Capacitor

A capacitor is a device used to store electrical charge and electrical energy. Capacitors are generally with two electrical conductors separated by a distance.

It consists of two electrical conductors (capacitor plates) separated by a distance. The space between the plates can be a vacuum or filled with an insulating material called a dielectric. The storage capacity of a capacitor is determined by its capacitance. Capacitors find various applications, such as filtering static in radio reception and storing energy in heart defibrillators. They consist of two conducting plates with a dielectric between them. When connected to a battery, the capacitor transfers a small amount of charge from the positive plate to the negative plate, resulting in opposite charges ($+\mathit{Q}$ and $-\mathit{Q}$) on the plates.

Both capacitors shown here were initially uncharged before being connected to a battery. They now have charges of $+\mathit{Q}$ and $-\mathit{Q}$ (respectively) on their plates. (a) A parallel-plate capacitor consists of two plates of opposite charge with area A separated by distance $\mathit{d}$. (b) A rolled capacitor has a dielectric material between its two conducting sheets (plates).

A system composed of two identical parallel-conducting plates separated by a distance is called a parallel-plate capacitor. The magnitude of the electrical field in the space between the parallel plates is $\mathit{E}=\frac{\mathit{\sigma }}{{\mathit{ϵ}}_0}$, where $\mathit{\sigma }$ denotes the surface charge density on one plate. Thus, the magnitude of the field is directly proportional to $\mathit{Q}$.

Capacitors with different physical characteristics (such as shape and size of their plates) store different amounts of charge for the same applied voltage $\mathit{V}$ across their plates. The capacitance $\mathit{C}$ of a capacitor is defined as the ratio of the maximum charge Q that can be stored in a capacitor to the applied voltage $\mathit{V}$ across its plates. In other words, capacitance is the largest amount of charge per volt that can be stored on the device:

The SI unit of capacitance is the farad (F), named after Michael Faraday (1791–1867). Since capacitance is the charge per unit voltage, one farad is one coulomb per one volt, or

By definition, a $1\mathit{F}$ capacitor is able to store $1\mathit{C}$ of charge (a very large amount of charge) when the potential difference between its plates is only $1\mathit{V}$. Capacitors can be produced in various shapes and sizes.

Calculation of Capacitance

We can calculate the capacitance of a pair of conductors with the standard approach that follows.

1. Assume that the capacitor has a charge $\mathit{Q}$.
2. Determine the electrical field $\overrightarrow{\mathit{E}}$ between the conductors. If symmetry is present in the arrangement of conductors, you may be able to use Gauss’s law for this calculation.
3. Find the potential difference between the conductors from

   ${\mathit{V}}_{\mathit{B}}-{\mathit{V}}_{\mathit{A}}=-\underset{\mathit{A}}{\overset{\mathit{B}}{\int }}\overrightarrow{\mathit{E}.}\mathit{d}\overrightarrow{\mathit{l}}$

   where the path of integration leads from one conductor to the other. The magnitude of the potential difference is then $\mathit{V}=\left|{\mathit{V}}_{\mathit{B}}-{\mathit{V}}_{\mathit{A}}\right|$.
4. With V known, obtain the capacitance directly from $\mathit{C}=\frac{\mathit{Q}}{\mathit{V}}$.

To show how this procedure works, we now calculate the capacitances of parallel-plate.

Parallel-Plate Capacitor

The parallel-plate capacitor consists of two identical conducting plates with surface area $\mathit{A}$, separated by distance $\mathit{d}$. When a voltage $\mathit{V}$ is applied, the capacitor stores charge $\mathit{Q}$. The capacitance of the capacitor, denoted by $\mathit{C}$, depends on $\mathit{A}$ and $\mathit{d}$. According to Coulomb's force, the force between charges increases with larger charge values and decreases with greater distance. Therefore, a larger plate area ($\mathit{A}$) allows for storing more charge, leading to a higher capacitance ($\mathit{C}$). Similarly, when the plates are closer together (smaller $\mathit{d}$), the attraction between opposite charges increases, resulting in a higher capacitance $\mathit{C}$.