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# Capacitor

A Capacitor is an arrangement for storing large amounts of electric current or electric energy in a very small space. Capacitors have a wide applications in fans, electric motors, mixers, electric juicers, flour mills and all electrical instruments which are rotating on a particular axis. Let us study more about the Capacitor in this section.

 Related Calculators capacitor reactance calculator Parallel Plate Capacitor Calculator

## How Does a Capacitor Work?

A Capacitor is a component used in the electronic circuit to hold the charges. It is made two conductors separated by an insulating medium. The two conductors are charged by connecting them with a battery.
The main feature of capacitors is that it block dc whereas it allows ac to flow. Hence they are used in alternating current circuits only.
It works on the following principle:

The charge flowing through a conductor is directly proportional to Voltage.

i.e         Q $\alpha$ V

Where, Q = Charge
and V = Voltage.

Q = CV

Here C is proportionality constant called as capacitance. So capacitance is

C = $\frac{Q}{V}$

The unit of capacitance is Coulomb / Volt or farad.

## Capacitor Symbol

The capacitor is represented as:

## Capacitors in Series

In series combination, the capacitors are arranged in series order. We know that the Capacitors are having two plates. The second plate of first Capacitor is connected to the first plate of second capacitor. similarly the second plate of the second capacitor connected to the first plate of the third capacitor and so on . Finally , the first plate of the first capacitor and the second plate of the last capacitor are connected to the opposite terminals of the battery.

Let us connect two Capacitor in series,

Let Q be the charge given to the left plate of the capacitor C1. By electrostatic induction, a charge -Q appears on inner side of the right plate of C1. Let V1, V2 be the potential difference across C1, C2 respectively.

Then, total potential is given by
V = V1 + V2 $\frac{Q}{C_{s}}$ = $\frac{Q}{C_{1}}$ + $\frac{Q}{C_{2}}$ $\frac{1}{C_{s}}$ = $\frac{1}{C_{1}}$ + $\frac{1}{C_{2}}$Where Cs is the effective Capacitance in series
The effective Series capacitance Cs is given by
Cs = $\frac{C_{1} C_{2}}{C_{1} + C_{2}}$For ‘n’ number of capacitors in series, the effective capacitance can be written as$\frac{1}{C}$ = $\frac{1}{C_{1}}$ + $\frac{1}{C_{2}}$ + $\frac{1}{C_{3}}$ + ………………+ $\frac{1}{C_{n}}$

## Capacitors in Parallel

In the Parallel combination, the capacitors are first arranged so that all the first plates of different capacitors are connected at one terminal and all the second plates of the capacitors are connected at another terminal . The two terminals are then connected to the two terminals of a battery.

Consider two capacitors of capacitance C1 and C2 connected in parallel to a battery of potential difference V.
The charges on each capacitor is given as
Q1 = C1V and Q2 = C2 V

The total charge drawn from the battery is

Q = Q1 + Q2 = C1V + C2V
If C is the combined capacitance of two capacitors then total charge drawn from the battery is Q = CV, the above equation changes to
CV = C1V + C2 VFor ‘n’ numbers of capacitors in parallel then combined capacitance is C = C1 + C2 + C3 ………………… + Cn

## Types of Capacitors

The Capacitors in series and parallel combination are the simplest basic form of Capacitors. But based on the Insulating materials, their combination in circuits and all, we can classify the capacitors in the following types:
1. Variable Capacitor
2. Multiple Capacitor
3. Paper Capacitor
4. Electrolyte Capacitor

## Variable Capacitor

As the name itself indicates, the capacitance of a variable capacitor can be varied gradually. This is achieved by adjusting the effective area included between the plates. The plates are usually made of brass or aluminium and are semi circular in shape.

A Variable capacitor consists of two sets of plates. One set of plates is fixed in position and is called the stator. The other set of plates can be rotated over the stator by rotating the piston. This set is called rotor. During rotation of the motor, the area common to the plates of stator and rotor is varied.
Variable capacitors are widely used in circuits in radio and T.V receivers.

