3.2 Capacitance

What is capacitance?

Capacitance (C) measures a conductor's ability to store charge per unit potential difference: C = Q/V. It depends only on the geometry of the conductors (size, shape, separation) and the material between them, not on Q or V themselves. Units are farads (F), where 1 F = 1 C/V.

Parallel-plate capacitor

For a parallel-plate capacitor, C = ε₀A/d, where A is plate area and d is separation. Applying Gauss's Law gives the field E = σ/ε₀ between plates, and integrating E over distance gives V = Ed. Combining Q = σA with V = Ed yields the formula, showing C increases with larger area and decreases with larger separation.

Energy stored in a capacitor

Charging a capacitor requires work since charge must be moved against an increasing field. The energy stored is U = ½QV = ½CV² = Q²/2C. This energy is stored in the electric field between the plates, and can be found by integrating the work dW = V dq from 0 to Q.

Key terms

Capacitance formula

C = Q/V, where Q is stored charge and V is potential difference; measured in farads (F).

Parallel-plate capacitance

C = ε₀A/d, derived from Gauss's Law field E = σ/ε₀ and V = Ed.

Energy stored in a capacitor

U = ½QV = ½CV² = Q²/2C, found by integrating dW = V dq from 0 to Q.

Farad

The SI unit of capacitance; 1 F = 1 C/V.

Does capacitance depend on Q or V?

No — capacitance depends only on the geometry of the conductors and the dielectric medium between them.

Try an AP-style question

A parallel-plate capacitor has plate area 0.01 m² and separation 1 mm. What is C? (ε₀ = 8.85 × 10⁻¹²)

1. 8.85 × 10⁻⁹ F
2. 8.85 × 10⁻¹² F
3. 8.85 × 10⁻¹¹ F
4. 8.85 × 10⁻¹⁴ F