Capacitance

A capacitor is a fundamental passive electronic component designed to store electrical energy in an electric field. The simplest form consists of two parallel conductive plates separated by an insulating material called a dielectric.

### Defining Capacitance

The ability of a capacitor to store charge is quantified by its capacitance (C). Capacitance is defined as the ratio of the magnitude of the charge (Q) on one of the plates to the potential difference (V) between the plates.

C = Q / V

The SI unit for capacitance is the farad (F), named after Michael Faraday. One farad is a very large unit, defined as one coulomb of charge stored per volt of potential difference (1 F = 1 C V⁻¹). In practice, capacitance is usually measured in smaller sub-multiples: microfarads (1 μF = 10⁻⁶ F), nanofarads (1 nF = 10⁻⁹ F), and picofarads (1 pF = 10⁻¹² F). The capacitance of a parallel-plate capacitor depends on the area of the plates, the distance between them, and the permittivity of the dielectric material.

### Energy Stored in a Capacitor

To charge a capacitor, work must be done to move charge from one plate to the other against the electrostatic force of the charge already on the plates. This work done is stored as electrical potential energy (W) in the electric field between the plates.

As charge is added to the plates, the potential difference across them increases linearly from 0 to a final value V. The average potential difference during the charging process is ½V. The total work done (and thus energy stored) is the total charge moved multiplied by the average potential difference:

W = Q × (½V) = ½ QV

By substituting the capacitance equation (C = Q/V), we can derive two other useful forms of the energy equation:

Using Q = CV, we get: W = ½ (CV)V = ½ CV²

Using V = Q/C, we get: W = ½ Q(Q/C) = ½ Q²/C

Graphically, the energy stored is represented by the area under a graph of potential difference (V) against charge (Q). Since V is directly proportional to Q, this graph is a straight line through the origin. The area of the triangle formed is ½ × base × height, which corresponds to ½ QV.

### Charging a Capacitor

Consider a circuit with a capacitor (C), a resistor (R), a switch, and a DC power supply of e.m.f. V₀. When the switch is closed, the charging process begins.

* Q(t) = Q₀(1 - e⁻ᵗ/ᴿᶜ)

* V(t) = V₀(1 - e⁻ᵗ/ᴿᶜ)

* I(t) = I₀e⁻ᵗ/ᴿᶜ

### Discharging a Capacitor

Now, consider a fully charged capacitor (with initial charge Q₀ and voltage V₀) connected across a resistor (R) through a switch.

* Q(t) = Q₀e⁻ᵗ/ᴿᶜ

* V(t) = V₀e⁻ᵗ/ᴿᶜ

* I(t) = I₀e⁻ᵗ/ᴿᶜ

### The Time Constant (τ)

In both charging and discharging, the rate of change is determined by the product of resistance and capacitance. This product is known as the time constant (τ) of the circuit.

τ = RC

The time constant has units of seconds and represents the time scale of the charging or discharging process.

* During discharging, the time constant is the time it takes for the charge, voltage, or current to decrease to 1/e (approximately 37%) of its initial value.

* During charging, it is the time taken for the charge or voltage to rise to (1 - 1/e) (approximately 63%) of its final maximum value.

A circuit with a large time constant (large R or C) will charge and discharge slowly, while a circuit with a small time constant will be much faster.