Electrodynamics Lecture 10: Understanding Dipolar Fields and Microscopic Origin of Dipole Moments
Explore the concepts of dipolar fields, microscopic polarizability, the Clausius-Mossotti equation, and electrostatic energy in dielectric media in Lecture 10 of Electrodynamics. Delve into topics such as microscopic origin of dipole moments in polarizable isotropic atoms/molecules and the alignment of molecules with permanent dipoles. Understand induced dipole moments and the field due to a collection of induced dipoles.
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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 10: Complete reading of Chapter 4 A. Microscopic polarizability macroscopic B. Clausius-Mossotti equation C. Electrostic energy in dielectric media 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 1
02/07/2014 PHY 712 Spring 2014 -- Lecture 10 2
Focus on dipolar fields: p Dipole moment : outside 3 p r r ' ' r ' d ) ( ( ) r single For extent the of : r Electrosta potential tic from dipole : p r 3 1 ( ) r = 4 r 0 Electrosta tic field from single dipole : ( ) 5 2 r p r r p 1 3 4 r ( ) r ( ) = 3 E p r 4 3 0 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 3
Microscopic origin of dipole moments Polarizable isotropic atoms/molecules Charge anisotropic molecules Polarizable isotropic atoms/molecules E +q equilibriu At E m q E : = 2 0 x 0 q m k=m 2 x = x 2 0 m 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 4
Polarizable isotropic atoms/molecules continued: +q E k=m 2 x equilibriu At m : = 2 0 E x 0 q m E q = x 2 0 m Induced dipole moment : 2 q = = 0 x E E p q mol 2 0 m 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 5
Alignment of molecules with permanent dipoles p0: E p0 freely a For rotating dipole average its moment electric an in field, estimated assuming Boltzmann a distributi on : cos / p E kT E cos d p e 0 0 = p mol cos / pE kT d e 2 0 1 p = 0 E E mol 3 kT 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 6
Field due to collection of induced dipoles 0 1 = + E E E 2 tot i ext 7 i 3 0 = + E E E ( ) site i j ext 6 j i 5 0 = E E tot i 4 9 8 Electrosta field tic from single dipole : ( ) 5 2 r p r r p 1 3 4 r ( ) r ( ) = 3 E p r 4 3 0 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 7
Field due to collection of induced dipoles -- continued 0 = + E E E tot i ext 1 2 i 7 3 0 = + E E E 6 5 ( ) site i j ext j i 4 9 0 = E E 8 tot i ( ) 5 2 r p r r p 3 1 4 r ( ) r ( ) r i = i + 3 E p E i i ext tot 4 3 0 ( ) 5 2 r p r r p 3 r 1 ( ) r j j j = + E E ( ) ext site i 4 i 0 1 V 1 1 = + = + E E p E P ( ) site i tot tot 3 3 0 0 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 8
Field due to collection of induced dipoles -- continued 1 = + E E P ( ) site i tot 3 1 2 0 7 = 0 3 p E mol site 6 5 0 1 1 = = + P p E P mol 4 9 tot 3 V V 8 0 E 0 = = tot P E mol 0 e tot V 1 mol V Claussius-Mossotti equation mol / 1 V = = 0 3 V e mol + / 2 1 mol 0 3 V 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 9
Example of the Clausius-Mossotti equation Pentane (C5H12) at various densities Density (g/cm3) Mol/m3 e/e0 3V*(e/e0-1)/(e/e0+2) 0.613 5.12536E+27 1.82 1.25646E-28 0.701 5.86114E+27 1.96 1.24084E-28 0.796 6.65544E+27 2.12 1.22536E-28 0.865 7.23236E+27 2.24 1.2131E-28 0.907 7.58353E+27 2.33 1.2151E-28 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 10
Re-examination of electrostatic energy in dielectric media 1 = 3 r r ( ) ( ) W d r mono 2 In terms D displaceme r mono of nt field : = ( ) 1 1 1 ( ) = = 3 3 3 D r D r r D r r ( ) ( ) ( ) ( ) ( ) W d r d r d r 2 2 2 1 = + 3 D r E r 0 ( ) ( ) d r 2 1 = 3 D r E r ( ) ( ) W d r 2 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 11
Comment on the Modern Theory of Polarization Some references: R. D.King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993) R. Resta, Rev. Mod. Physics 66, 699 (1994) R, Resta, J. Phys. Condens. Matter 23, 123201 (2010) Basic equations : = = + E = 0 tot bound mono P bound = D mono = + E D P 0 Note: In general P is highly dependent on the boundary values; often it is more convenient/meaningful to calculate P. 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 12
Comment on the Modern Theory of Polarization -- continued = bound = nuclear + electronic P bound bound e n = electronic P 0r w w 0 n n crystal V 02/07/2014 PHY 712 Spring 2014 -- Lecture 10 13
