Revise Capacitors and capacitance for Cambridge 9702 (19.1)

At its core, a capacitor is an electrical component designed to store electric charge and, by extension, electrical potential energy. The most common type is the parallel-plate capacitor, which consists of two conductive plates separated by an insulating material called a dielectric. This dielectric prevents current from flowing directly between the plates while allowing an electric field to form, storing energy.

The ability of a capacitor to store charge is quantified by its capacitance (C). It's defined as the amount of charge (Q) it can store per unit of potential difference (V) across its plates. The higher the capacitance, the more charge it can hold for a given voltage.

$C = \frac{Q}{V}$

The capacitance of a parallel-plate capacitor depends on its physical characteristics: the area of the plates, the distance between them, and the insulating material (dielectric) used.

$C = \frac{\varepsilon A}{d}$

Here, A is the area of overlap between the two plates, d is the separation between the plates, and ε (epsilon) is the permittivity of the dielectric material. Permittivity is a measure of how well a material can store energy in an electric field. It is given by $\varepsilon = \varepsilon_r \varepsilon_0$, where $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \text{ F m}^{-1}$) and $\varepsilon_r$ is the relative permittivity (or dielectric constant) of the material.

When capacitors are connected in series, they are linked end-to-end, forming a single path. In this configuration, the charge stored on each capacitor is the same, but the total potential difference applied across the combination is distributed among them. This means the overall storage ability is reduced.

$\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$

In a parallel connection, capacitors are connected side-by-side across the same two points in a circuit. This means each capacitor experiences the identical potential difference. The total charge stored is the sum of the charges on each capacitor, leading to a greater overall storage ability.

$C_{total} = C_1 + C_2 + \dots$

When a capacitor stores charge, it also stores electrical potential energy within the electric field between its plates. This energy can be released when the capacitor discharges. The amount of energy stored can be determined from the area under a charge-voltage (Q-V) graph, which for a constant capacitance, is a triangle.

$E = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$