Physics · Ch 2 — Electrostatic Potential and Capacitance

Capacitors and Capacitance

2.11

Capacitors and Capacitance

What is a Capacitor?

A capacitor is a system of two conductors separated by an insulator (dielectric). In the most common configuration, the two conductors carry equal and opposite charges: $+Q$ and $-Q$ . The potential difference between them is $V = V_1 - V_2$ .

The charge of the capacitor is $Q$ (the magnitude of charge on one plate). The total charge of the system is zero.
The conductors are charged by connecting them to the two terminals of a battery.

Why is Capacitance Defined?

The electric field $\mathbf{E}$ in the region between the conductors is directly proportional to the charge $Q$ on the plates. This follows from Coulomb’s law and the superposition principle: if $Q$ is doubled, $\mathbf{E}$ doubles at every point.

Since potential difference $V$ is the work done per unit positive charge to move a test charge from the negative to the positive plate against the field, $V$ is also proportional to $Q$ .

Therefore, the ratio $\frac{Q}{V}$ is a constant for a given capacitor:

C = \frac{Q}{V}

This constant $C$ is called the capacitance of the capacitor.

What Does Capacitance Depend On?

Geometrical factors: shape, size, and separation of the two conductors.
Dielectric material: the nature of the insulator between the conductors (discussed later in the chapter).

Capacitance is independent of $Q$ and $V$ .

SI Unit of Capacitance

The SI unit is the farad (F):

1\ \text{F} = 1\ \text{C V}^{-1}

In practice, the farad is a very large unit. Common submultiples are:

$1\ \text{mF} = 10^{-6}\ \text{F}$ (microfarad)
$1\ \text{nF} = 10^{-9}\ \text{F}$ (nanofarad)
$1\ \text{pF} = 10^{-12}\ \text{F}$ (picofarad)

Physical Meaning of Large Capacitance

From $C = Q/V$ , for a given $Q$ , a large capacitance means a small potential difference $V$ . This is important because:

High $V$ implies a strong electric field around the conductors.
A strong electric field can ionise the surrounding air, causing the charge to leak away (the capacitor discharges).

Dielectric Strength

The dielectric strength of a medium is the maximum electric field it can withstand without breaking down (losing its insulating property).

For air, dielectric strength is about $3 \times 10^6\ \text{V m}^{-1}$ . …

Figure 2.24A system of two conductors separated by an insulator forms a capacitor.

Fig. 2.24 — A system of two conductors separated by an insulator forms a capacitor.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure shows two irregularly shaped conductors placed side by side with a narrow gap between them. The left conductor is labelled Conductor 1 and carries a charge $+Q$ , with a potential $V_1$ written below it. A cluster of + signs is drawn along its right-facing surface. The right conductor is labelled Conductor 2 and carries a charge $-Q$ , with a potential $V_2$ written below it. A cluster of - signs is drawn along its left-facing surface. The opposite charges face each other across the insulating gap, which is the region between the conductors.

The physical idea the figure teaches is that a capacitor is formed by any two conductors separated by an insulator. The charges on the two conductors are equal in magnitude but opposite in sign ( $+Q$ and $-Q$ ). The potential difference between them is $V = V_1 - V_2$ . The electric field in the gap is proportional to $Q$ , and therefore $V$ is also proportional to $Q$ . This leads to the defining relation:

C = \frac{Q}{V}

where:

$C$ is the capacitance of the capacitor (a constant for a given geometry and dielectric),
$Q$ is the magnitude of charge on one conductor (the "charge of the capacitor"),
$V = V_1 - V_2$ is the potential difference between the two conductors. …

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