a hollow sphere of radius R. Find the potential inside the sphere.
(c) A particle is confined to the region x > 0 by a potential that increases linearly as
U(x) = U₀x
Find the mean position of the particle at temperature T.
Q. No. 7.
(a) When a gas expands adiabatically, its volume doubles while its absolute temperature decreases by a factor of 1.32. Calculate the number of degrees of freedom of the gas molecules.
(b) State and prove Ampere’s Law.
(c) Find the RMS speed of oxygen molecules at 0°C.
Q. No. 8.
(a) An ensemble of non-interacting spin-1/2 particles is in contact with a heat bath at temperature T and subjected to an external magnetic field. Each particle can occupy one of two quantum states of energies ±ε₀. If the mean energy per particle is −ε₀/2, find the free energy per particle.
(b) Derive the electromagnetic wave equation in vacuum and describe the properties of monochromatic electromagnetic waves.
(c) Discuss adiabatic demagnetization using thermodynamic (TDS) equations with mathematical details.
PAPER-II (Subjective) 80 Marks
PART-II
Attempt ONLY FOUR questions from PART-II, ALL Questions carry EQUAL marks.
(a) An electric dipole, comprising a positive charge +q and a negative charge −q, is placed on the x-axis. Each charge is at the same distance a from the origin. The total separation between the charges is 2a. Calculate the electric field E⃗ due to these charges along the y-axis at the point P, which is at a distance y from the origin. Assume y ≫ a. (ε₀ = 8.85 × 10⁻¹² C²/N·m²). (10)
(b) Write down the mathematical expression to evaluate the electric field E⃗ at a distance r from the source charge q in vector form. Sketch the graph of E as a function of r. (6)
(c) Define:
(i) Electric field
(ii) Electric dipole (4) (20)
Q. No. 3.
(a) Discuss the photoelectric effect and establish Einstein’s equation for the photoelectric effect. (10)
(b) Describe the inadequacy of the wave theory of light to explain the photoelectric effect. (6)
(c) A photon of energy 12 eV falls on a certain metal plate whose work function is 4.15 eV. Find the stopping potential. The mass and charge of the electron are 9.11 × 10⁻³¹ kg and 1.6 × 10⁻¹⁹ C, respectively, and the value of Planck’s constant is 6.64 × 10⁻³⁴ J·s. (4) (20)
Q. No. 4.
(a) Discuss intrinsic and extrinsic semiconductors. (10)
(b) Describe the properties of diamagnetic, paramagnetic and ferromagnetic materials. (6)
(c) Briefly discuss the Landé g-factor. (4) (20)
Q. No. 5.
(a) Four charged particles of charge q, 2q, 3q and 4q are at the corners of a square of side a arranged in the counter-clockwise direction. Determine:
(i) The electric field at the location of charge q.
(ii) The total electric force exerted on q. (10)
(b) A parallel plate capacitor has a plate separation of 1 mm. Calculate the surface area of each plate of the capacitor to obtain a capacitance of 1 F. Is it possible to produce such a capacitor in the laboratory? Comment.
(ε₀ = 8.85 × 10⁻¹² C²/N·m²). (5)
(c) Define:
(i) Capacitance
(ii) The unit of capacitance
(iii) Surface charge density (5) (20)
Q. No. 6.
(a) Set up the Schrödinger Wave Equation for a particle of mass m confined in a one-dimensional box which has perfectly rigid walls at x = 0 and x = L. Solve the differential equation to find the expressions for energy and the eigen wave functions of the particle. (10)
(b) Sketch the graphs for the first three eigen wave functions ψ₁, ψ₂ and ψ₃. (5)
(c) Plot the graphs for the probability densities corresponding to ψ₁, ψ₂ and ψ₃. (5) (20)
Q. No. 7.
(a) Discuss the motion of a charged particle of mass m, charge q and velocity v⃗ in a magnetic field B⃗ directed into the plane of the paper. (8)
(b) Discuss the atomic description of dielectrics. (6)
(c) Let d be the separation between the parallel plates of a capacitor of capacitance C in the absence of a dielectric material. A slab of dielectric constant K and thickness d/2 is placed between the plates. Calculate the capacitance in the presence of the dielectric material. (6) (20)
Q. No. 8.
(a) Discuss the properties of three subatomic particles and their corresponding antiparticles. (8)
(b) Explain in detail how γ-radiation can be detected. (6)
(c) How can we prove that an electron does not exist in the nucleus of an atom? (6) (20)
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