Linear Isotropic Dielectrics

Electric Susceptibility. In many substances, the polarization is proportional to the field,
            P = ε₀ χ_eE                                … (1)
Provided E is not too strong. The constant of proportionality, χe, is called the electric susceptibility of the medium. Materials which obey Eq. (1) are called linear isotropic dielectrics. E in eq. (1) is the total field. It may be due in part to free charges and in part to the polarization itself.

Permittivity, Dielectric Constant. In linear media, we have
        D = ε₀ E + P = ε0 E + ε₀ χ_e E = ε₀ (1+χ_e ) E = K ε0 E = εE
    . :    D = εE                                        … (2)

Here, ε = ε₀ (1 + χ_e)                                        … (3)
 ε is called the permittivity of the material.

K = 1 + χ_e = ε/ε₀ = dielectric constant = relative permittivity                        … (4)

Relation between Bound and Free Charge Densities

The volume bound charge in a homogeneous linear dielectric is proportional to the density of free charge:
            ρ_b = - ∇. P = - ∇. (ε₀ χ_e /ε D) = - (χ_e /1 + χ_e ) ρf                 … (5)