Capacitance

• Charge per volt; half QV under the line; 37% after RC, never zero.
• Charge per volt. Series like parallel resistors. Half QV, under the line. 37% after RC, and never zero.

• 19.1.1 I can define — define capacitance, as applied to both isolated spherical conductors and to parallel plate capacitors
• 19.1.2 I can recall — recall and use C = Q / V
• 19.1.3 I can — derive, using C = Q / V, formulas for the combined capacitance of capacitors in series and in parallel
• 19.1.4 I can use — use the capacitance formulas for capacitors in series and in parallel
• 19.2.1 I can determine — determine the electric potential energy stored in a capacitor from the area under the potential–charge graph
• 19.2.2 I can recall — recall and use W = ½QV = ½CV²
• 19.3.1 I can — analyse graphs of the variation with time of potential difference, charge and current for a capacitor discharging through a resistor
• 19.3.2 I can recall — recall and use τ = RC for the time constant for a capacitor discharging through a resistor
• 19.3.3 I can use — use equations of the form x = x₀e−t/RC where x could represent current, charge or potential difference for a capacitor discharging through a resistor

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “Capacitors in series add, like resistors in series.” Correct Series capacitors combine like parallel resistors: 1/C = 1/C1 + 1/C2, and the result is smaller than the smallest. It is parallel capacitors that add. See 19.1.3.
• “The time constant RC is the time for the p.d. to halve.” Correct After RC the p.d. has fallen to 1/e, about 37%, of its starting value. Halving takes 0.693RC. See 19.3.2.
• “A charged capacitor holds 2Q: Q on each plate.” Correct The plates carry +Q and −Q; the net charge is zero and “the charge stored” is Q, the charge on one plate. See 19.1.1.
• “The energy stored is QV.” Correct It is ½QV, the area of the triangle under the V–Q graph, because the early charge went on at a low p.d. The supply does QV of work; the other half is dissipated in the circuit's resistance. See 19.2.1.
• “In series, the larger capacitor takes the larger p.d.” Correct Series capacitors carry the same Q, so V = Q/C: the smaller capacitor takes the larger p.d. See 19.1.4.
• “A bigger resistor lets less charge flow off the capacitor.” Correct The same charge CV0 flows whatever the resistance; a bigger R only makes it flow more slowly. The area under the I–t graph is unchanged. See 19.3.1.
• “A graph of ln V against t has gradient −RC.” Correct ln V = ln V0 − t/RC, so the gradient is −1/RC, in s−1. See 19.3.3.
• “Capacitance is the charge a capacitor can hold.” Repair Capacitance is the charge stored per unit potential difference, C = Q/V. The charge a capacitor holds depends on the p.d. it is charged to.
• “A charged capacitor stores 2Q, Q on each plate.” Repair The plates carry +Q and −Q. The charge stored is Q, and the net charge is zero.
• “For an isolated sphere, V is the p.d. across the sphere.” Repair V is the sphere's potential relative to zero at infinity; there is no second plate. Hence C = Q/V = 4πε0r.
• “Capacitors in series add.” Repair Capacitors in parallel add. In series 1/C = 1/C1 + 1/C2: the reverse of resistors.
• “1/C = 1/10 + 1/40 = 1/8, so C = 0.125 µF.” Repair That sum is 1/C. Invert it: C = 8.0 µF, which is less than the smaller capacitor, as a series combination must be.
• “Series capacitors carry equal charge only if their capacitances are equal.” Repair Every capacitor in a series chain carries the same charge Q, whatever its capacitance: each pair of joined inner plates is an isolated conductor with zero net charge, so −Q on one plate means +Q on the other (conservation of charge). Equivalently, the same current flows in every part of the loop for the same time. It is the p.d.s that differ, V = Q/C.
• “The larger series capacitor takes the larger p.d.” Repair Same Q, so V = Q/C: the smaller capacitor takes the larger p.d.
• “The energy stored is QV.” Repair ½QV, the area of the triangle under the V–Q graph. The supply does QV of work, and half is dissipated in the circuit's resistance.
• “Doubling V doubles the energy stored.” Repair W = ½CV2, so doubling V quadruples it.
• “Energy released from 8.0 V to 4.0 V is ½C(8.0 − 4.0)2.” Repair Subtract the energies, not the p.d.s: ½C(8.02 − 4.02), the trapezium under the V–Q graph.
• “The capacitor is fully discharged after one time constant.” Repair After RC, 37% remains; after 5RC under 1%. In principle it never reaches zero.
• “The time constant is the time to halve.” Repair RC is the time to fall to 1/e (37%); halving takes 0.693RC.
• “The current falls steadily because the p.d. falls steadily.” Repair The p.d. falls exponentially, fast at first and ever more slowly, and I = V/R falls the same way.
• “A bigger resistance means less charge flows off.” Repair The same charge CV0 flows, more slowly; the area under the I–t graph is unchanged.
• “A graph of ln V against t has gradient −RC.” Repair The gradient is −1/RC, in s−1. A gradient of −0.0209 s−1 means RC = 47.9 s.
• “RC is in ohm farads, not seconds.” Repair Ω F = (V A−1)(C V−1) = C A−1 = s.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Interleave with the chapters that use this one. When you reach topic 21 (alternating currents), re-answer: why does a larger capacitor, or a larger load resistance, give a smoother rectified output? (Because RC is longer, so the p.d. falls less between peaks.) When you reach topic 23 (nuclear physics), re-answer: why is the half-life 0.693/λ, and why is the time to halve a capacitor's p.d. 0.693RC? (The same exponential, the same ln 2.) Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 19 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• Topic 19 is not in Paper 1 (AS topics only). A Paper 4 structured question can ask you to define capacitance (for plates or for a sphere), to show that the series or parallel formula follows from C = Q/V, to explain why series capacitors carry equal charge or why the stored energy is ½QV and not QV, and to sketch a discharge curve with its key features.
• Combined capacitance and the charge and p.d. on each capacitor of a network; the energy from W = ½CV2 or from an area on a V–Q graph; the time constant, a value at a time, or the time to reach a value in a discharge. Given on the sheet: the series and parallel formulas, x = x0e−t/RC and ε0. Recall: C = Q/V, W = ½QV = ½CV2 and τ = RC.
• A charged capacitor discharges through a known resistor; the p.d. (or current) is read at timed intervals. ln V against t is a straight line of gradient −1/RC, which gives C. The largest uncertainty is reading a falling p.d. at the instant a stopwatch shows the time; a data logger removes it. A Paper 5 analysis adds error bars on ln I and a worst acceptable line.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.