Capacitors

Devices that store energy in the form of an electric field. Theory, formulas, worked examples and interactive graphs.

Conceptual context

A capacitor is a device that stores energy in the form of an electric field. Unlike a battery, which converts chemical energy into electrical energy, the capacitor simply separates opposite charges on two nearby conducting surfaces —the plates— and maintains that separation as long as the applied voltage lasts.

It appears in filter circuits, camera flashes, cardiac defibrillators, computer memories, and smartphone touch sensors. Understanding the capacitor means understanding how the electric field stores and releases energy in a controlled manner.

The capacitor bridges electrostatics and electric circuits: its behavior depends on the static electric field between its plates, yet in AC circuits it becomes a reactive element that blocks DC and passes high-frequency current. This leaf covers the electrostatic side: how capacitance is defined, how energy is stored, and what physical factors determine the capacitance of a parallel-plate capacitor.

🟢 Essential level

When a capacitor is connected to a battery, electric charges redistribute: one plate accumulates positive charge and the other accumulates negative charge of equal magnitude. Between the two plates an electric field is trapped — this is the physical manifestation of the stored energy.

What defines a capacitor as a better or worse charge storage device is its capacitance $C$. A high-capacitance capacitor accumulates a lot of charge for a small voltage; a low-capacitance one needs high voltages to accumulate the same charge. Capacitance does not depend on the applied voltage or the stored charge: it is a fixed property of the device, determined by its geometry and its dielectric material.

The voltage $V$ between the plates is the direct cause of everything that happens in the capacitor. Higher voltage means more accumulated charge $Q$ and more stored energy $E$. The relationship is simple and elegant: charge grows proportionally with voltage, and energy grows quadratically.

🔵 Formal level

Basic formulas

The quantitative definition of capacitance is obtained by measuring how much charge $Q$ the capacitor accumulates per volt $V$ of applied voltage:

Definition of electric capacitance

This equation has three useful readings: it calculates $C$ by measuring Q and V, predicts $Q$ knowing C and V, or finds $V$ knowing C and Q. All three forms are exactly equivalent and derive from the same linear relationship.

Real student questions

Why does the capacitor energy formula carry a 1/2 factor? Because voltage is not constant during charging: it begins at zero and grows to the final value. The work at each charge increment is the instantaneous voltage times the elemental charge; integrating from zero to the final charge introduces the 1/2 factor.

Does capacitance change if I change the voltage? No. $C$ is a geometric and material property, independent of voltage. What changes is the stored $Q$ and $E$; $C$ itself does not change (except in nonlinear dielectrics).

Why do capacitors in series have lower capacitance than either one alone? In series they share the same $Q$ but voltage is distributed between them. Equivalent capacitance measures charge per total volt; distributing voltage reduces the charge accumulated for a given total voltage below what either capacitor alone would store.

Why does a capacitor block DC but pass AC? In DC, once charged to source voltage, current stops — charge cannot cross the dielectric. In AC, voltage changes sign constantly so the capacitor charges and discharges alternately; the external current appears to "pass through", though no charge actually crosses the dielectric.

Final synthesis

The capacitor stores energy in the electric field between its plates. Its capacitance $C$ is the central property: fixed for a given geometry and material, it determines how much charge $Q$ and energy $E$ the device stores for each applied voltage $V$. Energy grows with the square of voltage, making $V$ the most critical design parameter. The geometry of the parallel-plate capacitor links $C_0$ with $\epsilon$, area, and separation.