(References to) "Introduction to Electrodynamics" by David J. Griffiths 3rd ed., Prentice-Hall, ISBN X

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1 lectic Fields in Matte page. Capstones in Physics: lectomagnetism. LCTRIC FILDS IN MATTR.. Multipole xpansion A. Multipole expansion of potential B. Dipole moment C. lectic field of dipole D. Dipole in an electic field.. Chages in Mateials A. Polaization B. Displacement field C. Dielectic boundaies (Refeences to) "Intoduction to lectodynamics" by David J. Giffiths d ed., Pentice-Hall, 999. ISBN X 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

2 . MULTIPOL XPANSION lectic Fields in Matte page...a. MULTIPOL XAPANSION OF POTNTIAL stat fom V() d ρ(r) R 4π o R R [(R) ]/ [ R R ]-/ R [ R (R ) ] / R dvol d R dx dy dz ρ(r) expand fo >>R : Taylo seies in R << to fist ode: R [ R O(R ) ] each tem of the integand factos into a poduct of a function of times a function of R : ˆ V() d 4π o R ρ(r) 4π o d R ρ(r) R total chage Q dipole moment p to all odes: let α R, the angle between and R. Then cos α ˆ ˆR, and R [ R l whee P l (cos α) lim u l! u cos α( R ) ] / Pl ( cosα ) l l [ u cos α u ]/ is a Legende Polynomial of ode l R l 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

3 lectic Fields in Matte page...b. DIPOL MOMNT p d R ρ(r) R conside anothe chage distibution a R-a ρ'(r) ρ(r a) Then its dipole moment is R' R ρ(r) ρ'(r) p' d R ρ'(r) R d R ρ(r a) R change vaiables: R' R a, R R' a, dr dr' p' d R' ρ(r') (R' a) d R' ρ(r') R' a d R' ρ(r') p' p a Q total this is simila to the ule fo the moment of inetia, I' I a M total..c. LCTRIC FILD OF DIPOL Fa fom the distibution () q totalˆ 4π monopole (p ˆ)ˆp 4π o dipole multipoles... highe 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

4 lectic Fields in Matte page.4 If p lies at the oigin and points in the z diection, the potential is p pcosθ V (, θ ). 4π π The electic field of the dipole is 4 V : p p z z θ y θ φ V p cosθ 4π V p sinθ θ 4π V sinθ φ x Thus p (, θ ) ( cosθ ˆ sinθ ˆ). θ 4π zˆ cosθ ˆ sinθ ˆ θ Using, (, θ ) p 4π 4π (p ˆ)ˆ - p 4π ( cosθ ˆ zˆ) ( pcosθ ˆ p zˆ) 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

5 lectic Fields in Matte page.5..d. DIPOL IN AN LCTRIC FILD dipole moment p qd in unifom field pictue plane of, p NO NT FORC F q d θ p q F total foce F F both diected along total foce q q Instead of a net foce, thee is a toque on the dipole τ i i F i let vecto distance fom cente of dipole, ± ±d/, then τ d (q) d (q) q(d ) p toque on dipole τ p, diection to both and p, magnitude p sin θp the toque tends to make the dipole line up with the field wok done by field to align dipole dw τ dθ p sin θ dθ stat fom p θ enegy dθ p sin θ p cos θ π/ Alignment enegy U p, minimum enegy when p In non-unifom field F U (p ) p ( ) (p ) (p ) 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

6 lectic Fields in Matte page.6.. CHARGS IN MATRIALS Mateials ae made up of atoms and/o molecules. Atoms and molecules ae made up of electons and nuclei. The nuclei have: moe than 99.9% of the mass positive chages Ze (Z atomic numbe) The electons have: negative chages e most of the mateial's ability to espond to fields Chages divide into fee chages and bound chages Conductos have fee chages, insulatos don't. Most solid conductos ae cystalline (often micocystals). Solid insulatos: may be cystalline, o polymes (long molecules). Fluids ae insulatos unless they contain chaged ions. All mateials, even the vacuum, ae dielectics: they have bound chages which move in electic fields. 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

7 lectic Fields in Matte page.7..a. POLARIZATION A dielectic is an insulato chages can't move fa But they can move a little way in esponse to a field see "Oscillato Model" q the chages accumulate on the suface the induced chages q cause an attactive foce! the induced chages cause an induced field which educes the intenal field in a linea dielectic, intenal κ applied handy ule of thumb: o κ o The polaization affects the potential enegy 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

8 lectic Fields in Matte page.8 OSCILLATOR MODL OF DILCTRIC POLARIZATION static chages ±q sepaation d dipole moment qd educed mass m, sping constant k vibation fequency Ω k/m electic field balance of foces on negative chage: electic foce q estoing foce k d -q k q - d DIPOL MOMNT p qd q ( q/k ) polaization density P np χ(static)o whee n density of oscillatos pe unit volume (Giffiths: N) χ(static) electic susceptibility nq ok nq omω NOT: the desciption has to be modified when changes on a time scale of /Ω 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

