Capacitance

All three subtopics here are printed under "A Level subject content" in the 9702 syllabus, and every objective carries the tier "A Level". None of it is AS. It is examined on Paper 4. Paper 1 is the AS multiple-choice paper, and the whole 9702 bank on this site comes from Paper 1, so no practice is tagged to this topic.

Syllabus points

19.1 Capacitors and capacitance

• Define capacitance, as applied to both isolated spherical conductors and to parallel plate capacitors.
• Recall and use C = Q/V.
• Derive, using C = Q/V, formulae for the combined capacitance of capacitors in series and in parallel.
• Use the capacitance formulae for capacitors in series and in parallel.

19.2 Energy stored in a capacitor

• Determine the electric potential energy stored in a capacitor from the area under the potential-charge graph.
• Recall and use W = ½QV = ½CV².

19.3 Discharging a capacitor

• Analyse graphs of the variation with time of potential difference, charge and current for a capacitor discharging through a resistor.
• Recall and use τ = RC for the time constant for a capacitor discharging through a resistor.
• 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.

Capacitance

Capacitance is the charge stored per unit potential difference:

C = Q / V

measured in farads, where 1 F is 1 C V⁻¹. The farad is a very large unit, so practical capacitors are measured in microfarads, nanofarads and picofarads.

The definition applies to both cases the syllabus names:

• For a parallel plate capacitor, Q is the charge on one plate and V the potential difference between the plates.
• For an isolated spherical conductor, Q is the charge on the sphere and V its potential relative to infinity.

One point of language that is regularly marked wrong. A charged capacitor holds +Q on one plate and -Q on the other, so the net charge is zero. "The charge stored" always means the magnitude on one plate.

Capacitance is a property of the capacitor itself, fixed by its geometry and the material between the plates. Doubling the potential difference doubles the charge stored and leaves C unchanged, which is why a graph of Q against V is a straight line through the origin with gradient C.

Worked example. A 220 μF capacitor is charged to 12 V.

Q = 220 × 10⁻⁶ × 12 = 0.00264

so 2.64 × 10⁻³ C, or 2.64 mC.

Capacitors in parallel

Connected in parallel, every capacitor has the same potential difference across it, and the total charge is the sum of the individual charges.

Q = Q₁ + Q₂ + Q₃

Since Q = CV and V is common to all:

CV = C₁V + C₂V + C₃V

Dividing through by V:

C = C₁ + C₂ + C₃

Capacitances in parallel simply add.

Capacitors in series

Connected in series, the charge on each capacitor is the same, because the charge pushed onto one plate of the first capacitor induces an equal and opposite charge on the plate facing it, and so on down the chain. The potential differences add to the supply voltage.

V = V₁ + V₂ + V₃

Since V = Q/C and Q is common to all:

Q/C = Q/C₁ + Q/C₂ + Q/C₃

Dividing through by Q:

1/C = 1/C₁ + 1/C₂ + 1/C₃

So the combined capacitance in series is less than the smallest individual capacitance. That is the opposite of what resistors do, and confusing the two is the commonest error in this section. Resistors add in series; capacitors add in parallel.

Worked example. A 4.0 μF and a 12 μF capacitor in series.

1/C = 1/4.0 + 1/12 = 0.3333

so C = 3.0 μF, which is indeed smaller than either.

In parallel the same two would give

4.0 + 12 = 16

that is 16 μF.

Energy stored

Charging a capacitor takes work, because each extra charge has to be pushed onto a plate that is already charged and repelling it. The first charge arrives easily, the last is pushed against the full potential difference, so the mean potential difference during charging is half the final value.

That is what the area under a graph of potential difference against charge gives: a triangle of height V and base Q, so

W = ½QV

Substituting Q = CV, or V = Q/C, gives the other two forms:

W = ½CV² = ½Q²/C

The factor of ½ is the whole point, and it is the mark most often lost. Energy supplied by the battery during charging is QV; energy stored in the capacitor is ½QV. The other half is dissipated in the resistance of the circuit as the charging current flows, and no arrangement of components avoids it.

Worked example. How much energy is stored in a 470 μF capacitor charged to 9.0 V?

W = 0.5 × 470 × 10⁻⁶ × 9.0² = 0.019035

so about 1.9 × 10⁻² J.

Because W depends on V squared, doubling the potential difference gives four times the energy. If that capacitor were charged to 18 V instead:

0.019035 × 4 = 0.07614

that is 7.6 × 10⁻² J.

Discharging through a resistor

Connect a charged capacitor across a resistor and it discharges. All three quantities, charge, potential difference and current, fall exponentially:

Q = Q₀e^(-t/RC)

V = V₀e^(-t/RC)

I = I₀e^(-t/RC)

The reason all three share the same shape is that Q, V and I are proportional to one another at every instant: V = Q/C and I = V/R.

The shape of an exponential decay has one defining property, and it is what questions test: the quantity falls by the same fraction in equal intervals of time. The rate of decrease is proportional to how much is left, so the curve approaches the axis without ever reaching it.

The time constant

τ = RC

with τ in seconds when R is in ohms and C is in farads. Its meaning can be stated in three equivalent ways, and each is worth knowing because different questions ask for different ones:

• The time taken for the charge, potential difference or current to fall to 1/e of its initial value, which is about 37 per cent.
• The time taken to fall to about 63 per cent below its initial value.
• The time it would take to discharge completely if the initial rate of discharge were maintained, which is the intercept of the initial tangent on the time axis.

A larger R means a smaller current, so the capacitor takes longer to empty. A larger C means more charge stored at the same potential difference, so again it takes longer. Both increase τ.

Worked example. A 100 μF capacitor charged to 6.0 V discharges through a 47 kΩ resistor.

τ = 47000 × 100 × 10⁻⁶ = 4.7

so the time constant is 4.7 s.

The potential difference after 10 s:

10 / 4.7 = 2.128

V = 6.0 × e^(-2.128)

The exponential term is 0.1191, so

6.0 × 0.1191 = 0.7146

giving about 0.71 V.

Worked example, finding a time. How long until the charge falls to 20 per cent of its initial value?

0.20 = e^(-t/4.7)

Taking natural logarithms of both sides, ln 0.20 is -1.609, so

t = 1.609 × 4.7 = 7.562

that is about 7.6 s.

Reading the graphs

• Plotting V against t gives a decay curve. Its gradient at any point is proportional to the current, since I = V/R.
• Plotting ln V against t gives a straight line with gradient -1/RC and intercept ln V₀. This is the standard way of finding the time constant from experimental data, and of testing whether a decay really is exponential, since only an exponential gives a straight line on log axes.
• The area under an I against t graph is the charge that has flowed.

Common mistakes

• Saying a charged capacitor stores a net charge of Q. It stores +Q and -Q, so the net charge is zero.
• Adding capacitors in series as if they were resistors, and using the reciprocal formula for parallel.
• Getting a series combination larger than one of its components, which is always a sign the formula has been used the wrong way round.
• Forgetting to take the reciprocal at the end of the series calculation.
• Dropping the factor of ½ from W = ½QV.
• Saying the energy supplied by the battery equals the energy stored. Half is dissipated in the circuit resistance.
• Saying doubling the voltage doubles the stored energy. It quadruples it.
• Saying the time constant is the time for the capacitor to discharge completely. An exponential never reaches zero.
• Reading 1/e as 50 per cent rather than 37 per cent, or confusing the time constant with a half-life.
• Leaving capacitance in microfarads when calculating τ = RC.
• Plotting V against t and expecting a straight line, rather than ln V against t.