Molecular Field in a Dielectric

The electric field which is responsible for polarizing a molecule of the dielectric is called the molecular field, Em. This is the electric field at a molecular position in the dielectric. It is produced by all external sources and by all polarized molecules in the dielectric. It is produced by all external sources and by all polarized molecules in the dielectric with the exception of the one molecule at the point under consideration.

The molecular field can be calculated in the following way. Let us suppose that the dielectric sample has been polarized by placing it in the uniform electric between the plates of a parallel plate capacitor. It will be assumed that the polarization is uniform, and the P is parallel to the field producing it. Suppose we want to compute the field at position O of a molecule. We assume that this molecule is not present at all. We draw a sphere of radius r around O. We regard the dielectric as consisting of two parts:
(a)    The dielectric outside the sphere is treated as a continuum of dipoles. The part of the dielectric outside the sphere may be replaced by a system of polarization charges.
(b)    The molecules inside the sphere are treated as individual dipoles.



The molecular field at O consists of four components:
    E_m = E₀ + E₁ + E₂ + E₃                                … (1)

(1)    E0 is the external field due to the charged plates of the capacitor. It is given by
E₀ = σ/ε₀ = D/ε₀                                     … (2)
where σ is the free charge density on the capacitor plates.

(2)    E₁ is the depolarizing field due to the bound charges on the outer faces of the dielectric. It has the value,
E₁ = P/ε₀                                     … (3)

(3)    Let us evaluate E₂, the field at O due to the bound charges on the surface of sphere. Let dS be a surface element of the sphere with polar coordinates r, θ. The component of polarization P normal to dS is P cos θ. Therefore, charge per unit area over the element dS is P cos θ. The electric field at O due to the charge over area dS is 1/4πε₀ (P cos θ) dS/r². It s directed along r or OA.

The component of the field in the direction of E is
    (1/4πε₀ P cos θ dS /r²) cos θ = P cos² θ dS/4πε0 r²

The area of the ring element (shown shaded) is
    dS = 2π (r sin θ) (r dθ) = 2πr² sin θ dθ

The component of the field at O perpendicular to E due to this ring is zero, since such components are symmetrically distributed around the axis. The component of field along the direction of E is
    P cos²θ (2πr² sin θ dθ)/ 4πε0r² = P/2ε₀ cos²θ sin θ dθ

The field (E₂) at O due to the entire induced charge on the surface of the sphere is
    E₂ = P2ε₀ ∫ cos₂ θ sin θ dθ = P/3ε₀
Or     vectorially,     E₂ = P/3ε₀

(4)    E₃ is the field due to the polarized molecules within the sphere. For many practical cases of gases, liquids and cubic crystals, this term vanishes i.e.,
E₃ = 0

Eq. (1) now becomes,
            E_m = D/ε₀ – P/ ε₀ + P/3ε₀                             … (5)

Or            E_m = D – P/ ε₀ + P/3ε₀                             … (6)

Or            Em = E + P/3ε₀                                 … (7)
Where, E = (D – P)/ ε₀, is the macroscopic electric field in the dielectric.
Eq. (7) gives the electric field Em that acts upon a single molecule of the dielectric.

Clausis-Mossotti Relation. Under the action of this field E_m, the molecule displays an electric moment P in the direction of the field which is dependent upon the field strength, i.e.,
        p = α E_m
where α is the molecular polarizability. If there are n molecules per unit volume of the dielectric, then the polarization (electric moment/ volume) is obtained as,
        P = np = nα E_m
         = nα (E + P/ 3ε₀)                                … (9)
But by definition, P = (ε_r – 1) ε₀E
Eq. (9) now becomes
        (ε_r – 1) ε₀ E = nα [E + (ε_r – 1) ε₀E/3ε₀]

Or            α = 3ε₀/n (ε_r – 1)/(ε_r + 2)                            … (10)

This equation is called Clausius-Mossotti relation and gives α, the molecular polarizability.



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