9 lectic Fields in Matte page.9 BOUND CHARGS conside a suface ds in a dielectic ds apply electic field ds (Giffiths: da) chages move distance ddo see "Oscillato Model" chages within ddo coss suface volume of chages cossing suface (ddo) ds n density of bound chages pe unit volume (Giffiths: N) conclusions: chage moving though suface dq n q (ddo) ds P ds. at suface of dielectic, chages accumulate see "Suface Conditions" suface chage density whee ˆn unit nomal to suface σbound P ˆn P. inside dielectic, P() ρbound nq poof: conside any closed volume, then chage moving out of volume ds P dvol P but consevation of chage chage moving out of volume chage in volume dvol ρbound dvol P tue fo any volume integands equal 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

10 ..B. DISPLACMNT FILD lectic Fields in Matte page. Aim: to find electic field in tems of fee chages ρfee Poblem: comes fom all chages, bound and fee o () ρ() see "Gauss' Law" Solution: exploit elation between bound chages and P, P() ρbound see "Suface Polaization" add to Gauss' law: (o P) ρ ρbound ρfee define displacement field D o P D ρfee once we know D, can find fom elation between and polaization P, P np P χ(static)o see "Oscillato Model" substitute, setting Po as is almost always tue D o P o χ(static)o D κo whee κo and κ χ(static) pemittivity, χ polaizability, κ dielectic constant so the method is:. Find D fom fee chages. Find fom D see "Suface Conditions" see "Method of Images" see "Capacitos" Giffiths notation κ K, χ(static) χelectic 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

11 ..C. DILCTRIC BOUNDARIS lectic Fields in Matte page. the polaization chage accumulates at the suface of a dielectic suface chage density q σbound P ˆn P whee ˆn unit nomal to the suface see "Suface Polaization" This implies that the electic field is discontinuous: ecall flat sheet of chage choose "Gaussian pillbox" da q enclosed o left A ight A ight left σ o σa o to suface L σ A positive Apply same easoning to the integal fom of D ρfee Pependicula component of D is continous: D D ˆn same on both sides, same on both sides altenate agument fo : apply dl to path nea suface L ( left ight) 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

12 lectic Fields in Matte page. DILCTRIC SPHR dielectic sphee, chage adius R, cente at extenal electic field ẑ potential V() Vo z fa fom the sphee Axially-symmetic solutions to Laplace's equation, V : ( ) V(, θ) A n n B n / n P n (cos θ) outside n and V(, θ) ( C n n D n / n ) P n (cos θ) inside n (i.e. diffeent coefficients) To get the coefficients, use othogonality condition, d(cos θ) P m (cos θ) P n (cos θ) n - δ mn fo fixed. Condition at Inside the sphee the potential is finite D n fo all n. Condition at Outside sphee V( ) z V o whee V o constant. P,P cos θ V( ) V o P (cos θ) P (cos θ) A V, A ; All the othe A n have to be zeo. The B n ae not constained by these bounday conditions, but we can find them fom the continuity conditions. continued 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

13 Continuity of V at inteface lectic Fields in Matte page. DILCTRIC SPHR continued Coefficients inside, outside elated by continuity conditions at suface R : V same just inside and just outside, V(R,θ) C n R n P n (cosθ) (A n R n B n R n )P n(cosθ) n n V(R,θ) in V(R,θ) out [(C n R n ) A n R n B n R n n ]P n(cosθ) multiply by P m (cos θ) d(cos θ) and integate coefficient of each P n (cos θ) C n A n Continuity of D at inteface B n R n Similaly, the pependicula i.e. adial component of D o V has to be continuous at the suface, giving D out D in o ( V (, θ ) in V (, θ ) out ) R. multipole expansions [ ( C n n ) ( ) n A n n B n / n ] R P n (cosθ) coeff. of each P n zeo C n A n n n B n R n fo n > Solve togethe with equation fo continuity of V n B n ( )C n n R n, B n R n ( n ) n C n, n C n A n ( ) n C n A n n n ( ) -, so both B n and C n ae popotional to A n fo n >. continued 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

14 lectic Fields in Matte page.4 DILCTRIC SPHR continued fo n A C V unifom constant voltage Continuity of D pep [( C ) A B ] R B R D B, A C V unifom constant voltage fo n A, D C A, B R C R, fo n,,... A n B n C n D n Consequently, V (, θ ) V V cosθ R cosθ inside outside and zˆ zˆ R ( cosθ ˆ sinθ ˆ) θ inside outside Polaization P ( ) in ( ) zˆ ( ) zˆ 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee)

15 lectic Fields in Matte page.5 5 Oegon State Univesity Philip J. Siemens (evised by Yun-Shik Lee) Induced electic potential when 4 V p ˆ π, whee P p 4 R π θ θ π π θ cos cos ), ( R R V 4 4 Induced suface chage density θ σ cos ˆ P

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