Capacitors

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📘 Capacitors

Reference Textbook

📖 Reference Textbook:
University Physics Volume 2: Chapter 8 — Capacitance

Theory

What Is a Capacitor and What Is Capacitance?

A capacitor is an electronic component that stores electrical energy in an electric field, created between two conductors separated by an insulating material (called a dielectric) or by vacuum. In a parallel-plate capacitor, these conductors are two plates.

Capacitance is the measure of a capacitor's ability to store charge per unit voltage. Its SI unit is the farad (F).

Capacitors are widely used in various applications, including:

• Energy storage (e.g., in power supplies).
• Filtering (e.g., in electronic circuits to smooth out fluctuations in voltage).
• Timing circuits (e.g., controlling signal timing in oscillators).
• Signal coupling and decoupling (e.g., in audio equipment).

Their ability to store and release energy quickly makes them essential in electronics.

▶ Video: Capacitors — 1

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▶ Video: Capacitors — 2

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Capacitance Definition

The capacitance $C$ of a capacitor is defined as the ratio of the magnitude of the charge $Q$ stored on either plate to the magnitude of the potential difference $\mathrm{\Delta }V$ across the plates:

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

where:

• $C$ is the capacitance (in farads, F);
• $Q$ is the magnitude of the charge on either plate (in coulombs, C);
• $\mathrm{\Delta }V$ is the magnitude of the potential difference between the plates (in volts, V).

The plates carry equal and opposite charges, $+Q$ and $-Q$. The charge stored refers to the magnitude of the charge on either plate, rather than the net charge of the capacitor.

$1\mathrm{F}=1\mathrm{C}/\mathrm{V}$

Capacitance of a Parallel-Plate Capacitor

For a parallel-plate capacitor, the capacitance depends on the area $A$ of the plates, the separation $d$ between them, and the permittivity $\epsilon$ of the material between the plates.

If the space between the plates is completely filled with a uniform dielectric and edge effects are neglected,

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

where:

• $C$ is the capacitance (in farads, F);
• $\epsilon$ is the permittivity of the dielectric material (in farads per metre, F/m);
• $A$ is the area of one plate (in square metres, m²);
• $d$ is the separation between the plates (in metres, m).

The permittivity can be written as

$\epsilon =\kappa {\epsilon }_0$

where $\kappa$ is the dielectric constant (relative permittivity) and ${\epsilon }_0$ is the permittivity of vacuum. For vacuum, $\kappa =1$; for air, it is approximately 1.

A larger plate area increases the capacitance, while a larger separation decreases it.

Energy Stored in a Capacitor

The energy $U$ stored in a charged capacitor is

$U=\frac{1}{2}C(\mathrm{\Delta }V{)}^2$

where:

• $U$ is the energy stored (in joules, J);
• $C$ is the capacitance (in farads, F);
• $\mathrm{\Delta }V$ is the magnitude of the potential difference between the plates (in volts, V).

Using $Q=C\mathrm{\Delta }V$, equivalent expressions are

$U=\frac{1}{2}Q\mathrm{\Delta }V=\frac{Q^2}{2C}$

This energy is stored in the electric field.

Energy Density in a Capacitor

The energy density $u$ represents the energy stored per unit volume in the electric field between the plates.

For a linear dielectric,

$u=\frac{1}{2}\epsilon E^2$

where:

• $u$ is the energy density (in joules per cubic metre, J/m³);
• $\epsilon$ is the permittivity of the material (in farads per metre, F/m);
• $E$ is the electric field magnitude (in volts per metre, V/m).

For a uniform field between parallel plates, the energy density is also $u=U/(Ad)$.

Equivalent Capacitance in Series

For capacitors connected in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances:

$\frac{1}{C_{\mathrm{e}\mathrm{q}}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots +\frac{1}{C_n}$

where:

• $C_{\mathrm{e}\mathrm{q}}$ is the equivalent capacitance (in farads, F);
• $C_1,C_2,\dots ,C_n$ are the individual capacitances (in farads, F).

For initially uncharged capacitors charged in series:

• Each capacitor stores the same charge magnitude.
• The potential differences across the capacitors add to give the total potential difference.
• The equivalent capacitance is smaller than any of the individual capacitances.

Equivalent Capacitance in Parallel

For capacitors connected in parallel, the equivalent capacitance is the sum of the individual capacitances:

$C_{\mathrm{e}\mathrm{q}}=C_1+C_2+\cdots +C_n$

where:

• $C_{\mathrm{e}\mathrm{q}}$ is the equivalent capacitance (in farads, F);
• $C_1,C_2,\dots ,C_n$ are the individual capacitances (in farads, F).

For capacitors connected in parallel:

• Each capacitor has the same potential difference across it.
• The stored charges add to give the total stored charge.
• The equivalent capacitance is larger than any of the individual capacitances when two or more capacitors are connected.

▶ Video: Capacitors in Series and in Parallel — 1

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▶ Video: Capacitors in Series and in Parallel — 2

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RC Circuits: Charging and Discharging a Capacitor

The voltage across a capacitor changes over time as it charges or discharges through a resistor.

See RC Circuits for the relationships describing this behaviour.

Demonstrations

Build Your Own Capacitor

▶ Demonstration: Build Your Own Capacitor

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Capacitor Simulations

Explore capacitance, energy storage, and capacitor charging with these simulations:

• Charging and Discharging a Capacitor
• An Example of a Natural Capacitor
• Factors Affecting Capacitance
• PhET: Capacitor Lab

Additional Resources

• Charged Parallel Plates
• RC Circuit Simulator

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