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Capacitance

19. Capacitance

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Lesson summary

19. Capacitance — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Define capacitance, combine capacitors in series and parallel, calculate stored energy, and analyse the exponential discharge of a capacitor through a resistor.Capa…

  • Capacitance C
  • Charge storage
  • Supply potential
  • Construction
  • Dielectric
  • Same charge

Part 1 of 10Capacitors and capacitance

  • Capacitor = a device that stores energy by storing separated charge, used as a backup energy store, for example if power fails.
    • The symbol is two parallel lines perpendicular to the connecting wires.
  • Capacitance C = the charge stored per unit potential difference, C = Q/V.
  • Unit — the farad, F = C V−1; one farad is very large, so values are in µF, nF or pF, and a typical circuit capacitor is 47 µF.
  • Charge Q — it is the magnitude of the charge on one plate, because the capacitor is neutral overall.
  • Notation — the letter C is used both for capacitance and for the coulomb.

Part 2 of 10Isolated spherical conductor

  • Charge storage — a conducting sphere stores charge on its surface, where like charges repel and spread evenly.
  • Supply potential — more charge is stored when the supply potential is higher.
  • Capacitance — the surface charge acts like a point charge at the centre, so V = Q/(4πε0R).
    • Combined with C = Q/V this gives C = 4πε0R, which depends only on the radius.

Part 3 of 10Parallel-plate capacitor

  • Constructiontwo parallel conducting plates separated by an insulating dielectric, connected to a supply.
  • Chargingthe supply pushes electrons onto one plate (making it negative) and removes them from the other (positive).
  • Dielectricit stops charge flowing across, so equal and opposite charges build up on the plates.

Part 4 of 10Capacitors in series

  • Same charge — each capacitor in a series chain holds the same charge Q.
  • Shared p.d. — V = V1 + V2, with V1 = Q/C1 and V2 = Q/C2.
  • Result — dividing by Q gives 1/Ctotal = 1/C1 + 1/C2 + 1/C3 + …
    • The total capacitance is smaller than the smallest capacitor.
    • Two identical capacitors in series give half the capacitance of one.

Part 5 of 10Capacitors in parallel

  • Same p.d. — each capacitor in parallel has the same voltage V.
  • Shared charge — Q = Q1 + Q2, with Q1 = C1V and Q2 = C2V.
  • Result — Ctotal = C1 + C2 + C3 + …
    • The total capacitance is larger than any single capacitor.
    • Two identical capacitors in parallel give twice the capacitance of one.
ArrangementSame quantityCombined capacitance
SeriesCharge Q1/C = 1/C1 + 1/C2 + …
ParallelPotential difference VC = C1 + C2 + …
  • Resistors — these formulae are the opposite way round to those for resistors.

Part 6 of 10Combination circuits

  1. Combine every group of parallel capacitors first by adding their capacitances.
  2. Treat each group as one capacitor in the series arrangement and add the reciprocals.
  3. Take the reciprocal of the sum to find the total capacitance.
  • Check — the total must be less than the smallest series component.

Part 7 of 10Charging and the energy stored

  • Work in charging — as the plates charge, repulsion makes it harder to add more electrons, so the supply does more work per coulomb.
  • Graph — V is proportional to Q, so the V–Q graph is a straight line through the origin.
  • Energy stored = the area under the V–Q graph, a triangle, W = ½QV.
    • Using Q = CV gives W = ½CV2 and W = Q2/(2C).
  • Units — convert charge in µC and p.d. in kV to C and V to get joules.

Part 8 of 10Discharge through a resistor

  • Process — electrons flow from the negative plate round the circuit to the positive plate until the charges are equal and the p.d. is zero.
  • Current — it is large at the start, opposite to the charging direction, and falls to zero.
  • Exponential decay — current, p.d. and charge all fall exponentially, because the rate of fall is proportional to the amount remaining.
    • The graphs of I, V and Q against time have the same shape, starting on the y-axis and decreasing.
  • Resistance — a large resistance gives a small current and a slower discharge, and a small resistance discharges faster.

Part 9 of 10Time constant

  • Time constant τ the time for the charge, current or p.d. of a discharging capacitor to fall to 1/e (37%) of its initial value.
  • Equationτ = RC, in seconds, with R in ohms and C in farads.
  • From a graphfind 0.37 of the initial value and read the corresponding time.
  • Meaninga larger time constant means a slower discharge, and it lets similar quantities be compared.

Part 10 of 10Discharge equations

  • General form — X = X0e−t/RC, where X is charge, current or p.d.
    • Q = Q0e−t/RC, V = V0e−t/RC and I = I0e−t/RC.
  • Exponential constant — e ≈ 2.718 and 1/e = 0.368, giving the 37% in the time-constant definition.
  • Time — rearrange to X/X0 = e−t/RC and take natural logs, t = −RC ln(X/X0).
  • Proportional — the current, charge and p.d. are proportional to each other at any instant.

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