19.1aCapacitors and capacitance
Definitions
- Capacitance
- the charge stored per unit potential difference (C = Q/V).
- Farad
- the capacitance of a conductor that stores one coulomb of charge per volt (1 F = 1 C V-1).
- Capacitance of an isolated spherical conductor
- the charge on the sphere divided by its potential (relative to zero at infinity).
- Parallel-plate capacitor
- two conducting plates separated by an insulator (dielectric), carrying equal and opposite charges +Q and -Q.
Key points
- A capacitor stores charge (+Q on one plate, -Q on the other) and hence energy; its net charge is zero.
- Q is the charge on one plate, not the total of both.
- Q ∝ V for a capacitor, so a Q-V graph is a straight line through the origin with gradient C.
- Isolated sphere: V = Q/(4πε0r), so C = 4πε0r.
- Practical values are small: μF (10-6), nF (10-9), pF (10-12).
Parallel-plate capacitor charged by a cell: equal and opposite charges, a uniform field between the plates.
Q against V for a capacitor: a straight line through the origin whose gradient is C.
Watch out!
- Use the charge on one plate in C = Q/V; the total charge on a capacitor is zero.
- Convert μF to F (× 10-6) before calculating.
19.1bCapacitors in series and parallel
Definitions
- Combined capacitance
- the capacitance of the single capacitor that would store the same charge for the same p.d. as the combination.
Key points
- Parallel: the same p.d. across each; charges add, Q = Q1 + Q2, so CV = C1V + C2V.
- Series: the same charge Q on each capacitor; p.d.s add, V = V1 + V2, so Q/C = Q/C1 + Q/C2.
- Parallel total is larger than any single capacitor; series total is smaller than the smallest.
- The rules are the opposite of those for resistors.
- Two equal capacitors: parallel gives 2C; series gives C/2.
Series: each capacitor carries the same charge Q; the p.d.s add.
Parallel: each capacitor has the same p.d.; the charges add.
Watch out!
- Do not use the resistor rules: capacitors add in parallel, reciprocals add in series.
- After finding 1/C for series, remember to take the reciprocal.
19.2Energy stored in a capacitor
Definitions
- Energy stored in a capacitor
- the electric potential energy stored, equal to the area under the potential-charge graph.
Key points
- Adding a small charge ΔQ at p.d. V needs work ΔW = VΔQ: a thin strip under the V-Q graph.
- V ∝ Q, so the total area is a triangle: W = ½QV.
- Substituting Q = CV gives W = ½CV2 = ½Q2/C.
- Doubling V quadruples the energy stored (W ∝ V2).
- Charging from a battery: the battery supplies QV, the capacitor stores ½QV; the other half is dissipated in the resistance.
Energy stored = shaded triangle under the V-Q line; each thin strip is VΔQ.
Energy stored against p.d.: a parabola, W = ½CV2.
Watch out!
- Energy stored is ½QV, not QV: the p.d. rises from zero as the charge builds up.
- In W = ½CV2, square only the V.
19.3aDischarging a capacitor
Definitions
- Exponential decay
- a decrease in which the quantity falls by the same fraction in equal intervals of time.
Key points
- Discharging through a resistor, Q, V and I all decay exponentially with the same time constant.
- The current is greatest at the start (I0 = V0/R) because the p.d. is greatest then.
- Rate of discharge ∝ charge remaining: as Q falls, V and hence I fall, so discharge slows.
- Discharge current flows the opposite way to the charging current.
- Area under an I-t graph = charge that flowed.
- Gradient of a Q-t graph = current at that instant.
Charge (A) and discharge (B) circuit: switch to A to charge from the cell, then to B to discharge through R.
Discharge curve: after one time constant τ = RC, Q (or V or I) has fallen to 0.37 of its initial value.
Watch out!
- The decay never reaches zero: quote times as multiples of τ, not a "time to discharge fully".
- Each time constant removes the same fraction (63%), not the same amount.
19.3bThe time constant and log graphs
Definitions
- Time constant (τ)
- the time taken for the charge (or p.d. or current) to fall to 1/e (about 0.37) of its initial value; τ = RC.
Key points
- τ = RC has units of seconds (Ω × F = s).
- Larger R or larger C gives slower discharge.
- After about 5RC the capacitor is effectively fully discharged (under 1% remains).
- Taking logs: ln V = ln V0 - t/RC, so ln V against t is a straight line of gradient -1/RC.
- Time to halve: t½ = RC ln 2 ≈ 0.69RC.
- Find RC from a V-t graph: read the time for V to fall to 0.37V0.
ln V against t is a straight line: intercept ln V0, gradient -1/RC.
Each interval of τ multiplies the remaining charge by 0.37; each interval of 0.69RC halves it.
Watch out!
- Use natural logs (ln), not log10, when the gradient is to give -1/RC.
- Convert μF and kΩ before multiplying: 100 μF × 10 kΩ = 1 s.