Electricity and magnetism

Capacitors and capacitance

A capacitor stores charge and electric-field energy; its capacitance measures charge stored per unit potential difference.

A capacitor consists of two conductors separated by an insulator. Connected to a source, its plates acquire opposite charges and establish an electric field. For an ideal parallel-plate capacitor, capacitance increases with plate area and decreases with plate separation.

C=\frac{Q}{U},\qquad C=\varepsilon\frac{A}{d}\ (\text{parallel plates}),\qquad W=\frac{1}{2}CU^2=\frac{Q^2}{2C}

Definition: Capacitance

Capacitance $C$ is measured in farads (F); $Q$ is the magnitude of charge on either plate and $U$ is the voltage between them. In the linear regime, $C$ is set by geometry and dielectric, not by $Q$ or $U$ . For parallel plates, $A$ is plate area, $d$ separation, and $\varepsilon$ the permittivity of the material.

Adjust plate area, separation, and voltage to track capacitance, stored charge, and energy.

Combining capacitors

Capacitors in parallel share the same voltage, so their equivalent capacitance is the sum. In series, each carries the same charge and the voltage divides; reciprocal equivalent capacitance is the sum of reciprocals.

C_{\parallel}=C_1+C_2,\qquad \frac{1}{C_{\mathrm{s}}}=\frac{1}{C_1}+\frac{1}{C_2}

Example: Energy in a capacitor

A $100\,\mu$ F capacitor is charged to $12$ V. Find its charge and stored energy.

Solution

$Q=CU=100\times10^{-6}\times12=1.2\times10^{-3}$ C. $W=CU^2/2=7.2\times10^{-3}$ J.

When a dielectric fills the gap, its molecules polarize and reduce the internal field; at the same voltage, the capacitor can hold more charge. Distinguish a capacitor still connected to a source from an isolated one: the source keeps $U$ fixed, whereas isolation keeps $Q$ fixed. For example, doubling plate area while keeping separation and dielectric unchanged doubles capacitance according to $C=\varepsilon A/d$ .

Stored energy belongs to the electric field between the plates; it is not created energy. When the capacitor discharges through a load, that field energy becomes heat, light, or mechanical work.

Quick check

A capacitor of $C=5\,\mu$ F is placed across $10$ V. What is the magnitude of charge on either plate?

The equivalent capacitance of two capacitors in parallel is:

References

Young, Freedman (2019). University Physics