PPT11-Gauss Law in Dielectrics

PPT No. 11

* Gausss Law for Dielectric Materials, * Electric Displacement Vector D* Permittivity * Susceptibility * Dielectric Constant

Gauss Law for Dielectric Materials

Electrostatic field in the dielectric material is modified due to polarization and is not the same as in vacuum.

Hence the Gauss law

which is applicable in vacuum needs to be reconsidered for dielectric media.

Gauss Law for Dielectric Materials has two forms A) Integral B) Differential

A) Integral Form of Gauss Law

Fig. parallel-plate conductors (a) without (b) with dielectric


(I) Consider two parallel-plate conductors in vacuum having plane area S, separation d and vacuum between plates.

Let +q and q be the charges deposited on the plates.

Due to the charges, E0 is the uniform electric fielddirected from positive to negative plate (Fig. a).

Consider the Gaussian surface, In the form of a rectanglearound the upper conducting plate of positive charges.


Applying Gausss law the electric flux passing through the closed surface is given by

The field is

It is normal to the plate surfaces.


(II) Consider that a dielectric material of permittivity and dielectric constant k is filled completely between the plates

Charges q and +q are induced on the dielectric surfaces that are in the proximity of the plates having charges q and -q respectively. (Fig.b).

The induced charges set up an electric field Eindin the dielectric.The dielectric is polarized.

The dielectric remains as a whole electrically neutral as the positive induced surface charge must be equal to the negative induced surface charge.



In the presence of the dielectric the surface encloses two types of charge:

Free charge on the upper conducting plate is q and Induced charge on the top face of dielectric is -qdue to the polarization of the dielectric material

The net charge enclosed by the Gaussian surface around the (same upper conducting) plate (of positive charges +q) is q-q.



According to Gausss law

Field E in the dielectric is in the opposite direction to that of the applied electric field E0.

The effect of the dielectric is to weaken the original field E0 by the factor k= / 0.

Fig. E-Field Lines in a dielectrcic due to Applied Field E0 & Induced Field Eind


The magnitude of the net induced charge q` is always less than magnitude of the free charge q applied to the plates & is equal to zero if dielectric is absent



Where D = E = 0 k E (k= / 0).

The induced surface charge is purposely ignored on the right side of this equation, since it is taken into account fully byintroducing k on the left side.

D is called as the displacement vector.


The equation states that the surface integral of displacement vector Dover a closed surface = the free charge enclosed within the surface or

The outward flux of D over any closed surface S equals the algebraic sum of the free charges enclosed by S


This important equation, although derived for parallel plate conductors, is true in general.

It is the most general form of Gauss law.

The charge q enclosed by the Gaussian surface is the free charge only, which can be controlled and measured.

Hence this form of Gauss law is very useful.

B) Differential Form of Gauss Law

A dielectric material kept in an electric field is polarized.

It has bound or polarization charge density bdue to accumulation of bound charges


The applied electric field itself is created by transferring electric charges.

They are called as free charges (brought from outside e.g. conduction electrons in metals).

They give rise to f the charge density due to free charges i.e. not due to polarization.

The total charge density consists of two parts


According to Gauss' law in differential form

E is the total electric field due to both types (bound and free) charges.

Rearranging the expression and substituting


The differential form of Gauss Law in dielectrics

This is the differential form of Gauss Law in dielectrics.

D is termed as the electric displacement.

D has the same dimensions as P(dipole moment per unit volume).

This is Gauss law in Dielectrics In terms of D

It can be written in terms of Eusing relation D = E => . E = f

The other equation in electrostaticsx E = 0remains unchanged in dielectrics.


Gauss Law in Dielectrics

According to the divergence theorem the differential form changes to integral form as

The flux of D out of a closed surface S is equal to the total free charge enclosed within that surface.

Thus the statement of Gausss Law in integral form can be obtained from differential form.

