Week 6 Lex 1: Dielcetrics II

In this lecture we continue with our discussions about Electrostatics in a Dielectric medium. Having obtained the field at a point outside a dielectric material due to the surface and volume charges developing in the dielectric material due to Polarisation,
we now turn to study the internal microscopic field in the dielectric. For the field outside we used the potential for a pure dipole, however polarisation which is due to charge displacements involve physical dipoles. Outside the material since the field point is far away this does not matter as the dipole potential dominates ( atom is neutral so it has zero net charge and hence no monopole moment. The discreteness of the molecular dipoles is also negligible far away and a continuous Polarisation vector field is justified. This is however not the case inside.


The microscopic field inside the dielectric material is very complex and impossible to calculate. But we can focus on macroscopic or average behavior which is the one observable in experiments. The macroscopic or the large scale field is obtained as an average of the microscopic field over a region is large enough to neglect discreteness of molecular dipoles but small enough so that we do not lose all the field variations. This is roughly of the order of 1000 molecular lengths.


For the macroscopic field inside the dielectric, we consider a spherical surface
of radius R around that point. The field has two contributions  
E=E_out + E_in
(i) E_out from all the charges (molecular dipoles) outside S the average field is the field they would produce at the center. This is given by just the dipole potential for the dipole moment density outside integrated over the region OUTSIDE.
(ii) E_in from all the charges ( molecular dipoles) inside S the average field is the field due to a dipole of dipole moment p which is the total dipole moment within the sphere assuming the sphere S to be small enough so that the polarisation vector P is roughly constant.


Now the term left out of the integral in (i) is due to the field at the center of a uniformly polarised sphere. This is calculated in eg 4.2 in Griffiths and it is exactly the same as that of the average field inside S. So this term puts back in the region that was left out of the integral for V due to outside charges.

So the final relation is a potential V due to all dipoles inside the full volume of the dielectric material and hence the integral for V any now be written as over the full volume of the dielectric material. This analysis justifies the use of the V due to dipoles for the macroscopic field both inside and outside the dielectric material.
This is not matter how complicated the internal ( microscopic) field is, for macroscopic fields we can replace it with a smooth distribution of pure dipoles. The argument is independent of the spherical shape S used over here and holds for any general shape.


Gauss Law in Dielectrics: We now turn to the form of the Gauss law in a Dielectrics. As Gauss Law refers to the total enclosed charge only, the differential form for Gauss Law will now be
∇.E=ρ_free + ρ_bound
where ρ is the charge density due to the free charges and the bound charges respectively. Neglecting the surface bound charges, the volume bound chrage density
ρ_b=-∇.P where P is the Polarisation vector. Rearranging terms we get
∇.[ε₀ E +P]=ρ_free
or ∇.D=ρ_free where D=ε₀ E +P is called the Electric Displacement Vector .


The Displacement vector only refers to the free field charge density. We can now write down the integral form of the Gauss Law in Dielectrics by integrating both over some volume τ bounded by a closed surface S and then using the Divergence Theorem to convert the LHS to a surface integral over the closed surface S. So the closed surface integral
∫_S D. da =Q_free^enc
where Q is the total free charge enclosed within the closed surface S. The surface bound charges are ignored in the edge thickness approximation where the polarisation falls rapidly to zero at the edge of the dielectric and hence no surface charge density.


Although the displacement vector D sees only the free charges it is important to notice that it cannot be used as a substitute for the electric field E.
They are very different from each other,
(i) No Coulombs Law for D unlike that of E
(ii) Curl D is not zero unlike Curl E=0. ∇ x D=∇ ; X P.
So no potential formulation for D as its not a conservative field in general. The integral form is ∫ _C D.dl=∫ _C P.dl
by using Stokes theorem on the differential form.


The electrostatic boundary conditions on the Displacement Vector D can now be easily derived exactly in the same fashion as derived for the Electric Field E by using the corresponding integral forms ∫D.da=Q_free^enc and ∫ _C D.dl=∫ _C P.dl. This is worked out in Griffiths and is left as a reading exercise for the students. Please check Ex 4.8 in Griffiths


A certain class of Dielectric materials offers a simplification of these relations.
In these materials P, D are both proportional to the total Electric field E ( applied field + Polarisation field). So for these materials
D =ε E an P =ε₀χ_e E. Its easy to see that
ε=ε₀(1 + χ_e )
The ratio ε/ε₀=(1 + χ_e )=ε_r is called Relative Permittivity or Dielectric Constant.


However please note that even for Linear Dielectrics the Curl D is NOT ZERO unless the entire space is filled with a single homogeneous dielectric. The reason for the decrease of the Electric Field of a point charge in a dielectric medium as compared to a point charge in free space is now clear. The polarisation charges actually screen the point charge so its actual effective magnitude within the dielectric decreases and hence the corresponding electric field is reduced.