Solution

Statement

$6.4.1.$ What is an electrical capacitor? What is the capacitance of a capacitor? How does the definition of the capacitance of an isolated conductor differ from the definition of the capacitance of a capacitor?

Solution and answer

An electrical capacitor is a device consisting of two conductors (plates) separated by a dielectric, designed to accumulate electric charge and electric field energy.

The capacitance of a capacitor ($C$) is a physical quantity characterizing the ability of a capacitor to accumulate charge at a given voltage between its plates. It is defined as the ratio of the charge ($Q$) on one of the plates to the voltage ($U$) between them: $C = Q/U$. The SI unit of capacitance is the farad (F).

Isolated conductor:
This is a single conductor located far from other charged bodies.
Its capacitance is defined as the ratio of the charge ($Q$) on the conductor to its potential ($\phi$): $C = Q/\phi$.
The potential depends only on the charge and the geometry of the conductor (for example, a sphere of radius $R$ has a capacitance $C = 4\pi\varepsilon_0 R$).

Capacitor:
Consists of two conductors (plates) with charges equal in magnitude but opposite in sign.
Capacitance is defined as the ratio of the charge ($Q$) on one plate to the potential difference ($U$) between the plates: $C = Q/U$.
It depends on the geometry of the plates, the distance between them, and the properties of the dielectric.

An isolated conductor has capacitance relative to infinity (the potential is measured from zero at infinity).

A capacitor is characterized by the capacitance between two conductors, and its capacitance is usually much larger than that of an isolated conductor of comparable size.

The capacitance of a capacitor is defined through the potential difference between the plates, while the capacitance of an isolated conductor is defined through its own potential relative to infinity.