Capacitance and capacitors

What a capacitor is

A capacitor is a device that stores separated electric charge. The simplest capacitor consists of two conductors separated by an insulator or empty space. When connected to a battery, one conductor gains charge $+Q$ and the other gains charge $-Q$.

The potential difference between the conductors is $Delta V$. Capacitance is defined as

C=rac{Q}{Delta V}

Capacitance measures how much charge can be stored per volt of potential difference. The SI unit is the farad:

$1 F=1 C/V$

A farad is very large for many practical devices, so microfarads, nanofarads, and picofarads are common.

Capacitance depends on geometry

Capacitance depends on conductor geometry and the material between conductors. It does not depend on $Q$ or $Delta V$ separately for an ideal linear capacitor. If charge doubles, voltage doubles, and the ratio stays fixed.

This is similar to spring constant: force and displacement vary, but $k$ characterizes the spring.

Parallel-plate capacitor

Consider two large parallel plates of area $A$ separated by distance $d$, with vacuum between them. If the plates carry surface charge densities $+sigma$ and $-sigma$, the electric field between them is approximately

E=rac{sigma}{epsilon_0}

The potential difference is

$Delta V=Ed$

Since

$Q=sigma A$

we have

Delta V=rac{Qd}{epsilon_0 A}

Thus

C=rac{Q}{Delta V}=rac{epsilon_0 A}{d}

Large area increases capacitance; larger separation decreases capacitance.

Capacitors in parallel

Capacitors in parallel share the same voltage. Their charges add:

$Q_{total}=Q_1+Q_2+cdots$

Since $Q_i=C_iV$, the equivalent capacitance is

$C_{eq}=C_1+C_2+cdots$

Parallel combination increases capacitance, like increasing plate area.

Capacitors in series

Capacitors in series carry the same magnitude of charge, but voltage differences add:

$Delta V_{total}=Delta V_1+Delta V_2+cdots$

The equivalent capacitance satisfies

rac{1}{C_{eq}}=rac{1}{C_1}+rac{1}{C_2}+cdots

Series combination decreases capacitance, like increasing separation.

Battery connected versus isolated

A capacitor connected to a battery has fixed voltage. If its geometry changes while connected, charge can flow to keep $Delta V$ fixed.

An isolated charged capacitor has fixed charge. If its geometry changes, the voltage changes because $Delta V=Q/C$.

This distinction is important in energy and dielectric problems.

Practical uses

Capacitors are used to store energy, smooth voltage signals, filter frequencies, tune circuits, create timing delays, separate AC from DC, and provide bursts of power. They appear in power supplies, radios, computers, cameras, sensors, and electric vehicles.

The big idea

Capacitance measures charge stored per potential difference. It depends on geometry and material, not on charge alone. Parallel-plate capacitors reveal the basic relationship $C=epsilon_0A/d$, while series and parallel combinations show how capacitors can be arranged to control equivalent capacitance.