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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2022/paper-1/section-a-iii/7/ii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 1 / Section A(iii) / 7 / ii / Solution by

Codex  0 2026-10-20

For constant current and nearly constant capacitance,

I = C \frac{d V}{d t}, V (t) = V (0) \pm \frac{∣ I ∣}{C} t .

(1)

An ideal supercapacitor therefore has approximately linear charge and discharge ramps, making a triangular voltage–time trace. Series resistance adds an immediate voltage step when current starts and a step of magnitude

2∣ I ∣ R_{s}

when it reverses directly from

+ I

- I

. Leakage and voltage-dependent capacitance introduce curvature.

A secondary battery instead follows electrode chemical potentials as its state of charge changes; many chemistries have broad plateaus or sloping plateaus, with polarization separating the charge and discharge curves. This distinction is typical rather than universal: some batteries have steeply sloping profiles, while strongly potential-dependent pseudocapacitive materials can depart substantially from a triangle.

Figure 1.
Idealized supercapacitor and battery responses, with capacitance–potential alternatives
. Original schematic curves; the lower panels distinguish a constant differential capacitance, a double-layer minimum and a localized redox contribution. They are alternative models, not a universal measured curve. **Details**.

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