(a) What do you understand by the critical size of a reactor ? Explain the main features of nuclear reactors. 5+15=20 (b) What…

(a) Critical size and main features of nuclear reactors

Critical size is the smallest dimension of the reactor core (for a given fuel, moderator and geometry) at which the chain reaction is just self-sustaining, that is, k_eff = 1. The multiplication factor is k_eff = k∞P_NL, where k∞ = ηfpε is the infinite-medium value and P_NL is the non-leakage probability. Leakage from the surface falls as the core grows (surface/volume ∝ 1/size), so below the critical size k_eff < 1 and the chain dies out; above it k_eff > 1 and power rises. The fuel mass in this volume is the critical mass, which is a different quantity from the size. Main features: Fuel: natural uranium (PHWR), enriched U-235 (PWR, BWR) or Pu-239 (fast reactors), whose fission releases heat and 2-3 fast neutrons. Moderator: heavy water, graphite or light water slows fission neutrons from about 2 MeV to thermal energies (0.025 eV), where the fission cross section is large; natural uranium needs heavy water or graphite because they absorb few neutrons. Control rods: cadmium, boron or hafnium absorb neutrons to hold k_eff at 1, or to shut the reactor down; the delayed neutrons (about 0.65% for U-235) make this control practical. Coolant: water, heavy water, CO₂ or liquid sodium carries heat to a steam generator. Reflector: returns leaking neutrons and so lowers the critical size. Shield and containment: concrete and steel confine gamma rays, neutrons and fission products. Reactor types differ by the combination chosen, for example PHWR (natural uranium, heavy water) and PWR (enriched uranium, light water).

(b) Superconductivity, the Meissner effect and diamagnetism

Superconductivity: below a critical temperature T_c, the d.c. resistance of certain materials falls to zero, so a current persists without a voltage. The state is destroyed above a critical field H_c. Meissner effect (1933): when a superconductor is cooled below T_c in a magnetic field, the flux is expelled from its interior, B = 0, except in a surface layer of thickness λ_L ≈ 10-100 nm (from ∇²B = B/λ_L² and λ_L = √(m/μ₀n_se²)). This happens whether the field is applied before or after cooling. A merely perfect conductor would instead trap the flux present when it became perfect, so flux expulsion is a separate property of the superconducting state, not a result of zero resistance. Why diamagnetic: B = μ₀(H + M) = 0 inside requires M = −H, so the susceptibility is χ = −1, perfect diamagnetism. Surface screening currents, which flow within λ_L as the field is applied, produce the magnetisation that cancels the applied field. Expelling the flux costs energy μ₀H²/2 per unit volume, which is balanced by the condensation energy μ₀H_c²/2 of the superconducting state, so the state survives until H = H_c. Type-I superconductors expel flux completely up to H_c; type-II expel it up to H_c1 and are in a mixed state with quantised vortices up to H_c2.

(c)(i) Input and output impedances of the non-inverting amplifier

The circuit is a non-inverting amplifier (voltage-series negative feedback) with R_f = 220 kΩ and R_i = 10 kΩ. Feedback fraction: β = R_i/(R_i + R_f) = 10/230 = 0.0435. Loop gain: A_OL β = 200,000 × 0.04348 = 8696, so 1 + A_OLβ = 8697. Input impedance: Z_in(CL) = Z_in(1 + A_OLβ) = 2 MΩ × 8697 ≈ 1.7×10¹⁰ Ω (17 GΩ), because feedback reduces the differential voltage that drives the input current. Output impedance: Z_out(CL) = Z_out/(1 + A_OLβ) = 75/8697 ≈ 8.6×10⁻³ Ω, because feedback holds the output voltage fixed against load changes.

(c)(ii) Closed-loop voltage gain

A_CL = A_OL/(1 + A_OLβ) = 200,000/8697 = 23.0 (dimensionless), more exactly 22.997. Ideal check: 1 + R_f/R_i = 1 + 220/10 = 23 = 1/β, so the gain is set by the resistors and not by A_OL, as long as A_OLβ ≫ 1.