HSC Exam Practice

Sample response. The peak emf (ε₀) is the maximum value of the induced alternating voltage produced by the generator; it occurs when the rate of change of flux is greatest (coil parallel to the field). The equation is ε₀ = NBAω, where: N = number of turns (dimensionless); B = magnetic flux density (tesla, T); A = coil area (metres squared, m²); ω = angular frequency of rotation (radians per second, rad s⁻¹).

Marking notes. 1 mark for correct definition of peak emf linked to maximum rate of change of flux. 1 mark for correct equation ε₀ = NBAω. 1 mark for all four variables named with correct SI units (award only if at least three are correct).

Sample response. The flux through the coil is given by Φ = NBA cos(ωt) and is maximum when the coil is perpendicular to the magnetic field (ωt = 0). At this orientation, the flux is at a turning point, so its rate of change (dΦ/dt) is zero; by Faraday’s Law (ε = −dΦ/dt), the emf is also zero. The emf is maximum when the coil is parallel to the field (ωt = 90°), where flux is zero but decreasing most rapidly. Since one is a cosine function and the other is a sine function (its derivative), they are exactly 90° (one quarter-cycle) out of phase.

Marking notes. 1 mark for stating that flux is maximum when coil is perpendicular to B and that emf is zero at this instant. 1 mark for stating that emf is maximum when coil is parallel to B (flux = 0, rate of change is greatest). 1 mark for explicitly linking to Faraday’s Law: emf depends on rate of change of flux, not on the instantaneous value of flux itself.

Sample response. A slip ring is a continuous conducting ring; one ring connects to each end of the coil, maintaining electrical contact through brushes as the coil rotates. Slip rings do not reverse the current direction — the alternating current produced by the coil passes through unchanged to the external circuit. Slip rings are used in AC generators because the AC output must not be interfered with. A split-ring commutator is divided into two halves that swap the brush connections every half-turn; this reversal ensures that the current in the external circuit always flows in the same direction, producing a pulsating DC output. The commutator is used in DC motors and DC generators precisely to convert the naturally alternating internal emf into a rectified (DC) output.

Marking notes. 1 mark for correct description of slip rings (continuous, no current reversal). 1 mark for stating slip rings are used in AC generators. 1 mark for correct description of split-ring commutator (split, reverses current every half-turn). 1 mark for stating commutator is used in DC motors/generators and explaining the DC output produced.

Sample response (a). A = (6.0 × 10⁻²)(4.0 × 10⁻²) = 2.4 × 10⁻³ m². ω = 2πf = 2π(50) = 314 rad s⁻¹. ε₀ = NBAω = (80)(0.60)(2.4 × 10⁻³)(314) = 36.2 V. [1 mark for correct A in m²; 1 mark for correct ε₀ with units]

Sample response (b). Output frequency = 50 Hz. The output frequency equals the rotation frequency because each complete rotation of the coil produces exactly one complete cycle of the sinusoidal emf — the emf passes through a full positive and negative cycle (+ peak, zero, − peak, zero) in the time it takes for one full 360° rotation. [1 for 50 Hz stated; 1 for correct explanation linking one rotation to one output cycle]

Sample response. The claim is incorrect because replacing slip rings with a commutator does not increase the voltage. The peak emf is determined entirely by ε₀ = NBAω — the contact mechanism has no effect on the magnitude of the induced emf. What the commutator does is reverse the connection to the external circuit every half-turn, so that the current in the external circuit is always in the same direction. This rectifies the output from AC to pulsating DC; the peak value of the DC output is the same as the peak value of the original AC output. The student has confused the type of output with its magnitude.

Marking notes. 1 mark for correctly stating the claim is incorrect and that the contact mechanism does not affect peak emf (peak emf is set by N, B, A, ω). 1 mark for correctly explaining what the commutator actually does (reverses current direction every half-turn, producing DC). 1 mark for stating the peak value of the DC output is the same as the AC peak (no increase in voltage).

