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

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

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

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

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

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

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

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

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

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

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