1. Electric field of a Polarized Object
© Frits F.M. de Mul
E-field of a polarized object

2. Electric field of a Polarized Object
Available:
An external E-field: Eex.
Eex
A dielectric object will become polarized.
Question:
Calculate E-field produced BY (not: IN) the polarized object.
E-field of a polarized object

3. Electric field of a Polarized Object
Analysis and symmetry
Approach to solution
Calculations
Conclusions
E-field of a polarized object

4. E-field of a polarized object
Analysis and Symmetry Z Y X
Coordinate axes:
assume Z-axis // Eex
Eex
Result of polarization:
Dipole distribution:
n dipoles/m3 ;
each dipole moment p [Cm]
Assume: p // Eex ;
n and p homogeneous
E-field of a polarized object

5. E-field of a polarized object
Approach to solution
Eex Z Y X v
Question: calculate E-field in arbitrary point P outside v P
Approach:
= calculate potential V ;
= E from V by differentiation
Distributed dipoles:
dV- integration over volume elements dv filled with dp.
E-field of a polarized object

6. E-field of a polarized object
Calculations (1)
Dipole:
Eex Z Y X P Z' r er  dp dz
volume element dv with dp = np.dv = np.dS..dz ; with dS  Z-axis
E-field of a polarized object

7. E-field of a polarized object
Calculations (2) Z Z' dp
Eex
cross section through P and Z’-axis :  P r
dS  rb rt
zt : top
zb : bottom dz er r P Y X
E-field of a polarized object

8. E-field of a polarized object
Calculations (3) P rb rt r zt zb
dS 
dz-integration  dr-integration :
dr = -dz.cos  dz = -dr / cos r  dz dr P
E-field of a polarized object

9. E-field of a polarized object
Calculations (4)
dS zt  rt r P rb zb
“Polarization” P = np  dz r P dr
E-field of a polarized object

10. E-field of a polarized object
Calculations (5)
Eex Z Y P Z' X r er  dp dz P dS
P.dS = P.dS
bound charges: db = P.dS
E-field of a polarized object

11. E-field of a polarized object
Conclusions P
Eex Z Y X
rbottom
rtop + -
bound charges: db = P.dS
Conclusion: the field of a polarized volume is equivalent to that of bound surface charges (provided homogeneous polarization) + -
the end
E-field of a polarized object