Concept of Capacitor

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Concept of Capacity of a Conductor:

• Whenever an electric charge is deposited on a conductor, its potential increases. The deposited charge spreads over its surface.
• For any conductor, the electric potential (V) is directly proportional to the charge store (Q).

Hence,  Q   ∝   V

Q = CV

Where C is a constant known as Capacity of the conductor.

• The capacity of a conductor (C) is defined as the amount of charge required to make the potential 1 unit ( 1 Volt).

Capacity of a Conductor:

• A capacity of a conductor is given by

C = Q /V

Let V = 1 V, then C = Q

• Capacity of conductor (C) is defined as the amount of charge required to make the potential 1 unit ( 1 Volt)

S.I. unit of Capacity:

• S.I. Unit of capacity is farad (F)

We have  C = Q /V

S.I. unit of Capacity = S.I. unit of electric charge / S.I. unit of potential difference

∴  1 farad = 1 coulomb / 1 volt

• The capacity of a conductor is  1 farad if a charge of  1 coulomb raises its potential by 1 volt.
• The farad is very large unit hence smaller practical units are used. Smaller units are

1 microfarad ( 1 μ F ) = 10-6 F

1 nanofarad  ( 1 n F ) = 10-9 F

1 picofarad  ( 1 pF   ) = 10-12 F4.

Uses of Condensers or Capacitors:

• It is used in radio tuning circuit.
• It is used to smoothen pulsating current in rectifier circuit.
• It is used in an automobile to control spark plugs.
• It is used to store large charges in a nuclear reactor.

Principle of Parallel Plate Capacitor:

• Condenser or Capacitor is a device used for storing a large quantity of charge at a low potential. It is an arrangement of two conductors carrying equal and opposite charges separated by an insulating medium.
• A condenser consists of two identical metal plates separated from each other by air or some other dielectric. A charge + Q is deposited on one plate and other plate earthed. Due to this potential of the arrangement decreases and hence capacity increases.

• Consider figure (a). Let a charge  +Q is given to the plate A.  If VA is the potential of plate A due to charge Q, its initial capacity ( in the absence of earthed plate B)

C1 = Q / VA

• Consider figure (b). When another identical plate B is kept near A, negative charge i.e. -Q is induced in B at the rear side and +Q on the far side of the conductor B.
• Consider figure (c) Now the far side of plate B is connected to earth, the free positive charge on it flows to the earth and gets neutralised. Thus only negative charge is left on B. If -VB is the potential of conductor B, then the total potential of the arrangement is (VA -VB).
• New capacity of plate A is

C1 = Q / (VA -VB)

• Since (VA -VB)< VA  then C2 > C1 Thus instead of using a single plate., if the conductor plate is used with identical parallel plate its capacity increases. This is called the principle of a condenser.

Types of Capacitors:

• Depending upon the shape of the plates of a capacitor, the types of capacitors are as follows
• Parallel Plate Capacitor:

• A parallel plate condenser consists of two identical metallic plates P1 and P2. The plates are each of area A and distance ‘d’ apart.
• The space between two plates is filled with an insulting medium known as a dielectric. One of the plate (P1)is charged and the other plate (P2)is earthed.

• Cylindrical Capacitor:

• It consists of two coaxial cylinders of radii ‘a’ and ‘b’ respectively. The outer surface of the inner cylinder is positively charged and the outer side of the outer cylinder is earthed. The inner surface of outer cylinder acquires a negative charge.
• The space between the two cylinders is filled with a suitable dielectric material.
• Spherical  Capacitor:

• It consists of two concentric spheres of radii ‘a’ and ‘b’ respectively. The outer surface of the inner sphere is positively charged and outer side of outer sphere is earthed. The inner surface of outer sphere acquires the negative charge.
• The space between the two spheres is filled with a suitable dielectric material.

Expression for Capacity of Parallel Plate Condenser:

• A parallel plate condenser consists of two identical metallic plates P1 and P2. The plates are each of area A and distance ‘d’ apart. The space between two plates is filled with an insulating medium known as a dielectric. One of the plate (P1)is charged and the other plate (P2) is earthed.
• When a charge +Q is deposited on plate P1, an equal amount of negative charge, -Q is induced on earthed plate P2. This sets up a uniform electric field between the plates. The lines of force, therefore originate from P1 and terminate at P2.
• As the plate are very close to each other, the electric intensity near plates is given by,

E = σ /εok  ………. (1)

Where,   σ =  Charge per unit area  ( Surface charge density ), ε

εo  =  permittivity of free space.

k  =  dielectric constant of the medium.

• If ‘A’ the surface area of each phase, then the surface charge density is given by

σ =  Q /A   ……………… (2)

substituting for σ in equation (1) we get

• The positive charge  +Q on plate P1 produce a potential difference V with respect to plate P2. Therefore, in the uniform electric field, the relation between intensity and potential is given by

E  =  V / d   ……………. (4)

From (3) and (4) we ge

• This is an expression for the capacity of a parallel plate condenser with a dielectric. Therefore, the capacity of a parallel plate condenser increases with increase in surface area, dielectric constant but decrease in plate separation.

Expression for the capacity of a Spherical Condenser:

• The electric potential at a point outside a spherical condenser of radius ‘r’ carrying charge Q is given by

This is an expression or capacity of a spherical condenser

Effect of Dielectric Material on the Value of Capacity of a capacitor:

• The capacity of a parallel plate condenser with dielectric is given by

C =  A εok /d

• Where, A = Area of plates k = dielectric constant of the medium between the plates,  d = distance between the plates
• If the medium between the two plates is air. k = 1 Thus, the capacity of a parallel plate condenser with air and other medium  are given by

• Thus if the space between parallel plates of a condenser is filled with a dielectric material the capacity of the condenser increases.

Expression for Energy Stored in a Charged Capacitor:

• A charged condenser stores in it an electrical potential energy (U) equal to the work (W) required to charge it.
• When a condenser acquires charge, its potential increases. The increases in potential is directly proportional to the amount of charge acquired.
• Consider a condenser of capacity C while charging, let v be its potential due to the deposition of charge ‘q’. The capacity of the condenser is given by

• To deposit additional charge dq, work has to be performed against electric potential v. Hence work to be done (dW) to deposit an additional charge dq is given by

The total work done to charge the condenser full i.e. from 0 to Q is given by

This is an expression for energy stored in a charged condenser.

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