Part (a) — Calculated column. NBA = (50)(0.40)(8.0 × 10⁻³) = 0.160 V·s. Multiply by each ω: ω = 50 → 8.0 V; ω = 100 → 16.0 V; ω = 200 → 32.0 V; ω = 300 → 48.0 V; ω = 400 → 64.0 V. [1 mark for correct NBA constant; 1 mark for all five calculated values correct (accept ±0.1 V rounding)]

Part (b) — Relationship. Both measured and calculated peak emf increase proportionally with angular frequency — doubling ω doubles ε₀, tripling ω triples ε₀. This is a direct (linear) proportional relationship: ε₀ ∝ ω (since N, B, A are constant). A graph of ε₀ vs ω would be a straight line through the origin. [1 for identifying direct proportionality; 1 for stating it is linear through the origin or equivalent]

Part (c) — Discrepancy at high ω. One physical reason: at higher rotation speeds, the resistance of the coil wire causes a voltage drop when current flows, meaning the terminal voltage delivered to the measuring circuit is less than the induced peak emf ε₀ = NBAω. This discrepancy grows at higher ω because the induced emf (and hence the current drawn) increases with ω; a larger current produces a larger resistive voltage drop (V = IR), so the gap between the ideal theoretical emf and the measured output voltage widens as ω increases. [1 for a valid physical reason accessible from the lesson (coil resistance/back-emf/measurement loading); 1 for explaining why it worsens at higher ω]

Sample Band 6 response. The claim is valid and well-supported by both theory and design practice. Faraday’s Law states that the induced emf in a circuit equals the negative rate of change of magnetic flux through it: ε = −d(NΦ)/dt. In an AC generator, a rectangular coil of N turns and area A rotates at angular frequency ω in a uniform magnetic field B. As the coil rotates, the flux through it changes as Φ = NBA cos(ωt) (choosing t = 0 when the coil is perpendicular to B, maximising flux). Applying Faraday’s Law, ε = −d/dt[NBA cos(ωt)] = NBAω sin(ωt) = ε₀ sin(ωt), where the peak emf ε₀ = NBAω. This derivation shows that every component of the generator’s design — the number of turns, the field strength, the coil area, and the rotation speed — is directly determined by Faraday’s Law. A power station engineer wanting to increase peak emf without changing rotation speed could increase N (more turns means greater total flux linkage change per revolution, more emf per Faraday’s Law) or increase B by using stronger electromagnets (greater flux per unit area means greater rate of change of flux for a given coil geometry). The phase relationship between flux and emf is critical to understanding the generator output: the emf is zero when the coil is perpendicular to the field (ωt = 0), because at this instant flux is at its maximum turning point and dΦ/dt = 0 (Faraday’s Law gives zero emf). The emf is maximum when the coil is parallel to the field (ωt = 90°), where flux is zero but changing at the greatest rate — the coil sides are cutting field lines perpendicularly. Flux and emf are thus 90° out of phase: a cosine function for flux and a sine function for emf. The slip rings maintain this alternating output by making continuous, non-reversing electrical contact with the rotating coil. Unlike a split-ring commutator, which swaps the connections every half-turn to rectify the output to pulsating DC, slip rings allow the full AC waveform to pass to the external circuit unchanged. This is essential in power generation, where AC is required for transmission via transformers. In conclusion, the claim is correct: Faraday’s Law is not merely one factor among several — it is the fundamental physical principle from which the generator equation, the phase relationships, the design parameters, and the purpose of the contact mechanism all directly derive.

Marking criteria (8 marks).
1 — States Faraday’s Law correctly (ε = −dΦ/dt or ε = −NdΦ/dt) and applies it to show how changing flux in a rotating coil generates emf.
1 — Derives or states ε = NBAω sin(ωt) with reference to differentiating Φ = NBA cos(ωt) (or equivalent reasoning).
1 — Correctly identifies N and B (or N and A, or B and A) as two factors an engineer could increase to raise peak emf without changing rotation speed, with physical explanation for each.
1 — Correctly states emf is zero when coil is perpendicular to B and links to dΦ/dt = 0 at that instant.
1 — Correctly states emf is maximum when coil is parallel to B and links to maximum rate of change of flux / coil sides cutting field lines perpendicularly.
1 — Correctly distinguishes slip rings (continuous, pass AC unchanged) from split-ring commutator (reverses each half-turn, produces DC).
1 — Explains why AC generators use slip rings (AC output must not be rectified; needed for transmission via transformers).
1 — Reaches an explicit evaluative judgement integrating Faraday’s Law as the foundational principle from which generator design features all derive.