Week 5 Lex 3: Electrostatics in Material Media. ( Dielectrics)

Having covered a number of issues for Electrostatics in free space we now move to the study of Electrostatics in a material media like dielectrics/insulators. The basic difference with conductors in this case is that in a dielectric charges have restricted mobility unlike free charges in a conductor. So the charges cannot move freely but they can be displaced a little from their equilibrium position
which makes the atom neutral.


When a dielectric material is placed in an external electric field a separation of the +ve and -ve charges take place so that there is a field inside the dielectric material which opposes the external field such that the net field inside is reduced. Due to the charge separation surface charges appear on the dielectric.


At the microscopic level the centers of the +ve and -ve atomic chrage distributions are displaced by the external applied field ( like the problem of two spherical charge distributions +ρ and -ρ displaced by a small distance d done in a DIPA). The atom is now said to be polarized and develops a small dipole moment p which is proportional to the applied field E such that p=α E and α is called the atomic polarisability. For a molecule the situation is more complex but once more a dipole moment develops but the polarisability α is now direction dependent and is a rank 2 tensor.


Apart from this certain molecules due to their asymmetric structures have a charge distributions with a FROZEN-IN PERMANENT dipole moment p. These are called POLAR molecules. In an uniform external field E these permanent dipoles feel a torque N=p X E which tends to orient the dipole moment vector in the direction of the external field ( for a non uniform field the dipole experiences a force F=(p.∇) E. Note that (p.∇)is a scalar differential operator which operates on each component of the vector field E. This is the operator version of scalar multiplication of a vector.)


So the microscopic picture that emerges for a dielectric from the above consideration is that a dielectric material either develops induced dipoles or has permanent dipoles ( for polar molecules)
both of which tend to orient along the direction of the applied external field. Some of this orientation will even stay after the field is removed especially for Polar molecules. So in a dielectric material
we have a large number of dipoles oriented in the same direction. This results in a dipole moment per unit volume which is called the Polarization vector P .


Now it is possible to calculate the field due to such a Polarised material by finding the Potential due to a small volume dτ' which has a dipole moment Pdτ' and integrating over the entire volume. Manipulation of this volume integral using product rules and the Divergence theorem show that the potential at some point due to a Polarized material is due to two contributions.

(i) A surface BOUND chrage density σ_b=P. n cap
(ii) A volume BOUND charge density ρ_b=-∇. P


This mathematical result is confirmed by a physical analysis of the microscopic dipoles canceling each other in the volume resulting in a surface bound charge density and a volume bound charge density when the cancellations are not complete in the volume due to unequal dipoles ( non constant dipole moment /unit volume which is non uniform Polarisation). The expression match exactly confirming the mathematical analysis using Vector Calculus.