Capacitance

This guide covers Topic 19, Capacitance, in full — subtopics 19.1 Capacitors and capacitance, 19.2 Energy stored in a capacitor and 19.3 Discharging a capacitor — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content, building directly on uniform electric fields from Topic 18.

Before studying this

This resource assumes uniform electric fields from Electric Fields, and D.C. circuit analysis from D.C. Circuits: Kirchhoff’s Laws and Potential Dividers.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 19

19.1 Capacitors and capacitance — defining capacitance as C = Q/V; recalling and using C = Q/V; deriving, using the formula for capacitors in series and in parallel, and using these formulas for capacitor networks; describing the action of a capacitor in a simple circuit; applying the definition C = Q/V to an isolated spherical conductor to derive C = 4πε₀r.

19.2 Energy stored in a capacitor — recalling and using the fact that the area under a potential–charge graph is the energy stored; deriving, using the area under a potential–charge graph, the equation W = ½QV = ½CV²; recalling and using W = ½QV = ½CV².

19.3 Discharging a capacitor — analysing graphically the discharge of a capacitor through a resistor; recalling and using τ = RC for the time constant of a discharging (or charging) circuit; recalling and using equations of the form x = x₀e^(−t/RC) where x could represent current, charge or voltage in a discharging capacitor circuit.

Capacitors and capacitance

A capacitor stores charge on two conducting plates separated by an insulator. Its capacitance C is defined as the charge stored per unit potential difference across it:

C = Q/V

Capacitance is measured in farads (F), where 1 F = 1 C V⁻¹.

The capacitance of an isolated sphere

The definition C = Q/V applies just as well to a single isolated charged conducting sphere as to a pair of plates. An isolated sphere of radius r carrying charge Q has surface potential V = Q/(4πε₀r) (from the electric potential of a point charge, since the sphere behaves as a point charge at its centre for points on or outside its surface), so its capacitance is:

C = Q/V = 4πε₀r

showing that the capacitance of an isolated sphere depends only on its radius — larger spheres store more charge for the same potential.

Capacitors in series and parallel

For capacitors connected in series, the reciprocals of the individual capacitances add:

1/C = 1/C₁ + 1/C₂ + ...

For capacitors connected in parallel, the capacitances themselves add directly:

C = C₁ + C₂ + ...

This is the reverse pattern from resistors: resistors in series add directly, while capacitors in series add reciprocally, and vice versa for parallel.

Energy stored in a capacitor

As a capacitor charges, the potential difference across it rises as charge accumulates, so the work done to add each successive increment of charge is not constant. Plotting potential difference V against charge Q gives a straight line through the origin, and the energy stored is the area under this graph (a triangle):

W = ½QV

Substituting Q = CV gives an equivalent form:

W = ½CV²

Worked example. A 470 μF capacitor is charged to 12 V. The energy stored:

W = ½CV² = 0.5 × 470 × 10⁻⁶ × 12² = 0.0338 J

Discharging a capacitor

When a charged capacitor discharges through a resistor, the current, charge and voltage all decay exponentially with time:

x = x₀ e^(−t/RC)

where x can represent current I, charge Q, or voltage V, and x₀ is its initial value. The quantity RC is called the time constant, τ:

τ = RC

The time constant is the time taken for the quantity to fall to 1/e (about 37%) of its initial value, and gives a measure of how quickly a capacitor discharges through a given resistance: a larger RC means slower discharge.

Charging a capacitor (beyond 19.3 — background only, not examinable)

Subtopic 19.3 is titled “Discharging a capacitor” and the syllabus coverage above only requires the discharge form x = x₀e^(−t/RC); the charging equations below are included for context and conceptual completeness only, not as recall or exam-required material.

Charging is the reverse process. Charge and p.d. rise towards their final value:

Q = Q₀(1 − e^(−t/RC))

while the current decays exponentially, starting at its maximum value V₀/R (when the capacitor is uncharged and offers no opposing p.d.) and falling towards zero as the capacitor’s own p.d. increasingly opposes the supply:

I = I₀ e^(−t/RC)

Current always decays exponentially in both charging and discharging — the difference is what charge and p.d. do: they rise during charging and fall during discharging. The time constant τ = RC has exactly the same meaning in both cases, and is independent of the capacitor’s initial charge.

Common mistakes

• Using the resistor-style formula for capacitors in series or parallel — the rules are reversed compared with resistors.
• Using W = QV instead of W = ½QV, forgetting the factor of ½ that comes from the linearly rising voltage during charging.
• Assuming discharge is linear rather than exponential — current, charge and voltage all fall off exponentially, never reaching exactly zero in a finite time.
• Confusing the time constant τ = RC with the time for complete discharge — τ is the time to fall to 1/e of the initial value, not the total discharge time.
• Assuming charge or current rises linearly during charging — the current decays exponentially even while the charge is rising, since it is the rate of charge delivery that falls as the capacitor’s own p.d. increasingly opposes the supply.

Quick revision checklist

• C = Q/V, and combining capacitors in series (reciprocal) and parallel (direct)
• C = 4πε₀r for an isolated spherical conductor
• W = ½QV = ½CV² for energy stored, from the area under a V-Q graph
• x = x₀e^(−t/RC) for exponential discharge of current, charge or voltage
• τ = RC as the time constant, with the same meaning in charging and discharging

Extension, beyond 19.3 (background only, not examinable): Q = Q₀(1 − e^(−t/RC)) for charge rising during charging, while current still decays exponentially.

Related resources

• Electric Fields — the previous A Level topic
• Magnetic Fields — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.