In this article, we shall study the concept of the capacity of conductor, capacitors and their types, and energy stored in the capacitor.

Concept of Capacity of 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 is given by

C = Q /V

Let V = 1 V, then C = Q

The 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 conductor is 1 farad if a charge of 1 coulomb raises its potential by 1 volt.

The farad is a very large unit hence smaller practical units are used. Smaller units are

1 microfarad (1 μF) = 10^-6 F

1 nanofarad (1 nF) = 10^-9 F

1 picofarad (1 pF) = 10^-12 F4.

Uses of Condensers or Capacitors:

• It is used in the radio tuning circuit.
• It is used to smoothen pulsating current in the 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 V_A is the potential of plate A due to charge Q, its initial capacity ( in the absence of earthed plate B)

C₁ = Q / V_A

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 the earth, the free positive charge on it flows to the earth and gets neutralized. Thus only negative charge is left on B. If -V_B is the potential of conductor B, then the total potential of the arrangement is (V_A -V_B).

New capacity of plate A is

C₁ = Q / (V_A -V_B)

Since (V_A -V_B)< V_A  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 insulating medium known as a dielectric. One of the plates (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 the 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 the outer side of the outer sphere is earthed. The inner surface of the 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 /ε_ok  ………. (1)

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

ε_o  =  permittivity of free space.

k  =  dielectric constant of the medium.

Now ‘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 produces 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 an increase in surface area, dielectric constant but decreases 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 ε_ok /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 increase 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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In this article, we shall study two simple capacitors circuits: capacitors in series and capacitors in parallel. We shall derive an expression for equivalent of such combination.

Equivalent Capacity of Condensers in Series:

A number of condensers are said to be joined in series if they are joined end to end and each condenser acquires the same charge. In series combination, the plates of each condenser carry a charge of the same magnitude. However, the potential difference across the whole combination is the sum of the potential difference across individual condensers.

Consider three condensers of capacities C₁, C₂ and C₃ respectively, then the total potential drop is given by :

V = V₁ + V₂ + V₃   ….(1)

This is the terminal p.d. of the cell. In series combination, the charge on each condenser is the same i.e. Q.

For the 1st condenser C₁= Q/V₁   Hence V₁ = Q/C₁

For the 2nd condenser, C₂= Q/V₂   Hence V₂ = Q/C₂

For the 3rd condenser, C₃= Q/V₃   Hence V₃ = Q/C₃

Substituting these values in equation (1)

V = Q/C₁ + Q/C₂ + Q/C₃

∴  V = Q(1/C₁ + 1/C₂ + 1/C₃)  …………….. (2)

If the three condenses are replaced by an equivalent condenser of capacity ‘C’, then for equivalent or effective capacity of the condenser.

C = Q/V hence V = Q/C

Substituting in (2), we get,

Q/C = Q(1/C₁ + 1/C₂ + 1/C₃)

∴  1/C = 1/C₁ + 1/C₂ + 1/C₃

In general, when ‘n’ condenser of capacity C₁, C₂, C₃,…., C_n are connected in series, then their equivalent capacity is given by

Thus when a number of condensers are connected in series the reciprocal of their resultant capacity is the sum of reciprocals of their individual capacities.

The resultant capacity of series combination is always less than the least capacity in the combination.

Equivalent Capacity of Condensers in Parallel:

A number of condensers are said to be joined in parallel if they are connected between two common points so that the potential difference across each condenser is the same. However, each condenser acquires a charge depending on its capacity.

Let us consider three condensers of capacity C₁, C₂ and C₃ connected in parallel across a cell of terminal potential difference  V. One plate of each condenser is connected to the positive terminal of the cell and the other plate is connected to the negative terminal. The total charge + Q supplied by the cell gets distributed over the condensers.

If Q₁, Q₂ and Q₃ are the amounts of charge accumulated by condenser C₁, C₂ and C₃ respectively, then charge supplied by the cell is given by

Q   = Q₁ + Q₂ + Q₃    ……….. (1)

In parallel combination p. d. across each condenser is the same i.e. This is also the terminal p. d. of the cell.

For the 1st condenser C₁= Q₁/VHence Q₁ = C₁V

For the 2nd condenser, C₂= Q₂/VHence Q₂ = C₂V

For the 3rd condenser, C₃= Q₃/VHence Q₃ = C₃V

Substituting these values in equation (1)

Q = C₁V + C₂V + C₃V

∴  Q = V(C₁ + C₂ + C₃)    ……….. (2)

If the three condensers are replaced by an equivalent condenser of capacity C, then for equivalent or effective condenser,

C = Q/V hence Q = CV

Substituting in (2), we get,

CV = V(C₁ + C₂ + C₃)

C = C₁ + C₂ + C₃

In general, when ‘n’ condenser of capacities C₁, C₂, C₃,  …… Cn are connected in series, then their equivalent capacity is given by

Thus when a number of condensers are joined in parallel, their effective capacity equals the sum of the capacities of the individual condensers.

The resultant capacity of the parallel combination is always greater than the greatest capacity in the combination.

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