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

Critical size and reactor features. Critical size is the minimum size of a reactor core for which neutron production equals neutron loss by leakage and absorption, so the effective multiplication factor k_eff is unity. If k_eff is below one, the chain reaction dies away; if above one, power rises. For a given fuel composition, geometry and reflector, there is a corresponding critical mass and critical volume, with critical mass equal to density times critical volume. A sphere minimises surface-to-volume leakage, so it has the lowest critical mass for the same material. A reactor is a controlled fission system. In the fuel, enriched U-235 or Pu-239, fission releases heat and fast neutrons. A moderator, such as heavy water or graphite, slows neutrons to thermal energies where fission probability is higher; Indian heavy-water reactors such as CIRUS and Dhruva, and the PHWR fleet, use heavy water moderator, while PHWRs also use heavy water coolant with natural uranium fuel. Control rods of cadmium or boron absorb neutrons to keep k_eff at unity or shut down the chain reaction. A coolant carries heat to a steam generator or turbine, while biological shielding and containment protect against gamma and neutron radiation. Thus the essential features are fuel, moderator, control system, coolant, shielding and safety systems arranged to maintain a stable, controllable k=1.

Superconductivity and Meissner effect. Superconductivity is a phase below a critical temperature Tc in which DC resistance becomes zero and persistent currents can flow without voltage. The Meissner effect is the expulsion of magnetic flux from the interior when a material enters the superconducting state, so B≈0 inside except within a London penetration depth. The effect is observed whether the field is applied before or after cooling, showing that flux expulsion is a property of the superconducting state rather than a memory of the normal state. Type-I superconductors expel flux completely until a critical field Hc, above which superconductivity is destroyed; Type-II superconductors have Hc1 and Hc2, expelling flux below Hc1 and allowing quantised vortices between Hc1 and Hc2. A superconductor is diamagnetic not because zero resistance alone forbids flux, but because the Meissner state is the thermodynamically stable state: in a weak field, surface screening currents predicted by the London equations lower the magnetic free energy by expelling flux. A perfect conductor could trap pre-existing flux, but a true superconductor minimises Gibbs free energy and therefore shows perfect diamagnetism.

Op-amp feedback analysis. For the given non-inverting amplifier, the feedback network samples the output voltage and returns a fraction to the inverting input. The feedback fraction is β = R1/(R1+Rf) = 10/(10+220) = 0.04348. The loop gain is βAOL = 0.04348 × 200,000 = 8695.65, so 1+βAOL = 8696.65. Since the inverting input is at βVout and the differential input is Vd = Vin − βVout, while Vout = AOL Vd, solving gives Vout/Vin = AOL/(1+βAOL). Voltage-series feedback therefore increases input impedance: the current drawn from the source is Vd/Zin(OL), so Zin(CL)=Zin(OL)(1+βAOL)=2×10^6×8696.65≈1.74×10^10 Ω, i.e. 17.4 GΩ. It reduces output impedance: with the input source shorted, the feedback loop makes the output appear lower by the same loop factor, Zout(CL)=Zout(OL)/(1+βAOL)=75/8696.65≈8.6×10^-3 Ω. The closed-loop gain is ACL=AOL/(1+βAOL)=200,000/8696.65≈23.0, which agrees with the ideal non-inverting result 1+Rf/R1=23. The large loop gain means the finite-gain correction is small: the exact gain is 22.997, only about 0.013 per cent below the ideal value. This is why the datasheet values are transformed by feedback: the source sees a much larger resistance than the bare op-amp, and the load sees a much smaller source resistance. Hence negative feedback makes the amplifier’s gain set by the passive network while making its input very high and output very low.