(a) A force F⃗ is given by F⃗ = x²y x̂ + zy² ŷ + xz² ẑ. Determine whether or not the force is conservative. 10 marks (b)…

(a) Is the force conservative?

A force defined on all space is conservative if and only if ∇×F = 0 everywhere; then the work is path-independent and F = −∇V. F = (x²y, zy², xz²), so Fx = x²y, Fy = zy², Fz = xz². (∇×F)x = ∂Fz/∂y − ∂Fy/∂z = 0 − y² = −y² (∇×F)y = ∂Fx/∂z − ∂Fz/∂x = 0 − z² = −z² (∇×F)z = ∂Fy/∂x − ∂Fx/∂y = 0 − x² = −x² Hence ∇×F = −(y² x̂ + z² ŷ + x² ẑ), which is non-zero except at the origin. The force is not conservative: the work done depends on the path, and no scalar potential exists.

(b) Gravitational self-energy of the Earth

Treat the Earth as a uniform sphere of mass M and radius R, built up shell by shell. When a sphere of radius r has been assembled, its mass is m = (4/3)πr³ρ, and a shell of thickness dr has mass dm = 4πr²ρ dr. Bringing it from infinity needs work dU = −G m dm / r = −(16/3)π²Gρ² r⁴ dr. Integrating from 0 to R: U = −(16/15)π²Gρ²R⁵. With ρ = 3M/(4πR³), this is U = −3GM²/(5R). Numbers: GM² = 6.67×10⁻¹¹ × (6×10²⁴)² = 6.67×10⁻¹¹ × 3.6×10⁴⁹ = 2.40×10³⁹ J·m U = −3 × 2.40×10³⁹ / (5 × 6.4×10⁶) = −2.25×10³² J. The Earth's gravitational self-energy is about −2.25×10³² J (the energy needed to disperse it to infinity); this is the uniform-density approximation, and the real, centrally dense Earth is more tightly bound (about −2.5×10³² J).

(c) Consequences of the Lorentz transformation for length and time

For frame S′ moving with speed v along x, x′ = γ(x − vt), t′ = γ(t − vx/c²), with γ = 1/√(1 − v²/c²) ≥ 1. Time dilation: a clock at rest in one frame shows proper time Δt₀; an observer in the other frame measures Δt = γΔt₀, so moving clocks run slow. Evidence: cosmic-ray muons (proper lifetime 2.2 μs) reach the ground because their lifetime in the Earth frame is γ times longer. Length contraction: a rod of proper length L₀ along the motion is measured as L = L₀/γ; dimensions perpendicular to the motion are unchanged. Relativity of simultaneity: events simultaneous in S (Δt = 0) separated by Δx have Δt′ = −γvΔx/c² in S′, so simultaneity is relative. The effects are reciprocal and become appreciable only when v is comparable to c; for v ≪ c, γ → 1 and the Galilean results are recovered. Because simultaneity is relative, length contraction and time dilation do not contradict each other.

(d) Snell's law from Huygens' principle

Huygens' principle: every point of a wavefront is a source of secondary wavelets, and the new wavefront is their common tangent (envelope). Let a plane wavefront AB in medium 1 (speed v₁) meet the plane interface at A, with the incident ray making angle i with the normal; B reaches the interface at C. In time t, B travels BC = v₁t. In the same time the wavelet from A spreads in medium 2 (speed v₂) to a sphere of radius AD = v₂t. The tangent from C to this sphere is CD, the refracted wavefront. The angle between the incident wavefront and the interface equals i, and that between the refracted wavefront and the interface equals r. In right triangles ABC and ADC, sharing the hypotenuse AC: sin i = BC/AC = v₁t/AC, sin r = AD/AC = v₂t/AC. Dividing: sin i / sin r = v₁/v₂. Since μ = c/v, v₁/v₂ = μ₂/μ₁. Hence sin i / sin r = v₁/v₂ = μ₂/μ₁. For a denser medium 2, v₂ < v₁, so r < i: the ray bends towards the normal.

(e) Three- and four-level pumping schemes

Lasing needs a population inversion between two levels, and a metastable upper level so that stimulated emission can compete with spontaneous decay. Three-level scheme (ruby laser): levels E₁ (ground), E₂ (metastable) and E₃ (pump band). Pumping takes atoms from E₁ to E₃; they decay rapidly and non-radiatively to E₂, which accumulates population. Laser transition: E₂ → E₁. Because the lower level is the ground state, more than half of all atoms must be pumped to E₂ before inversion, so the threshold is high and operation is often pulsed. Four-level scheme (Nd:YAG, He-Ne): levels E₁ (ground), E₂ (lower laser level), E₃ (metastable upper level) and E₄ (pump band). Atoms go from E₁ to E₄, decay quickly to E₃, and lase to E₂, which is nearly empty because it empties fast to E₁. Inversion between E₃ and E₂ is reached with only a small pumping, giving a low threshold and continuous operation. In both, a photon from spontaneous emission stimulates further identical photons (same frequency, phase, direction); mirrors of the optical cavity feed back, and when gain exceeds losses a coherent beam emerges. The four-level scheme is more efficient.