RTMNU-CAPACITOR-DIELECTRICS- B.Sc.-I

UNIT-II

CAPACITOR

Introduction A capacitor is a device which stores electric charge. Capacitors vary in shape and size, but

the basic configuration is two conductors carrying equal but opposite charges. Capacitors have many important applications in electronics. Some examples include storing electric potential energy, delaying voltage changes when coupled with resistors, filtering out unwanted frequency signals, forming resonant circuits and making frequency-dependent and independent voltage dividers when combined with resistors.A capacitor consists of two metal plates separated by a nonconducting medium known as the dielectric medium or simply the dielectric, or by a vacuum. It is represented by the electrical symbol shown in the figure.

Capacitance of a capacitorIf a potential difference is maintained across the two plates of a capacitor (for example, by connecting the plates across the terminals of a battery) a charge +Q will be stored on one plate and - Q on the other. The ratio of the charge stored on the plates to the potential difference V across them is called the capacitance C of the capacitor.

Thus Q CV

If, when the potential difference is one volt, the charge stored is one coulomb, the capacitance is one farad, F. Thus, a farad is a coulomb per volt. It should be mentioned here that, in practical terms, a farad is a very large unit of capacitance, and most capacitors have capacitances of the

order of microfarads, .The dimensions of capacitance are

Capacitors in ParallelThe potential difference is the same across each, and the total charge is the sum of the charges on the individual capacitor. Therefore:

Capacitors in Series The charge is the same on each, and the potential difference across the system is the sum of the potential differences across the individual capacitances. Hence

V

Capacity of a Parallel Plate Capacitor

Consider a parallel plate capacitor consisting of two conducting plates having area A and separated by a small distance in vacuum or air as shown in the fig. Let and be the surface charge density on the plates due to charges +Q and –Q respectively.Neglecting the edge effect (fringing effect near the edges of the plates) the electric field between theplates can be calculated using Gauss’s law

i.e. or

-------------(1)

where is the permittivity of the free space ( ).

Now the work done W required to carry a test charge from one plate to other plate is equal to the

product of the force and the distance d (as the electric field between the plates is uniform)

i.e. work done the potential difference V between the plates

-----------(2)

Now magnitude of charge on either plate is given by Now capacity of the parallel plate arrangement is given by

(Using (1) and (2))

---------------------(3)Thus capacity of a parallel plate capacitor depends on the size and the geometrical arrangement of the plates . C of a parallel plate capacitor is 1. directly proportional to the area of the plates 2. Inversely proportional to the separation between the plates.

V

3. If the space between the plates is completely filled with a dielectric of dielectric constant K then

capacity is given by i.e. capacity increases by K times when the space between the capacitor plates completely filled with the dielectric. Example: A parallel plate capacitor is to be designed with a voltage rating 1 KV using a material of

dielectric constant 3 and dielectric strength about .For safety we would like, the field never to exceed, say 10 % of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pf ?

Solution: Maximum permissible voltage V =

Maximum permissible electric field E = 10% = Now potential difference between the parallel plate capacitor

or Now capacity of a parallel plate capacitor with dielectric constant K is given by

or

Dielectrics“…As our mental eye penetrates into smaller and smaller distances and shorter and shorter times, we find nature behaving so entirely differently from what we observe in visible and palpable bodies of our surroundings that no model shaped after our large-scale experiences can ever be "true". A completely satisfactory model of this type is not only practically inaccessible, but not even thinkable. Or, to be precise, we can, of course, think of it, but however we think it, it is wrong.”

Erwin Schrödinger

A dielectric is a material which does not conduct electricity ie. a dielectric a basically an insulator. In these materials all the electrons are bound to the nuclei of the atoms. Thus there are no free electrons to carry the current. However when an external electric field is applied these charges gets separated through a small distances creating bound charges. An ideal dielectric is a perfect insulator .Real dielectric have

feeble conductivities nearly times smaller than that of the good conductors. Glass, plastic, mica, oil, water, bakelite etc. are some examples of the dielectric constants.

According to modern theory neutral atoms in their ground state consist of a central positively nucleus surrounded by a spherically symmetric cloud of equal negative charge of smoothly varying density. Thus for an atom in ground state, the centre of gravity of its –ve charge lies exactly at its nucleolus which is taken to be point positive charge. The dipole moment of an atom is therefore zero.For a molecule the positive charge is supposed to concentrate at the nuclear points and the negative change forming a cloud of smoothly varying density. around constituent nuclei. Depending upon the shape of the cloud and variation of charge density inside it the molecules can be classified into two types non-polar molecules and polar molecules.

1. Non-polar moleculesThe molecules in which the centre of gravity of positive charges exactly coincides with that of the negative charges and there are no dipoles are called as non-polar molecules. Net dipole moment is zero

because dipole length is zero. Familiar examples are etc.

