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Capacitance

Capacitance:

    If we go on adding charge to a given body its potential goes on increasing i.e. charge on the body and potential are directly proportional to each other.

Charge µ Potential

or                             Q = CV

    where C is the constant of proportionality called the capacitance of the body. We can also define capacitance as the ratio of charge on the body to its potential. The capacitance of a capacitor depends upon:

(i)     size and shape of the conductor and nature of medium surrounding the conductor

(ii)     it also depends on the position of charges present in the neighborhood. It however is not dependent on the material of which conductor is made off.

Units of Capacitance:

    The SI unit of capacitance is Farad and capacitance is said to be one farad if

a charge of one coulomb is sufficient to raise the potential through one volt.


    The cgs unit of capacitance is statfarad and capacitance is said to be one statfarad if a charge of one e.s.u. is sufficient to raise the body potential by one statvolt.

1 Farad = 9 x 1011 stat farad

Capacity of Isolated Spherical Conductor::

    Let us consider a charge spherical body of radius r insulated from other charged bodies. If total charge on the body is q , then potential on the surface of sphere     =

    Capacity,


In cgs system,


or capacitance of a body is numerically equal to its radius.

Capacitor and Its Principle:

    A capacitor consists of two conductors separated by a certain distance with insulating medium called dielectric in between. Its main function is to increase the ability of body to take up charge. The basic principle is that the capacity of an insulated charged conductor is increased appreciably by bringing it near an earth connected uncharged conductor.

    Consider a plate A having charge +Q and potential V when another uncharged plate B is brought near this charged plate. Negative charge is induced on the inner side of this plate and positive charge is induced on the outer side of the plate B.

    The negative charge tries to decrease the potential and positive charge tends to increase it. On the whole there is net decrease in the potential of A because negative charge is near to the plate as compared with the positive charge.

On the other hand, if we connect the outer side of plate B to earth, the free positive charge on the surface disappears thereby causing a further reduction in potential. Hence as V goes on decreasing, to bring it back to the original potential we have to add lot more charge to it and thus capacity of the system increases further.

    As C = Q/V, if V decreases capacitance increases.

Parallel Plate Capacitor

:

    Parallel plate capacitor consists of parallel plates of conducting material seperated by certain fixed distance. The space in between the two plates consists of some insulating material called dielectric.

    Consider two such plates of area A with distance `d’ between them. Then the electric field between the two oppositely charged plates will be,


    Imagine a point P in between the two plates, potential difference between the two closely situated points around A is,

dV = E dr

    Potential difference between two plates,


    If instead of air there is some medium between the plates of capacitor, then


where er is the relative permittivity of the medium.

Potential Energy of Capacitor :

    Whenever a charge is added to the plates of capacitor, it increases its potential. To add more charge to it, we have to do work against coulombic repulsive force. This work done gets stored in the form of potential energy. If at any instant charge on the plates of the capacitor is q, the work done to add additional charge dq is given by,


    Total work done in increasing the charge from 0 to Q, we get,


    If V is the final potential of capacitor, then,


Force between Two Parallel Plates:

    Consider a parallel plate capacitor with distance between the plates of capacitor ‘d’. To increase this distance from d to d + Dd, the work has to be performed which is equivalent to the change in the potential energy,


    Also work done is given by,

dW = F Dd

    Equating the two values,


Substitute (ii) in (i),


Grouping of Capacitors:

Series of Grouping:

    Capacitors are said to be connected in series if second plate of first capacitor is connected with first plate of second capacitor and so on. The charge on the plates of all the capacitors is same, but potential difference will be different across different capacitors such that,

V = V1 + V2 + . . . . . . . . .+Vn


For two capacitors, C1
and C2,


Parallel Grouping:

    Capacitors are said to connected in parallel if positive plate of all the capacitors is connected to one point and negative plate to the other point. The potential difference across all the capacitors is same but the charge on the plates of capacitors is different, i.e.,

Q = Q1 + Q2 + . . . . . . . . . + Qn

Cp V = C1 V + C2 V + . . . . . . . . . + Cn V

Cp = C1 + C2 + . . . . . . . . . . + Cn

    i.e. the net capacitance is the sum of individual capacitance of all the capacitors.

Capacity of a Spherical Condenser

:

    Consider any two spherical shells of radius r1 and r2. The inner sphere is given a charge q and outer sphere is earthed. If inner sphere is given a positive charge q, there will be negative charge on the inner side of outer shell. If P be any point lying between two shells, then dV is the potential difference between two points situated a distance dr apart around P, then

dV = Edr where E = kq/r2

Potential difference V between A and B is,


When inner sphere is earthed,

    If instead of earthing outer sphere, we give a charge q to outer sphere and earth the inner sphere. If the charge induced on the inner sphere be q1, hence a charge +q1 will be present on inner surface of outer sphere, while +q2 is distributed over outer surface.

q = q1 + q2

    Thus two condensers are formed (i) between spheres A and B having capacity C1 given by 4pe0 ab/(b-a) (ii) between outer sphere and earth having capacity 4pe0b.

    Net Capacity,


Capacitance of Parallel Plate Capacitor with Dielectric Between Its Plates:

    Before finding the capacitance with dielectric between its plates we must know the behaviour of dielectric in the presence of electric field.

Polar and Non Polar Molecules :

    Polar molecules are those which are formed by the combination of two atoms having different electronegativities or we can say that molecules in which the centre of gravity of positive and negative charge do not coincide. As on the whole molecule is to be neutral therefore the magnitude of positive charge is equal to the magnitude of negative charge. Thus this system resembles a dipole and possesses a dipolemoment called its natural dipolemoment. For eg. HCl

    Non Polar Molecules are formed by joining atoms having same electronegativity. In these molecules centre of gravity of positive and negative charge coincide. Thus they do not possess dipolemoment of their own. But when non polar molecule is placed in an electric field, positive and negative charges experiences force in two opposite direction. Thus, molecules now also resembles a dipole and have dipolemoment which is called their induced dipolemoment. At some stage the electric force pulling the charges apart and the electrostatic attractive force balance each other and molecules is said to be polarised.

    Now, if a non polar dielectric slab is placed in an electric field, the atoms get polarised in the direction of . If q is the charge induced in any atom with d be distance between the two charges then total induced dipolemoment will be where is the dipolemoment and is the dipolemoment per unit volume called electric polarisation. As field acts on a dielectric a layer of positive charge is formed on the one side and a layer of negative charge on the other, this positive and negative layer generates an induced electric field . Thus net field inside the dielectric is,


Also,


    Also the ratio of applied electric field to reduced electric field

is called the dielectric constant of the medium. The polarisation is also

found to be proportional to E¢ or



    where K is called the dielectric constant of the material.

Capacitor with Dielectric Slab:

    Consider a parallel plate capacitor with plate area A and distance between capacitor plates d. Its capacitance is given by,


    Also the potential difference between the plates is given by,

    V = Ed where E is the electric field between capacitor plates. Now if we introduce a dielectric slab of thickness t between the plates of capacitor, the potential difference V¢ between plates

V¢ = E (d t) + (E Ei) t


    If instead of dielectric conducting plate is present between the plates of capacitor, then E Ei = 0 or

for conductors.