# Capacitors

## Capacitors and capacitance - charge and unit of charge.

A capacitor is a device used to store electrical energy.

The plates of a capacitor is charged and there is an electric field between them. The capacitor will be discharged if the plates are connected together through a resistor.

### Charge of a Capacitor

The charge of a capacitor can be expressed as

Q = I t (1)

where

Q = charge of capacitor (coulomb, C, mC)

I = current (amp, A)

t = time (s)

The **quantity of charge** (number of electrons) is measured in the unit Coulomb - *C - *where

1 coulomb = 6.24 10^{18}electrons

The smallest charge that exists is the charge carried by an electron, equal to *-1.602 10 ^{-19} coulomb*.

#### Example - Quantity of Electricity Transferred

If a current of *5 amp* flows for *2 minutes,* the quantity of electricity - *coulombs* - can be calculated as

*Q = (5 A) (2 min) (60 s/min)*

* = 600 C *

or, in electrons:

*(600 C) ( 6.24 10^{18} electrons / C)*

* = 3.744 10 ^{21} electrons*

### Electric Field Strength (Dielectric Strength)

If two charged plates are separated with an insulating medium - a dielectric - the electric field strength (potential gradient) between the two plates can be expressed as

E = U / d (2)

where

E = electric field strength (volts/m)

U = eletrical potential (volt)

d = thickness of dielectric, distance between plates (m)

#### Example - Electric Field Strength

The voltage between two plates is* 230 V* and the distance between them is *5 mm*. The electric field strength can be calculated as

*E = (230 V) / ((5 mm) (10 ^{-3} m/mm))*

* = 46000 volts/m*

* = 46 kV/m*

### Electric Flux Density

Electric flux density is the ratio between the charge of the capacitor and the surface area of the capacitor plates:

D = Q / A (3)

where

D = electric flux density (coulomb/m^{2})

A = surface area of the capacitor (m^{2})

### Charge and Applied Voltage

Charge in a capacitor is proportional to the applied voltage and can be expressed as

Q = C U (4)

where

C = constant of proportionality orcapacitance(farad, F,µF)

### Capacitance

From *(4)* the capacitance can be expressed as

C = Q / U (5)

One farad is defined as the capacitance of a capacitor when there is a potential difference across the plates of one volt when holding a charge of one coulomb.

It is common to use* µF (10 ^{-6 }F)*.

#### Example - Voltage over a Capacitor

A *5 µF* capacitor is charged with *10 mC*. The voltage across the capacitor can be calculated by modifying *(4)* to

*U = Q / C *

* = (10 mC) (10 ^{-3} C/mC) / ((5 µF) (10^{-6} F/µF)*

* = 2000 V*

* = 2 kV*

### Absolute Permittivity

The ratio of electric flux density to electric field is called absolute permittivity - *ε* - of a dielectric and can be expressed as

ε = D / E (6)

where

ε = absolute permittivity (F/m)

The absolute permittivity of free space or vacuum - also called the electric constant - *ε _{0}* - is

*8.85 10*.

^{-12}F/m### Relative Permittivity

Relative permittivity - also called the dielectric constant *ε _{r}* - is the ratio between the flux density of the field in an actual dielectric -

*ε*- and the flux density of the field in absolute vacuum -

_{}*ε*

*.*

_{0} *ε _{r} = ε / ε_{0} (7a)*

The actual permittivity can be calculated by *transforming (7a) to*

ε = ε_{r}ε_{0}(7b)

### Parallel Plate Capacitor

The capacitance of a plate capacitor - as shown in the figure above - is proportional with the area A of the plate. The capacitance can be expressed as

*C = ε _{r} ε_{0} A / d (8)*

*where *

*A = area of plate (m ^{2})*

*d = thickness of dielectric, distance between plates (m)*

For a plate capacitor with multiple plates the capacitance can be expressed as

*C = ε _{r} ε_{0} A (n - 1) / d (8b)*

*where *

*n = number of plates*

### Example - Capacitance of a Plate Capacitor

The capacitance of a plate capacitor with area *5 cm ^{2}*,

*10*plates and distance

*0.1 mm*between the plates - with ceramic dielectric with relative permittivity

*30*between the plates - can be calculated as

*C = ( 8.85 10^{-12} F/m)_{} (30) (5 cm^{2}) (10^{-4} m^{2}/cm^{2}) (10 - 1) / ((0.1 mm) (10^{-3} m/mm)) *

* = 11 10 ^{-9} F*

* = 11 pF*

### Typical commonly used Capacitors

Typical capacitors are

- variable air capacitors
- mica capacitors
- paper capacitors
- ceramic capacitors
- plastic capacitors
- titanium oxide capacitors
- electrolytic capacitors