It can be derived from first principles also. (Refer to Integral form)

Susceptibility

For most linear, isotropic, homogeneous dielectric materials, the polarization P is is aligned with and directly proportional to the average electric field intensity Eso that the ratio of the two, P/E, is a constant.

e = P/E in centimetre-gram-second (cgs) systeme = P/(0E) in metre-kilogram-second (mks) system

(0 = Electric constant)

In MKS, the constant of proportionalityis usually written as 0e

(to make e dimensionless).

Susceptibility

e expresses an intrinsic property of the material called as the Electric.

The relation between vectors P and E becomes

(In an anisotropic material, the polarization P and the field Eare not necessarily in the same direction.

Then the Electric Susceptibilityof the medium is a tensor quantity instead of e)

Susceptibility

Susceptibility

The electric susceptibility e of a dielectric material is a measure of the ease of polarization in applied electric field

It is also called as dielectric susceptibility.

It is a dimensionless parameter having positive value.

Due to the difference in definition, the value of the electric susceptibility of a given material in the mks system is 4 times its value in the cgs system

e in the mks system = 4 times e in the cgs system

Susceptibility

Permittivity

The relation between Electric Displacement D & the polarization density P by

The displacement field D is also proportional to electric field intensity ED = E where = r 0

The new constant is called the Permittivity of the material

Permittivity is directly related to electric susceptibility .

D = E

Permittivity

Permittivity

Permittivity is a measure of the ease /degree to which molecules of some materials polarize (align) under the influence of an electric field.

Permittivity () is a physical quantity that describes how an electric field affects and is affected by a dielectric medium,

It is determined by the ability of a material to polarize in response to the field & reduce the field inside the material.

Permittivity

Permittivity relates to the ability of a material to transmit (or "permit") an electric field.

In a capacitor, an increased permittivity allows the same charge to be stored with a smaller electric field (and thus a smaller voltage) which results in an increased capacitance.

In SI units, permittivity is measured in farad per meter (F/m);

0 = 8.85 1012 F/m is the Permittivity of vacuum or free space.

Permittivity

Susceptibility e and Relative permittivity

Relation between r the relative Permittivity & susceptibility r = 1 + e

[e = P/(0E)]

e = r - 1.

e = r - 1.

The relation between electric susceptibility of a medium e to its relative Permittivity r implies that

the Electric susceptibility is a measure of that part of the relative permittivity which is due to the material itself.

Susceptibility e and Relative permittivity

In the case of a vacuum,

Susceptibility determines the electric Permittivity of the material.

It has influence on many physical phenomena in the medium

Dielectric constant

*The dielectric constant is a characteristic quantity of a given dielectric substance.

The dielectric constant is the relative Permittivity of a dielectric material i.e. the ratio of the Permittivity of a substance to the permittivity of free space.

* It is denoted by k k = / 0

Dielectric constant

k = / 0

Since the dielectric constant is just a ratio of two similar quantities, it is dimensionless.

* The dielectric constant of free space / vacuum is 1.

It is a measure of the extent to which a material concentrates electrostatic lines of flux.

Dielectric constant

It can also be expressed as the ratio of the electric field without the dielectric (Eo) to the net field (E) with (i.e. in the presence of) the dielectric

= Eo / E

E Eo , so the dielectric constant 1.

Dielectric constant

As the dielectric constant increases, the electric flux density increases, if all other factors remain unchanged.

This enables objects such as sets of metal plates, to hold their electric charge for long periods of time, and/or to hold large quantities of charge.

Dielectric constant

= C / Co, where Co is the capacitance with no dielectric between the plates

Materials with high dielectric constants are useful in the manufacture of high-value Capacitors.

is an important parameter in characterizing Capacitors

In a region completely filled by material of dielectric constant k, all electrostatics involving the permittivity constant 0 is to be modified by replacing 0 by k0

Dielectric constant

Documents

PPT11-Gauss Law in Dielectrics