Capacitors and Capacitance

Definition

A capacitor is a system of two conductors separated by an insulator. In practice the two conductors carry $+Q$ and $-Q$ and are held at potentials $V_1$ and $V_2$, with $V=V_1-V_2$. The "charge of the capacitor" means $Q$, the charge on one conductor; the total charge of the device is zero. Even a single conductor can be treated as a capacitor, with the second conductor taken at infinity.

Why $Q/V$ is a constant

The field in the region between the conductors is proportional to $Q$ — this follows from Coulomb's law and superposition, so doubling the charge doubles the field everywhere. The potential difference is the work per unit charge in carrying a test charge from one conductor to the other against that field, so $V$ is proportional to $Q$ too. Their ratio is therefore independent of both:

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

• $C$ depends only on the geometrical configuration — shape, size and separation of the two conductors — and on the nature of the insulating medium between them. It does not depend on the charge given or the voltage applied.
• SI unit the farad: $1\ \text{F}=1\ \text{C V}^{-1}$. Dimensional formula $[\text{M}^{-1}\text{L}^{-2}\text{T}^{4}\text{A}^{2}]$.
• The farad is impractically large; the working units are $1\ \mu\text{F}=10^{-6}\ \text{F}$, $1\ \text{nF}=10^{-9}\ \text{F}$, $1\ \text{pF}=10^{-12}\ \text{F}$.
• Circuit symbol: two parallel line segments for a fixed capacitor; the same with an arrow drawn across it for a variable capacitor.

Reading a $q$–$V$ graph

Since $C=\dfrac{q}{V}$, a plot of charge against potential difference for a capacitor is a straight line through the origin whose slope is the capacitance. Given two lines $A$ and $B$ for two capacitors with the same plate separation but different plate areas, the steeper line is the larger capacitance. Because $C\propto A$, the capacitor with the greater plate area has the greater capacitance: the steeper line $A$ represents $C_2$ and the shallower line $B$ represents $C_1$.

Why large capacitance matters

From $V=\dfrac{Q}{C}$, a large $C$ holds a large charge at a modest potential difference. That matters because a large $V$ means a strong field around the conductors, and a strong enough field ionises the surrounding air, which then carries charge across and neutralises the plates — the charge leaks away.

• The maximum field a medium can withstand without breaking down is its dielectric strength. For air it is about $3\times10^{6}\ \text{V m}^{-1}$, so a gap of about $1\ \text{cm}$ breaks down at roughly $3\times10^{4}\ \text{V}$.
• So there is a limit to the charge a given capacitor can hold without significant leakage, and a capacitor is designed with $C$ high enough to keep $V$, and hence the field, below the breakdown limit.
• Beyond storing charge, capacitors are key elements of ac circuits.

Diagram to describe in words

Two irregular conductor blobs facing each other, the left marked $Q_1=+Q$ at potential $V_1$ and the right marked $Q_2=-Q$ at potential $V_2$, with a few field lines running from the positive to the negative conductor across the insulating gap, and $V=V_1-V_2$ labelled between them. Beside it, a $q$ against $V$ graph with two straight lines through the origin of different slopes, the steeper one annotated "larger $C$".

Standard results