Relation Between Polarization Vector and Density of Polarisation

Consider a polarized dielectric slab placed in a uniform electric field E. The dielectric surface is perpendicular to P. q_p and - q_p are the net charges on the surface. Let σ_p be the density of polarization charge on the surfaces, P the polarization vector, A the area of cross-section and l the length of the slab. Then,

Dipole moment of the slab = PAl                                … (1)
We may also consider this slab as a single large dipole with surface charges q_p (= σ_p A) separated by a distance l.

Dipole moment = charge x length = σ_p Al                            … (2)
Comparing Eqs. (1) and (2)
        P = | σ_p |

Thus the polarization P is numerically equal to the surface charge density σ_p.

In the case when the slab edge is not normal to the polarization vector, the surface charge density is given as
            σ_p     = P.n                            … (4)
where n is the unit (normal) surface vector.



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