Expression for Internal field in the Case of Liquids and Solids (One Dimensional) - Civil Engineering (CE) PDF Download

Expression for Internal field in the case of Liquids and Solids (One dimensional):
 Definition of Internal Field or Local Field:
When dielectric material is placed in the external electric field, it is polarized creating electric dipoles. Each dipole sets electric field in the vicinity. Hence the net electric field at any point within the dielectric material is given by “The sum of external field and the field due to all dipoles surrounding that point”. This net field is called internal field or Local field.

Expression for Internal field:
Consider a dipole with charges ‘+q’ and ‘q’ separated by a small distance ‘dx’ as shown in fig. The dipole moment is given by µ = qdx. Consider a point ‘P’ at a distance ‘r’ from the center of dipole.
The electric field ‘E’ at point ‘P, can be resolved into two components.
(1) The Radial Component along the line joining the dipole and the point ‘P’ is given by 
(2)The Tangential component or Transverse component perpendicular to the Radial component is given by 

Where ‘θ  is the angle between the dipole and the line joining the dipole with the point ‘P’, ‘ε0’ is permitivity of free space and ‘r’ is the distance between the point and dipole. Consider a dielectric material placed in external electric field of strength ‘E’.

Consider an array of equidistant dipoles within the dielectric material, which are aligned in the field direction as shown in the figure.

Let us find the local field at ‘X’ due all dipoles in the array.
The field at ‘X’ due to dipole ‘A’ is given by EX_A = E_r + E₀

The field at ‘X’ due to dipole ‘B’ is given by EX_B = E_r + E_θ

Hence the total field at ‘X’ due to equidistant dipoles ‘A’ and ‘B’ is given by

Similarly the total field at ‘X’ due to equidistant dipoles ‘D’ and ‘F’ is given by

The net field at ‘X’ due to all dipoles in the array is given by

In 3-dimensions the above equation can be generalized by replacing 1/a³ by ‘N’ (where ‘N’ is the number of atoms per unit volume) and 1.2/π by γ (where γ is called Internal Field Constant).

but polarization P = NαE ∴ Ei = E + γ P/ε₀

Since γ, P and ε0 are positive quantities E_i > E. For a Cubic Lattice γ = 1/3 and the Local field is called Lorentz field,It is given by
E_L = E + P/3ε₀

Clausius-Mosotti Relation
Consider an Elemental solid dielectric material. Since they dont posses permanent, dipoles, for such materials, the ionic and orientation polarizabilities are zero. Hence the polarization P is given by
P = Nα_eE_L where E_L is Lorentz field

Where ‘N’ is the number of dipoles per unit volume, αe is electronic polarizability, ε₀ is permitivity of free space, and E is the Electric field strength.
The polarization is related to the applied field strength as given below

where ‘D’ is Electric Flux Density and ε_r is Dielectric Constant. Equating equations (4.1) and (4.2)

The above equation is called Clausis-Mosotti relation. Using the above relation if the value of dielectric constant of the material is known then the electric polarizability can be determined using