Home › Learning Hub › AS & A Level Physics › A2 18–19
A2 18–19

Electric fields and capacitance

A Level · syllabus topics 18 and 19 · theory on Paper 4; the practical skills behind it on Paper 5

Topic 18 is Topic 13 with a change of sign and a change of constant. Learn the two side by side and you halve the work — but respect the differences, because the exam tests exactly those.

🎯What you need to be able to do

  • Understand that an electric field is a field of force, and define electric field strength as force per unit positive charge.
  • Represent electric fields by field lines, and describe the field of a point charge, of a uniformly charged sphere and between charged parallel plates.
  • Recall and use \( E = V/d \) for the uniform field between parallel plates.
  • Recall and use Coulomb’s law, \( F = Q_1Q_2/4\pi\varepsilon_0 r^{2} \), and \( E = Q/4\pi\varepsilon_0 r^{2} \) for a point charge.
  • Define electric potential, use \( V = Q/4\pi\varepsilon_0 r \), and relate field strength to the potential gradient.
  • Use \( E_p = Qq/4\pi\varepsilon_0 r \) for the electric potential energy of two point charges.
  • Define capacitance and use \( C = Q/V \), for an isolated conductor and for a parallel-plate capacitor.
  • Derive and use the formulae for capacitors in series and in parallel.
  • Deduce from the area under a potential–charge graph that the energy stored is \( W = \tfrac{1}{2}QV \), and use \( W = \tfrac{1}{2}CV^{2} = \tfrac{1}{2}Q^{2}/C \).
  • Analyse the discharge of a capacitor through a resistor using \( x = x_0e^{-t/RC} \), and understand the time constant \( \tau = RC \).

📚The physics

Gravitational and electric fields compared. Both are inverse-square laws with a potential that goes as \( 1/r \). The differences: gravity acts on mass and is only attractive, so gravitational potential is always negative; the electric force acts on charge and can be either sign, so electric potential is positive near a positive charge and negative near a negative one. And the constants differ by about twenty orders of magnitude — \( 1/4\pi\varepsilon_0 \) is \( 8.99 \times 10^{9} \) while \(G\) is \( 6.67 \times 10^{-11} \) — so that between two protons, once the charges and masses are put in, the electric repulsion is about \( 10^{36} \) times the gravitational attraction.

Field and potential. \( E = -\mathrm{d}V/\mathrm{d}r \): the field is the negative gradient of the potential. Where the potential is uniform the field is zero. Inside a hollow charged conductor the potential is constant and equal to its surface value, so the field there is zero — the principle behind every Faraday cage.

Left: radial field lines from a positive point charge with circular equipotentials, closer together near the charge. Right: parallel, evenly spaced field lines between charged plates with equally spaced flat equipotentials.
Field lines cross equipotentials at right angles; the field is strongest where the equipotentials crowd together.

Capacitance is charge stored per volt, in farads. A farad is enormous; real capacitors are measured in µF, nF and pF. Capacitors in parallel add directly (\( C = C_1 + C_2 \)), the opposite of resistors — think of it as increasing the plate area. In series the reciprocals add.

Energy stored. On a \(V\)–\(Q\) graph the gradient is \( 1/C \) and the area under the line is the energy. Because the line is straight through the origin, the area is a triangle: \( W = \tfrac{1}{2}QV \). The factor of one half is there because the first charge moves across at almost no p.d. while the last moves across the full \(V\).

P.d. against charge for a 470 microfarad capacitor: a straight line to 12 volts at 5.64 millicoulombs. The triangle beneath, one half Q V, is 0.034 joules.
Energy stored \( = \) area \( = \tfrac{1}{2}QV = 0.034 \) J for the worked example’s capacitor.

Discharge. Charge, current and p.d. all decay exponentially with the same time constant \( \tau = RC \), which is the time to fall to \( 1/e \approx 37\% \) of the initial value. After \( 5\tau \) the capacitor is more than 99% discharged. Plotting \( \ln V \) against \(t\) gives a straight line of gradient \( -1/RC \), which is how \(\tau\) is measured experimentally.

✏️Worked example

A 470 µF capacitor charged to 12 V is discharged through a 22 kΩ resistor. Find the time constant, the energy stored initially, and the p.d. after 15 s.

\( \tau = RC = 22 \times 10^{3} \times 470 \times 10^{-6} = 10.3 \) s.

\( W = \tfrac{1}{2}CV^{2} = 0.5 \times 470 \times 10^{-6} \times 12^{2} = 0.034 \) J.

\[ V = V_0e^{-t/\tau} = 12 \times e^{-15/10.3} = 12 \times 0.233 = 2.8\ \text{V} \]
Left: p.d. decaying exponentially from 12 volts with time constant 10.3 seconds, marking 4.4 volts at one time constant and 2.8 volts at 15 seconds. Right: natural log of p.d. against time is a straight line from 2.48 with gradient minus 0.097 per second.
The worked example’s discharge, and the straight-line \( \ln V \) graph used to measure \( RC \).
The mark people actually lose here is dropping the micro. Entering 470 rather than \( 470 \times 10^{-6} \) gives a time constant of 10 300 000 s, and the answer to the last part comes out as 12 V because the exponent is effectively zero — a result that looks harmless on the page. Convert every prefix to a power of ten in a separate line before substituting. The other loss is squaring the wrong thing in the energy formula: it is \( \tfrac{1}{2}CV^{2} \), not \( \tfrac{1}{2}(CV)^{2} \).

🔭See it happen

Charge a large electrolytic capacitor and discharge it through a resistor with a voltmeter and a stopwatch: students can find \(\tau\) from the 37% point directly and then check it against \(RC\). For fields, a Van de Graaff and a few paper streamers show field lines leaving a conductor perpendicular to its surface, and a shaving-foam-and-oil dish with semolina grains gives the classic dipole pattern in about a minute.

🔗Go deeper — other people’s work

The links below are not mine. They are here because they are good, and they may move or disappear without warning.

  • PhET, Charges and Fields and Capacitor Lab — the second shows the stored energy changing as you move the plates.
  • The Physics Classroom, “Static Electricity”.
  • Isaac Physics, “Electric Fields” and “Capacitors”.