2. Polar molecules The molecules in which the centre of gravity of positive charges does not coincide with that of the negative charges are known as polar molecules. Such molecules, therefore constitutes a permanent dipole

and have a dipole moment. Familiar examples are etc. These molecules consist of

dissimilar atoms and their dipole moment is of the order of which means a separation

of between the centers of the positive and negative charges of magnitude one electronic

charge ( ).

Dielectric Constant or Specific Inductive Capacity

The capacity of a parallel plate capacitor in vacuum (free space or air) is given by . Now if the space between the plates is completely filled with a dielectric material it is observed that capacity increases by a factor .i.e. new capacity

= .

This factor is called as the dielectric constant of the medium.Thus dielectric constant of a medium is defined as

Also Coulomb’s law for force of attraction or repulsion between two charges of magnitude separated by a distance r in free space is given by

But when the same charges are placed in some other medium of permittivity , the force of attraction or repulsion is found to be

It is observed that the force between the charges in dielectric medium is reduced by a factor .

i.e. or from which it can be shown that

Where is called as relative permittivity of the medium. Thus dielectric constant of the medium can also be is defined as

i.

ii.

Dielectric Strength ( )

In a dielectric if the applied electric field is increased beyond a certain value, the field will be able to pull out the electrons from the molecules. As a result there will be free electrons and the material becomes a conductor. This called dielectric breakdown. The maximum electric field strength which a dielectric can withstand without breakdown is called the dielectric strength of medium or material .

Dielectric in an Electric Field Consider a dielectric material made up of non-polar molecules. These molecules do not have any electric

dipole moment by themselves. However when an external electric field is applied, the centre of positive charges is pulled in the direction of electric field ,while the centre of negative charges is pulled in the direction opposite to the direction of electric field . As a result these non-polar molecules turns into polar molecules and do acquire a dipole moment due to induction. The induced dipole moment is found to increase with the strength of the applied external electric field and almost independent of temperature. The induced polar molecules are aligned in the direction of external field and cause the surface charges to be induced on the opposite faces of the dielectric. This phenomenon is called as the polarization of

dielectric. The induced charges produce an internal electric field ( always) inside dielectric in

the direction opposite to the externally applied electric field. So the resultant electric field is less than

the applied external electric field .The resultant electric field is given by

If K is the dielectric constant of the dielectric material then the electric field inside the deictic material is reduced by a factor of dielectric constant K

or In case of polar dielectrics, molecules are permanent dipoles and have net dipole moments. However in absence of external electric field these dipoles are randomly oriented in all possible directions due to thermal agitations. So net dipole moment of these dielectrics in the absence of external electric field is zero .When an external electric field is applied these molecular dipoles experience torque acting on them and tend to align in the direction of electric field. This alignment of dipoles is opposed by the random thermal motion of the molecules. The alignment of the dipoles is directly proportion to the applied external electric field

and is inversely proportional to the temperature. If is the intensity of externally applied electric

field and is the internal electric intensity due to polarization of dielectric. Then resultant

electric field intensity is

Electric Polarization Vector ( )When dielectric is placed in an electric field, its molecules become electric dipoles and the dielectric is

said to polarized. When a dielectric is polarized the induced charge appears on the surface area the dielectric material and therefore electric polarization is defined as ‘the induced surface charge per unit area’.

i.e. multiplying and dividing Numerator and denominator with d, the thickness of the dielectric material slab.

(since )

Now = induced electric dipole moment Thus

Thus polarization P is also defined as the induced dipole moment per unit volume.Since dipole moment is a vector quantity polarization is a vector quantity. The direction of polarization

vector is same as the direction of resultant electric field inside the dielectric. The polarization is

.

Capacity of a parallel plate capacitor partially filled with a dielectric

Consider a parallel plate capacitor having capacity in free space. Let d be the separation and A be the area of each plate. Capacity of parallel plate capacitor in free space is given as

---------------(1)

Let be the charge on each plate. The electric field in the free space between the plates of the capacitor can be calculated by Gauss’s law

or

or -------------------(2)Suppose a dielectric slab of thickness t and dielectric constant K is inserted between the plates as shown

in the figure. The electric field remains same as in the free space between the dielectric slab and the plate. However electric field inside the dielectric is reduced by a factor K i.e. inside the dielectric electric field E is given by

----------------------(2)

Now the work done required to carry a test charge from one plate to other is equal to work done in

the free space between the dielectric slab and capacitor plates and work done inside the dielectric.

total work done

i.e. ( )

Therefore potential difference between the plates is

Now, Capacity of a parallel plate capacitor with dielectric slab is given by is given by

------------------(3)

or ----------------(4)

From (3) it is clear that effect o introducing a slab of thickness t is the same as decreasing the free

space by an amount . Thus the capacitance of a parallel plate capacitor is increased when the space between the plates is filled with a dielectric. Special Case:Capacity of a parallel plate capacitor completely filled with a dielectricIf the space between the plates is completely filled with a dielectric then

Substituting in (4) we get

or

Thus the capacity of a parallel plate capacitor is increased by a factor dielectric constant when the space between the plates is completely filled with a dielectric.Note : If the space between the plates is partially filled with a conducting slab then effective separation between the plates is reduced to (d-t) .Hence the capacity of a parallel plates increases.