## Paper Capacitor

A Paper soaked in wax or oil is used as the dielectric in the paper capacitor. It is placed in between tin foils that serve as capacitor plates. This is shown in figure. In order to increase the capacitance to a large extent, waxed paper and strips of metal foils are arranged alternatively as shown in figure.

Finally, the entire capacitor can be rolled and sealed in a cylinder as shown. As these capacitors occupy small space and are cheaper in cost, they are widely used in radio circuits and in laboratories.

## Energy Stored in Capacitor

The Energy stored in a Capacitor is given by
W = $\frac{1}{2}$ C V2 or
W = $\frac{1}{2}$ (V Q) ($\because$ C = $\frac{Q}{V}$)Where
W = Energy stored in a Capacitor
C = Capacitance of a Capacitor
V = Potential difference across the Capacitor
Q = Charge on capacitor.

## Charging a Capacitor

Let Q and I be the charge on the capacitor and the current in the circuit at any time t.
Then using the ckt, we get,
RI = E - $\frac{Q}{C}$When Capacitor is charged fully, curent in the circuit will be zero.
i.e Q = Q0                                                    $\therefore$ E = $\frac{Q_{0}}{C}$or                       R $\frac{dQ}{dt}$ = $\frac{Q_{0} - Q}{C}$or                 $\frac{dQ}{Q - Q_{0}}$ = $\frac{1}{CR}$ dt
Integrating we get,                                    - log(Q0 - Q) = $\frac{1}{CR}$ t + KWhere K is a Constant of integration.
When t = 0, Q = 0 and K = - loge Q0$\therefore$ - loge(Q0 - Q) = $\frac{1}{CR}$ t - loge Q0orlog(Q0 - Q) - log Q0 = - $\frac{1}{CR}$ torlog $\frac{Q_{0} - Q}{Q_{0}}$ = - $\frac{1}{CR}$ tTaking antilog,
$\frac{Q_{0} - Q}{Q_{0}}$ = $e^{-\frac{1}{CR} t}$or                                          Q = Q0 (1 - $e^{-\frac{1}{CR} t}$)
This represents the equation for Charging of a Capacitor.

## Discharging a Capacitor

On releasing the key, the circuit becomes closed without the battery. The capacitor begins to discharge itself.
RI + $\frac{Q}{C}$ = 0or
R $\frac{dQ}{dt}$ + $\frac{Q}{C}$ = 0or
$\frac{dQ}{Q}$ = - $\frac{1}{CR}$ dtIntegrating we get,
logeQ = - $\frac{1}{CR}$ t + KWhere K is a Constant.
At time t=0, Q = Q0, $\therefore$ K = logeQ0
logeQ = - $\frac{Q}{Q_{0}}$ + logeQ0or
loge $\frac{Q}{Q_{0}}$ = - $\frac{1}{CR}$ tor$\frac{Q}{Q_{0}}$ = - $\frac{1}{CR}$ tor
$\frac{Q}{Q_{0}}$ = $e^{\frac{-t}{CR}}$
Therefore the discharge in the Capacitor at time t is given by
Q = Q0 $e^{\frac{-t}{CR}}$
Which is the equation for the decay of Charge in CR circuit.

## Capacitor Codes

Below is given the Capacitor codes which helps to identify the capacitor.

## Uses of Capacitors

Capacitors find many applications in electrical and electronic circuits. They are used as:

• Filters: Capacitor blocks direct current and allows alternating current. So they are used as filters in the circuit.
• Tuning circuits: Tuning circuits are used in communication system to catch and transmit electromagnetic waves. of certain frequency. For a tuning circuit, capacitors along with inductor is connected in series are used. A tuning circuit allows catching a particular radio frequency. Radio, TV receivers uses this principle.
• Oscillating circuit and Amplifier circuit: Capacitors are used in production of AC signal or in timer circuits. The oscillating circuits has a capacitor connected either with resistor or inductor. capacitors are used as components in all types of amplifier circuit to increase the given input signal.
• Power supplies: Capacitors are used to reduce voltage fluctuations in electric power supplies as filters, to transmit regulated pulsed signal and to provide necessary time delay.

## Electrolytic Capacitor

These are widely used when high capacitance are required .Capacitance of the order of 103 $\mu$ F can be easily obtained with electrolyte capacitors of small volumes.