§ 19.1 Capacitors and capacitance
Key ideas
- Capacitance is the charge stored per unit potential difference, C = Q/V, in farads.
- Capacitors in parallel add; in series the reciprocals add (giving less than the smallest).
Equations
C = Q / Vcapacitance = charge ÷ p.d.F
parallel: C = C₁ + C₂series: 1/C = 1/C₁ + 1/C₂F
Watch out: the combination rules are the opposite of resistors: capacitors add in parallel and combine reciprocally in series. Reaching for the resistor rule is the usual slip.
§ 19.2 Energy stored in a capacitor
Key ideas
- The energy stored is the area under the potential-charge graph, a triangle because V is proportional to Q.
Equations
W = ½ Q V = ½ C V²energy stored (also ½Q²/C)J
Fig. 1 · Energy stored is the shaded triangle under the V-Q line: because V rises in step with Q, the area is ½QV.
Watch out: charging a capacitor from a battery is only 50% efficient: half the energy moved is lost in the resistance, which is why the stored energy is ½QV, not QV.
§ 19.3 Discharging a capacitor
Key ideas
- Through a resistor, the charge, p.d. and current all decay exponentially with the same time constant.
- The time constant τ = RC is the time to fall to 1/e (about 37%) of the start; after about 5τ the capacitor is effectively discharged.
Equations
x = x₀ e^(−t/RC)decay of Q, V or I during dischargeno unit
τ = R Ctime constant of the discharges
Fig. 2 · Exponential discharge: after one time constant τ = RC the charge has fallen to 0.37 of its initial value, and the same curve describes V and I.
Watch out: the decay never quite reaches zero, so quote a time constant or a number of half-lives, not a "time to fully discharge". Each τ removes the same fraction, not the same amount.