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E-field of a polarized object1 Electric field of a Polarized Object © Frits F.M. de Mul

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E-field of a polarized object2 Electric field of a Polarized Object Question: Calculate E-field produced BY (not: IN) the polarized object. Question: Calculate E-field produced BY (not: IN) the polarized object. A dielectric object will become polarized. Available: An external E-field: E ex. Available: An external E-field: E ex. E ex

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E-field of a polarized object3 Electric field of a Polarized Object Analysis and symmetry Approach to solution Calculations Conclusions

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E-field of a polarized object4 Analysis and Symmetry Z Y X Coordinate axes: assume Z-axis // E ex Coordinate axes: assume Z-axis // E ex E ex Assume: p // E ex ; n and p homogeneous Assume: p // E ex ; n and p homogeneous Result of polarization: Dipole distribution: n dipoles/m 3 ; each dipole moment p [Cm] Result of polarization: Dipole distribution: n dipoles/m 3 ; each dipole moment p [Cm]

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E-field of a polarized object5 Approach to solution Approach: = calculate potential V ; = E from V by differentiation Approach: = calculate potential V ; = E from V by differentiation E ex Z Y X v v Question: calculate E-field in arbitrary point P outside v P Distributed dipoles: dV- integration over volume elements dv filled with dp. Distributed dipoles: dV- integration over volume elements dv filled with dp.

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E-field of a polarized object6 Calculations (1) E ex Z Y X P Z' r erer dp dz volume element dv with dp = np.dv = np.dS..dz ; with dS Z-axis Dipole:

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E-field of a polarized object7 Calculations (2) E ex Z Y P Z' X r erer dp dz P r dS cross section through P and Z-axis : rbrb rtrt z t : top z b : bottom

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E-field of a polarized object8 Calculations (3) P rbrb rtrt r ztzt zbzb dS dz-integration dr-integration : dr = -dz.cos dz = -dr / cos r dz dr P

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E-field of a polarized object9 Calculations (4) P rbrb rtrt r ztzt zbzb dS r dz dr P Polarization P = np

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E-field of a polarized object10 Calculations (5) E ex Z Y P Z' X r erer dp dz P dS P.dS = P.dS bound charges: d b = P.dS

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E-field of a polarized object11 Conclusions the end bound charges: d b = P.dS P E ex Z Y X r bottom r top + - Conclusion: the field of a polarized volume is equivalent to that of bound surface charges (provided homogeneous polarization)

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