Electrostatics. Dielectric Polarization and Boundary Conditions

Introduction to Electrostatics. The Dielectric
Polarization and Dielectric Boundary conditions



1 Dielectric Polarization
Dielectrics are materials that lack free electrons; however, their behavior is altered when sub-
jected to an electric field. The electric field exerts a force on each charged molecule, causing
the positive charges to move in one direction and the negative charges to move in the opposite
direction. As a result, the centers of positive and negative charges shift from their average po-
sitions in opposing directions. This effect is referred to as polarization. To measure the impact
of an applied electric field on a dielectric, we introduce a vector known as polarization (P),
which is defined as the dipole moment per unit volume. The unit of polarization is C/m2 .
Taking all the elementary dipoles in a given volume, the polarization is
N
∑ Qj dqj
j=1
P = lim (1)
∆v→0 ∆v
P1 = Q1 d1 , P2 = Q2 d2 etc are small dipole moments in the volume considered.




1

, Consider a dielectric material with dipole moment P per unit volume.
The potential at point O, which is located outside the dielectric material, resulting from an
elemental dipole moment Pdv0 is

P.R̂dv0
dV = (2)
4πε0 R2

where R02 = (x − x0 )2 + (y − y0 )2 + (z − z0 )2 . R is the distance between the volume element dv0
at (x0 , y0 , z0 ) (source element) and O (x,y,z) (the field point). Now, we have

1
= [(x − x0 )2 + (y − y0 )2 + (z − z0 )2 ]−1/2 (3)
R
So,

(x − x0 )x̂ + (y − y0 )ŷ + (z − z0 )ẑ
 
1 R̂
∇ =− 0 2 0 2 0 2 3/2
=− 2 (4)
R [(x − x ) + (y − y ) + (z − z ) ] R

Similarly, the gradient with respect to the primed co-ordinates is
 
0 1 R̂
∇ = 2 (5)
R R

Therefore
 
P.R̂ 0 1
= P.∇ (6)
R2 R

2