Microscopic picture of a dielectric in a uniform electric field-review:

In an electric field, the atomic nuclei and electrons of the dielectric material experience forces in the opposite directions. We are familiar that the electrons in a dielectric can't move freely as in a conductor. Therefore each atom becomes a tiny dipole having the positive and negative charge centers slightly separated. Taking the charge separation as 'a', the charge as 'q' the dipole moment 'p' in the direction of field related by the atom or molecule:

p = qa

The equation above represents the dipole moment induced in the atom or molecule by the field. Therefore, we call it the induced dipole moment. When there are 'n' such dipoles in an element of volume 'V' of the material, we can state the polarization vector 'P' as the (dielectric) dipole moment per unit volume as:

p = npV/V

In the dielectric, the charges neutralize one other, the negative charge of one atom or molecule is neutralized through the positive charge of its neighbor. Therefore in the bulk of the material, the electric field generates no charge density however only a dipole moment density. Though, at the surface this charge cancellation is not complete, and polarization charge densities of opposite signs come out at the two surfaces perpendicular to the field. Now what is the effect of the appearance of polarization charges?

The effect of this is that the electric field within the dielectric is less than the electric field causing the polarization. The polarization charges give mount to an electric field in the opposite direction. This field opposes the electric field causing the polarization.

Therefore we conclude that within the dielectric, the average electric field is less than the electric field causing the polarization. Though, the macroscopic or average field is not a reasonable measure of the local field responsible for the polarization of each and every atom.

Let us represent the field at the site or position of the atom or molecule as the local field.

Definition of Local Field:

It is the field on a unit positive charge kept at a place or site from which the atom or molecule has been eliminated provided the other charges remain unchanged.

The degree of the charge separation based on the magnitude of the local field. Therefore we conclude that the induced dipole moment 'p' is directly proportional to the local field 'E_loc'. Therefore we have:

P = α E_loc

Here, 'α' is the constant of proportionality and is termed as atomic or molecular Polarisability and E_loc the local field.

Determination of local field: electric fields in cavities of a Dielectric

The polarization of dense materials like liquids and most of the solids changes the electric field within the material. The field experienced by an individual atom or molecule based on the polarization of atoms in its immediate vicinity. The real value of the field differs rapidly from point to point. Much close to the nucleus it is extremely high and it is relatively small in between the atoms or molecules. By obtaining the mean of the fields over a space having an extremely large number of atoms one gets the average value of the field.

Clausius-Massotti Equation:

In a liquid we would anticipate or suppose an individual atom to be polarized through a field acquired in a spherical cavity instead of by the average (that is, macroscopic) field.

Therefore, by using: P = α E_loc and E_loc = E + (P/3ε_o)

P = n α E_loc

P = n α E + (P/3ε_o)

The above can be rewritten as:

P = [nα/{1- (nα/3ε_o)}] E

The susceptibility 'χ' was stated by the equation:

P = ε_oχE

Therefore, [(nα/ε_o)]/[1 - (nα/3ε_o)]

The equation above symbolizes the relation between the susceptibility and atomic or molecular Polarisability. This is one of the forms of the Clausius-Massotti Equation.

Polarization in a Gas:

Dissimilar the atoms or molecules of a liquid or solid it is likely to consider the atoms or molecules of a gas as far apart and independent. We can overlook the field due to the dipoles on the immediate neighborhood of an individual molecule. Therefore the local field causing polarization is the average or macroscopic field 'E'. Thus we can write:

P = ε_oχE = np

Here, 'n' is the number of molecules per unit volume. When we consider only an individual atom or molecule and write the dipole moment 'p' as:

P = ε_o α E

Here, 'α' is termed as the atomic Polarisability. Thus 'α' consists of the dimensions of volume and roughly equivalents the volume of an atom.

We can associate α or χ to the natural frequency of oscillation of electrons in the atom or molecule. When the atom is positioned in an oscillating field 'E' the centre of charge of electrons follows the equation:

m (d²x/dt²) + m ω_o²x = qE

Here, 'm' is the mass of electron of charge 'q'; m ω_o²x is the restoring force term and qE is the force from outside field - this equation is similar as the equation of forced oscillation. If the electric field differs with the angular frequency 'ω' then,

x = qE/m(ω_o² - ω²)

For our rationales in the electrostatic case ω = 0 that signifies that:

x = qE/mω_o²

And the dipole moment 'P' is:

P = qx = q²E/mω_o²

From the equation P = ε_o α E we can write the atomic Polarisability as:

α = q²/ε_omω_o²

And P/E = ε_o = εχ (ε_o -1) = ε_o n α

Relation between Polarisability and Relative Permittivity:

α = (3ε_o/n) [(ε_r - 1)/(ε_r + 2)]

Relation between the Polarisability and Refractive Index:

For the dielectric, the refractive index 'µ' stated as the ratio of the speed of light in vacuum to the speed in the dielectric medium, can be represented to be equivalent to √ε_r

µ² = ε_r

α = (3ε_o/n) [(µ² -1)/(µ² + 2)]

The equation above represents the relation between Polarisability and refractive index. This relation is termed as the Lorentz-Lorenz formula.

Role of Dielectric Capacitor in our practical life:

Dielectrics encompass quite a few applications. Dielectrics are employed broadly in capacitors. However the real needs differ based on the application; there are some features that are desirable for their utilization in capacitors. A capacitor must be small, encompass high resistance, be capable of being employed at high temperatures and encompass long life. From a commercial viewpoint it must as well be cheap. Particularly prepared thin kraft paper, free from holes and conducting particles, is employed in power capacitors where withstanding high voltage stresses is more significant than incurring the dielectric losses. Moreover, the Kraft paper is saturated having an appropriate liquid like chlorinated diphenyl. This raises the dielectric constant and therefore decreases the size of the capacitor. Moreover, the breakdown strength is raised.

Moreover to paper capacitors for general purpose, other kinds of capacitors are employed. In the film capacitors, thin film of Teflon, Mylar or polythene is employed. These not only decrease the size of the capacitor however as well has high resistivity. Teflon is employed at high frequencies as it consists of low loss. In electric capacitors, an electrolyte is deposited on the impregnating paper. The size of such a capacitor is small as the film is extremely thin. Polarity and the maximum operating voltage are significant specifications for such capacitors.  

A few ceramics can be employed as temperature compensators in the electronic circuits. High dielectric constant materials, where small variations in the dielectric constant having temperature can be tolerated, help miniaturize capacitors. Barium titanate and its amendments are the best illustrations of such materials.

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