Example1:The distance between the plates of a parallel plate capacitor of a capacitance is .A slab of

dielectric constant and thickness is inserted between the plates. What is the capacitance of the system? [RTMNU Summer-2007, 2 Marks]

Solution: We know that capacity of a parallel plate capacitor partially filled with a dielectric is

Here t

i.e.

But original capacity without dielectric is

Electric Displacement (D)

Electric field can be represented by electric lines of force These lines of force originates from the positive charges and terminates at the negative charges. Suppose one lines of force emanates from a unit charge. Then number of lines force coming out from charge will be equal to the magnitude of the charge i.e. .Imagine a hallow sphere of radius r with a charge at its centre. The total number of lines of force immerging out of the surface of hallow sphere normally will be equal to .

Now the surface area of the hollow sphere is .The number of lines of force coming out through a unit

area is .Maxwell named this quantity as the electric displacement D

In general electric displacement D is equal to the surface charge density of free charges.

i.e. Also electric intensity E at a point at a distance r from the charge is given by

we may write

Thus Where is the permittivity of the medium.Electric displacement is a vector quantity and is directed along the same direction as electric intensity.

The SI unit of is

Note: The electric displacement vector represents a partial field in the sense that its sources are free charges (NOT bound charges). It has nothing to do with the induced charges due polarization of the dielectric materials.

Relation between , and

Consider polarization of a dielectric slab placed between the plates of a parallel plate capacitor. Let

and be the charges on the plates of the capacitor then the surface charge densities on the plates of the capacitor is

and Now the electric displacement is equal to the surface charge density due to free charges.

The electric field intensity in the free space between the capacitor plates is

---------------(1)

Due to polarization of the dielectric charges are induced on the opposite faces of the dielectric. Let

and be the induced charges on the faces of the dielectrics. Consequently the induced surface charge densities is

and The electric field inside the dielectric due to induced charges is

----------(2)

This electric field is in the direction opposite to the to electric field due to free charges. Hence the resultant electric field inside the dielectric is

i.e.

or

Now is the induced surface density on the face of a dielectric and is equal to the magnitude of the

electric polarization .

or ----------(3)In vector form

------------(4)

In the free space where no polarization charges are present = 0, (4) may be written as

-------------(5)

Gauss’s Law in term’s of electric displacement vectorConsider the situation in which the charge is distributed over a volume V such that is the volume charge density. Then the charge enclosed by surface enclosing volume V is

-----------(1)Now According to Gauss’s law in electrostatics electric flux through closed surface enclosing volume V is

Where is the permittivity of the medium.Above equation may be written as

or --------(2)

But

Therefore in terms of (2) may be written as

------------(3)

This is Gauss’s Law for electrostatics in terms of displacement vector

Using (1)

-------------(4)

Applying Gauss’s divergence theorem surface integral in LHS of (4) may be written as volume integral as

Since the volume is arbitrary this is true for all the volumes therefore integrands must be equal.

---------------(5)

or

or ----------------(6)

(5) and (6) are differential forms of the Gauss’s law in electrostatics.

Schematic Representation of , and

Boundary conditions Satisfied by and

(A) Boundary conditions satisfied by Let’s consider a boundary separating the two medium as shown in the fig Construct a cylindrical Gaussian pillbox of height and base area

Let Average normal component of displacement vector to the bottom of the box in medium 1. It

is inward normal.

Average normal component of displacement vector to the face of the box in medium 2By making the height of the cylinder approaching to zero ( ), the contribution to the total outward flux from the curved surface can be made zero.Now, by Gauss’s law the net outward flux through the cylindrical Gaussian pillbox is

Where is the total charge enclosed by the ccylindrical Gaussian surface.

But

( )

Where Charge per unit area on the boundary of the two medium.

Thus the normal component of the displacement vector changes at the charged boundary between the two dielectric by an amount equal to the surface charge density .

However if the boundary is free from charge i.e. , then

i.e. Normal component of the displacement vector is CONTINEOUS across the charge free boundary between the two dielectrics.

(B) Boundary Conditions Satisfied by

Fig

Let’s consider two Electric fields and in the two media. Here we will make use of the fact that electrostatic force field is conservative in nature i.e. around any closed path, the potential difference vanishes. In other words the line integral of electric field around any closed path is zero.

Consider a rectangular path ABCD as shown in the fig.

