Capacitance

• Capacitance C = Q/V, unit farad (F = C V⁻¹); defined for parallel-plate capacitors and isolated spheres.
• Parallel: C = C₁ + C₂ + …; series: 1/C = 1/C₁ + 1/C₂ + ….
• Energy stored W = ½QV = ½CV² = area under the V–Q graph.
• Discharge: x = x₀ e^−t/RC for Q, V or I; time constant τ = RC.

C = Q/V

A parallel-plate capacitor has two conducting plates separated by an insulator. When charged, one plate holds +Q and the other −Q; “the charge on the capacitor” means Q. Capacitance is the charge stored per unit potential difference. For an isolated spherical conductor, V is its potential relative to infinity (zero), so a sphere of radius r has C = 4πε₀r.

Parallel: each capacitor has the same p.d., and the total charge is the sum: Q = Q₁ + Q₂ gives C = C₁ + C₂. Series: each capacitor carries the same charge Q, and the p.d.s add: V = V₁ + V₂, so Q/C = Q/C₁ + Q/C₂ and 1/C = 1/C₁ + 1/C₂. These are the reverse of the resistor rules.

Area under V–Q

Charging moves charge against the p.d. already on the plates, so work is done. On a graph of p.d. V against charge Q (a straight line through the origin, gradient 1/C), the work done moving a small charge ΔQ is VΔQ, a thin strip. The total energy is the area under the line: a triangle, ½QV.

When a capacitor is charged from a battery of e.m.f. V, the battery supplies QV but the capacitor stores only ½QV; the other half is dissipated in the circuit resistance.

Exponential decay, τ = RC

time constant

Taking logs gives ln x = ln x₀ − t/RC, a straight line with gradient −1/RC, a common way to find RC experimentally. The time to halve is τ ln 2 = 0.693RC.