DIELECTRIC AND BOUNDARY CONDITIONS

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced toward the field and negative charges shift in the opposite direction. This creates an internal electric field which reduces the overall field within the dielectric itself.

DIPOLE MOMENT ELECTRIC FIELD

What is Dielectric Polarization? a non polar dielectric placed in some external electric field. Center of positive charge of individual molecules is pulled automatically in the same direction as that of electric field towards the plate having negative charge. Similarly, the centre of the negative charged electrons is dragged in the opposite direction of the electric field, towards the plate having positive charge. So the centres of positive as well as negative charges are set apart..

Due to this the molecules are deformed from their original shape and hence separate at last. So, due to the above process each molecule gets some dipole moment. After sometime these molecules will get polarized when the forces of attraction between the centres of positive and negative charges and the force due to electric field will reach at some stable state

Boundary Conditions

8 Fundamental Laws of Electrostatics in Integral Form Conservative field Gauss’s law Constitutive relation

9 Fundamental Laws of Electrostatics in Differential Form Conservative field Gauss’s law Constitutive relation

10 Fundamental Laws of Electrostatics The integral forms of the fundamental laws are more general because they apply over regions of space. The differential forms are only valid at a point. From the integral forms of the fundamental laws both the differential equations governing the field within a medium and the boundary conditions at the interface between two media can be derived.

11 Boundary Conditions Within a homogeneous medium, there are no abrupt changes in E or D. However, at the interface between two different media (having two different values of , it is obvious that one or both of these must change abruptly.

12 Boundary Conditions (Cont’d) To derive the boundary conditions on the normal and tangential field conditions, we shall apply the integral form of the two fundamental laws to an infinitesimally small region that lies partially in one medium and partially in the other.

13 Boundary Conditions (Cont’d) Consider two semi-infinite media separated by a boundary. A surface charge may exist at the interface. Medium 1 Medium 2 x x xx ss

14 Boundary Conditions (Cont’d) Locally, the boundary will look planar x x x ss

15 Boundary Condition on Normal Component of D Consider an infinitesimal cylinder (pillbox) with cross-sectional area  s and height  h lying half in medium 1 and half in medium 2: ss  h/2 x x x ss

16 Boundary Condition on Normal Component of D (Cont’d) Applying Gauss’s law to the pillbox, we have 0

17 Boundary Condition on Normal Component of D (Cont’d) The boundary condition is If there is no surface charge For non-conducting materials,  s = 0 unless an impressed source is present.

18 Boundary Condition on Tangential Component of E Consider an infinitesimal path abcd with width  w and height  h lying half in medium 1 and half in medium 2:  h/2 ww a b c d

19 Boundary Condition on Tangential Component of E (Cont’d) a b c d

20 Boundary Condition on Tangential Component of E (Cont’d) Applying conservative law to the path, we have

21 The boundary condition is Boundary Condition on Tangential Component of E (Cont’d)

22 Electrostatic Boundary Conditions - Summary At any point on the boundary, – the components of E 1 and E 2 tangential to the boundary are equal – the components of D 1 and D 2 normal to the boundary are discontinuous by an amount equal to any surface charge existing at that point

23 Electrostatic Boundary Conditions - Special Cases Special Case 1: the interface between two perfect (non-conducting) dielectrics: – Physical principle: “there can be no free surface charge associated with the surface of a perfect dielectric.” – In practice: unless an impressed surface charge is explicitly stated, assume it is zero.

24 Electrostatic Boundary Conditions - Special Cases Special Case 2: the interface between a conductor and a perfect dielectric: – Physical principle: “there can be no electrostatic field inside of a conductor.” – In practice: a surface charge always exists at the boundary.