Let AB and CD each have length and BC and AD being negligibly small.

Now

i.e. Since paths BC and DA are negligibly small

Thus

Now and

Where Tangential components of and

Tangential components of

(As can not be zero) Thus tangential components of same on both sides of a boundary between the two dielectrics.

Or The tangential components of electric fields are continuous across the boundary.

Example :Let two isotropic dielectric media be separated by a charge free plane boundary as shown in

the figure .Let the permittivities be .Then show that

.where and are dielectric constants of the two medium.Solution: As shown in the figure medium 1 and medium 2 are separated by a charge free boundary

i.e. surface charge density on the boundary is zero. The boundary conditions on and are

----------(1)

and ----------(2)Now from fig.

i.e. normal components of is

and

so that (1) can be written as

--------------(3)

Also

and

so that --------------(4)Taking ratios of (4) and (3)

But and

i.e.

So that

If and are relative permittivities (dielectric constant) then

and

Or

Molecular PolarizabilityWhen a dielectric is placed in an external electric field its molecules becomes electric dipoles oriented in the direction of the field. Thus the dielectric acquires a net dipole moment and its molecules are polarized.For linear dielectrics induced dipole moment is directly proportional to the electric field intensity causing polarization.

i.e.

where is constant of proportionality to known as the polarizability.

If then Thus polarizability is defined as the induced dipole moment per unit electric field strength.

SI unit of polarizability is .Contribution to the total polarizability is due to following types

1. Electronic Polarizability The displacement of the electron cloud relative to the nucleolus in an atom constituting the molecules induces the dipole moment in a molecule. This is called electronic polarizability.

2. Ionic polarizability

If we have a solid dielectrics whose molecules are made up of ions there is relative motion of positive and negative ions resulting induced dipole moment. This is called ionic polarizability.

3. Orientational or dipolar polarizability If there are molecules with permanent dipole moments, randomly oriented, they tend to align in the direction of the applied electric field producing a net dipole moment. This is called orientational or dipolar polarizability. Thus total dielectric polarization may be considered as sum of these three contributions

Subscripts e, i, or o stands for electronic, ionic and orientational polarizability respectively. 32-

Larentz Local Field (Internal molecular field)When a dielectric solid is placed in an external electric field a molecule or an atom of solid experiences not only the external field but field produced by the dipoles as well. This resultant field experienced by the atom or molecule is called internal molecular field or Larentz local field or simply local field.

Calculation of Local FieldConsider a dielectric, uniformly polarized by placing it in a uniform electric field between two oppositely charged parallel plates. In order to calculate the local field experienced by the atom consider a small spherical cavity with the atom for which local field is to be calculated at its centre. The radius of the cavity is chosen large enough. The part of the dielectric external to the spherical cavity can be replaced by a system of charges induced at the surface of cavity.

The net local field at the point at the centre of the spherical cavity is given by

----------------(1)where

1. externally applied electric field due to the

charges on the capacitor plates. 2 Electric field at O due to induced charges on

the plane surface of the dielectric

3. Field at O due to polarized charges on the surface of the cavity.

4. Field at O due to permanent atomic dipoles . But in the present case i.e. for nonpolar isotropic

dielectric

Thus local field

Now, effectively produces a net field .So that

----------------(2)

Calculation of Consider a small elementary area ds surrounding a point A on the surface of the cavity of angular width and at an

angle with the direction of field .

The polarization vector will be parallel to

the component of normal to area is

Since polarization is induced surface charge per unit area, the total charge on area dS is

the electric intensity at O, due to this elementary

charge is

------------(3)

This field is directed along OA. Resolving this field

intensity along the applied field (parallel ) and perpendicular to it we have the component

along the field

i.e. along the field

The perpendicular component of at O is zero because they cancel out each other.

If the area be a ring shaped element as shown in fig. of radius and width , on the surface of sphere, then area of this element is

Thus parallel component of due to this ring is

the field at O due to the entire induced charge on the spherical cavity is :

Substitute

when and when

or --------(4)

(2) becomes -------(5)

This (5) is known as Larentz relation gives the actual field acting at the position a molecule at O within the dielectric.

Clausius- Mossotti EquationClaussius and Mossotti tried to correlate the dielectric constant (macroscopic parameter) to the molecular polarizability (microscopic parameter) of a non-polar dielectric. This relation is known as Claussius –Mossotti equation.The dipole moment of a single molecule is,

Now, if there are n molecules per unit volume then polarization P is given by

------------(6)But we know that in a dielectric

(In dielectric medium of permittivity )

or

(6) may be written as

------------------(7)Molecular polarizability is given by

-----------------(8)This equation is called as Claussius-Mossotti equation. The claussius-Mossotti equation holds best for dilute substances such as gases. For liquids and solids, this relation is only approximately correct.