The capacitor is a passive device, which can store electric charge into its conducting surface (plates) when connected with a voltage source and discharge the same stored charge when voltage source is not connected.
Simply, a capacitor has a capacity to store electrical charge into it.

Construction: The capacitors consist of two or more conducting surfaces separated by some insulating material which is known as dielectric. That means if we arrange, two conducting plate, separated by the layer of dielectric medium, it will be able to store electric charge.

Working principle: When we connect two parallel plates with a voltage source, like one plate with the positive and other with negative of the voltage source, we will see that, plate connected with positive terminal of source, become positive and plate connected with negative terminal become negative, and there is no current flow through it as it is electrically separated by some insulating material.
After a certain time, (known as charging time) when two plates hold maximum charge, it works as charge storage device. Now if we disconnect the voltage source and connect a load, we will see that the capacitor works as a voltage source, as current flows through the load. This process will continue until the charges of the plate become zero, known as discharging time. Thus a capacitor acts as a rechargeable battery.
Through above discussion we have seen that the capacitor work as open circuit in case of d.c source.

But what actually happens when connected with a d.c. source?
When a parallel plate capacitor is connected with a d.c. source, i.e. plate A with positive terminal of battery and plate B with negative terminal of battery, at first with instant a current flows through the capacitor. Plate A is charged with positive charge or proton and plate B is charged with negative charge or electron. Now when connected with battery, with a flash the electrons withdrawn from plate A and goes to plate B, which shows the momentary flow of electrons from plate A to plate B. After that the potential difference established between two plates and no current can flow.So, the capacitor acts as sort momentarily at initial condition and then act as open circuit in case of d.c. source.

Capacitor connected with a.c source:
When we connect a capacitor with an a.c voltage source, we may see that a.c current flows through the capacitor. But what actually happens?
We know that, a.c voltage means, the voltage which changes with time periodically into positive and negative cycle. When such voltage apply to the capacitor plate, during positive half one plate come into contact with positive polarity stores positive charge, another plate store negative charge as it become touched with negative polarity. But at next moment i.e. the next half of cycle, the same plate which stores positive charge attached with negative polarity and store negative charge. Similarly the other plate which stores negative charge now stores positive charge and this process continues. By this way the capacitor plates charged and discharged alternatively in case of a.c. and act as sort circuit. And no current flows between two plates due to insulating material.

Property by which, a capacitor can store electric charge is known as capacitance.
It is the ratio, of charge that can be stored and the applied voltage.
If Q coulomb of charge at one plate of capacitor by applying V volt,
Then Capacitance, C = Q/V = Charge/ Potential difference, coulomb/volt.
As per Michael Faraday, the unit of capacitance is Farad (Coulomb/volt).

Why maximum available capacitors are rated in micro or Pico farad?
As we have seen that capacitance, C = Q/V = charge/ potential difference. That means 1 farad capacitance requires 1 coulomb charge by applying 1 volt between its plates, which is difficult in practical cases. So capacitors are rated in micro or Pico farad.
1µF = 10-6 F.
1nF = 10-9 F.
1pF = 10-12 F.

Capacitance of a parallel plate capacitor:
Now we consider two parallel plates 1 and 2 for making a parallel plate capacitor. The area of each plate is A m<2 and two plates are separated by dielectric material at a distance of d.
The capacitance is directly proportional to the area of cross section, i.e. A.
So, C α A .
Capacitance is also inversely proportional to the distance of parallel plate.
So, C α 1/d .
As per two equation,
C α A/d .
Or, capacitance, C = Є0 ЄrA/d farad………..in case of medium.
Where Є0 Єr = permittivity of dielectric medium.
Or, C = Є0A/d farad………..in case of air.
Actually the equation is C = Є0 Єr (N-1)A/d farad.
Where N = number of parallel plate.
As we discussed with two parallel plate, C = Є0 Єr(2-1)A/d farad.
Or, C= Є0 ЄrA/d farad.
So, in case of multi plate capacitor, C = Є0 Єr(N-1)A/d farad.
In case of cylindrical or single core cable, the capacitance, C = 2π Є0 Єr/2.3log10(r2/r1) farad.
In case of l meter length of a cable, C = 2π Є0 Єrl/2.3log10(r2/r1) farad.
Where r2= outer surface radius and r1= inner surface radius.
In case of transmission line, for length of l meter, C = π Є0 Єr/2.3 log10(d/r) farad. Where d = distance from centre to centre of the wire and r = radius of each wire.

Energy stored in capacitor:
The capacitor store energy in the dielectric medium from external source like battery. That means it stores the energy from other source and releases it when discharging.
At first stage, when capacitor is in uncharged condition, connected to a battery, a little work is done for transferring charge from one plate to another. Then with every charge, a little work is done against the repulsive force of each plate. But as battery voltage is higher, it puts same charge on the plate and thus the charge on the plate of capacitor grows up.
The formula is W = ½ CV2 joules. If C is in farad and V is in volt.

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