Physics 9702 · Paper 4

From Φ–t Graphs to E–t Graphs

Faraday's law is a gradient statement, so every flux–time graph hides an e.m.f.–time graph inside it. The sketches are marked feature by feature.

Ramps become blocks

The e.m.f. graph is the gradient of the flux graph. Steady ramps give flat-topped blocks; constant flux gives zero e.m.f.; and the falling ramp gives a block of the opposite sign — each of those three features is its own sketch mark.

Three features, three marks (9702/42/O/N/23 Q7(b)(ii)): a steady ramp in $B$ gives a constant e.m.f. (a flat-topped block); wherever the flux is constant, the e.m.f. is zero; and a falling ramp gives a block of the opposite sign. The same checklist ran a 2024 solenoid question (9702/42/F/M/24 Q6(a)(iv)).
Reading numbers off: the e.m.f. at any instant is $NA \times \text{gradient}$ . For maximum e.m.f., pick the steepest segment — a 2023 Hall-probe-calibrated field graph gave a whole mark just for choosing the right 1.4 s stretch, leading to 0.027 V (9702/42/F/M/23 Q6(c)).

Sinusoids shift by a quarter cycle

Sinusoidal field: the e.m.f. is the (negative) gradient, so it runs a quarter-cycle out of step — zero where B peaks, maximum where B changes fastest (crossing zero). A 2024 sketch mark hung on the zeros being exactly at the peaks.

A sinusoidal field induces a sinusoidal e.m.f. shifted by a quarter period: E is zero exactly where B peaks (gradient zero) and largest where $B$ crosses zero (gradient steepest). A 2024 fourteen-marker hung a sketch mark on the zeros being at the peaks (9702/42/M/J/24 Q7(b)).
The same question's number chain shows the method: draw a tangent at the steepest point of the B–t curve (a named mark), multiply by $NA$ (the ×340 was a named mark) to get the peak rate of change 0.82 Wb s−¹ — and by Faraday that is the peak e.m.f., 0.82 V. Written as $V = A\sin(Bt)$ , the constants were $A = 0.82$ V and $B = 2\pi/T = 3100\ \text{rad s}^{-1}$ .

E–t is the gradient of Φ–t: ramps → blocks, plateaus → zero, rise vs fall → opposite signs, sinusoid → sinusoid shifted a quarter cycle.

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From Φ–t Graphs to E–t Graphs

Faraday's law is a gradient statement, so every flux–time graph hides an e.m.f.–time graph inside it. The sketches are marked feature by feature.

Ramps become blocks

Three features, three marks (9702/42/O/N/23 Q7(b)(ii)): a steady ramp in $B$ gives a constant e.m.f. (a flat-topped block); wherever the flux is constant, the e.m.f. is zero; and a falling ramp gives a block of the opposite sign. The same checklist ran a 2024 solenoid question (9702/42/F/M/24 Q6(a)(iv)).
Reading numbers off: the e.m.f. at any instant is $NA \times \text{gradient}$ . For maximum e.m.f., pick the steepest segment — a 2023 Hall-probe-calibrated field graph gave a whole mark just for choosing the right 1.4 s stretch, leading to 0.027 V (9702/42/F/M/23 Q6(c)).

Sinusoids shift by a quarter cycle

A sinusoidal field induces a sinusoidal e.m.f. shifted by a quarter period: E is zero exactly where B peaks (gradient zero) and largest where $B$ crosses zero (gradient steepest). A 2024 fourteen-marker hung a sketch mark on the zeros being at the peaks (9702/42/M/J/24 Q7(b)).
The same question's number chain shows the method: draw a tangent at the steepest point of the B–t curve (a named mark), multiply by $NA$ (the ×340 was a named mark) to get the peak rate of change 0.82 Wb s−¹ — and by Faraday that is the peak e.m.f., 0.82 V. Written as $V = A\sin(Bt)$ , the constants were $A = 0.82$ V and $B = 2\pi/T = 3100\ \text{rad s}^{-1}$ .

E–t is the gradient of Φ–t: ramps → blocks, plateaus → zero, rise vs fall → opposite signs, sinusoid → sinusoid shifted a quarter cycle.