external to the region of interest image charges
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1 externl to the region of interest imge chrges
2 x0 q q x 1 q 4 0 xy q ' xy ' 1 q 4 0 xn yn ' q ' xn y 'n ' 4 0 x q q ' y ', y y ' q n yn ' y ' 2 q ' y 'n ' n y ' 0 y, q ' y q qq q
3 0 xx q y 2 y 2 2 y cos 2 3 q q q y y ' y1 2 y 2 F q2 4 0 y y 2 2 2
4 F d F 2 q q d q 2 y cos y y cos d q y y
5 Q q q QqQ Qq 4 0 x q xy y q Q y q x x2 y 2y q F q y 2 Q q 3 2 y 2 2 y y y y
6 y q Qq Qq y 11 2 q Q
7 Q q V x 1 q 4 0 xy q y x x2 y V F q y V 1 3 q y 2 y 2 4 2y 0 y 2 2 y q Vq 4 0 V y 11 2
8 E Q R 2 RQ QR Q 4 0 r 2 R 2 2 r R cos Q r 2 R 2 2 r R cos Q r 2 R r R cos Q r 2 R r R cos
9 Rr Q R 2 r 3 r 2 cos Q R 3 r E0 r cos 2 E z D R Q2 2 R 4 0 E rr 3 0 E 0 cos
10 Gx,x' 1 xx ' x ' x ' 2 x 2 x ' 1 x 2 x ' 2 2 x x ' cos x 2 x ' x x ' 2 cos G GGn G 2 x 2 n 'x ' x x cos 3
11 outside x 1 x 2 2, ',' 4 d ' x x cos 3 where coscos cos 'sin sin ' cos ' interior x 1 4 G x 2 2 n 'x ' x x cos 3, ',' 2 x 2 x x cos 3 d'
12 x,, V d' 1 x d cos' 0 x x cos 3 1 V x d ' z cos cos' 0 zv 1 z2 2 z z 2 2 V z 3 V 2 2 z 2 x 2 2 d cos ' 3 x x cos d cos ' x x cos 3 2 x x cos 3 2 z V x d ' 4 x d cos'12cos cos 32 0 x x 2 2
13 12 cos cos 32 6 cos35 3 cos d ' 0 1 cos d cos' cos 0 2 d ' 0 1 cos 3 d cos' 4 cos3cos2 x,, 3 x2 2 V x x 2 V cos 7 12 cos cos 2 in 2 x cos3 3 2 cos in 2 x 2 x
14 b orthonormlu n b U n U m d m n f n1 Fourier series n1 n U n completeness of the function set b U n ' f ' d ' U n n b U n f d U n ' U n ' closure reltion or completeness n m x sin for m0, 2 cos 2 m x for x 2, 2
15 f x 1 2 A A 2 m x 0 cos B m m sin 2 m x m1 where 2 A m 2 2 f x cos 2 m x d x, b cd U n V m f, n m, b, U m x U m, x f x 1 m i 2 m x A m e A m 1 2 B m i 2 m x ' e f x sin 2 m x n m U n V m n m b d c d d U n V m f, b U n x U m x d x U n, x U m, x d xmn m n Fourier integrl U m x 1 i 2 m x e m0,1, 2, x2, 2 f x ' d x ' d x
16 2 m m A m k 2 d m Ak 2 d k f x 1 Ak e i k x d k Ak 1 e i k x f x d x e i kk ' x 1 d xkk ', e i k x x ' d k x x ' 2 2 orthogonlity condition completeness reltion xk
17 2 2 x 2 2 y 2 z 0 2 x, y, z X x Y y Z z 1 d 2 X X d x 1 d 2 Y 2 Y d y 1 d 2 Z 2 Z d z X d 2 X d x 2 2, 1 Y d 2 Y d y 2 2, 1 Z d 2 Z d z e i x e i y e 2 2 z
18 x0, y, z0 Xsin x x, y0, z0 Ysin y x, y, z00 Zsinh 2 2 z x, y, z0 x, yb, z 0 n n m m b n m sin n x sin m y sinh n 2 m 2 z x, y, z n, m1 n, m1 A n m n m A n m sin n x sin m y sinh n 2 m 2 z x, y, zcv x, y V x, y n, m1 A n m 4 b sinhc n 2 m 2 0 A n m sin n x sin m y sinh n 2 m 2 c d x 0 b d y V x, y sinn x sin m y
19
20 z e i x e y x0, y x, y0 x, y0 x, y0v e y sin x n 1 A n e n y or n sin n x A n 2 n x 0 x, 0 sin 4 V 1 for n odd n 0 for n even d x
21 x, y n odd 4 V y e x y x sine i n odd 4 V n odd ln1 Z Z Z2 2 n odd Z n n 1 2 n y 4 V n e sin x i n 4 V n e x i y Z n i n Ze Z3 3 Z Z ln 1 Z ln 1 Z 1 Z 1 Z x, y 2 V 1 Z ln sin n x for y x i y
22 1 Z 1 Z 1Z 2 2 i Z 1 Z 2 x, y 2 V tn1 sin the phse of the rgument of ln 1 Z 1 Z 2 Z tn1 1Z 2 x sin sinh y 0 tn 1 x sinh y 2
23 , R R R d d d R d 1 d d d R d 1 v2, v B d 2 d 0 d 2 d v2 R v b v A cosv B sinv, 0 b 0 ln n1 but for V v0 R 0 b 0 ln A 0 B 0 n n sinn n b n n sinn n
24 b n b n b, 0,V b 0 B 0 b A0, sin 0 m,,v 1 sin E 1 1 sin E cos m1,2, m,v m sin m m1 m, 0, 0 E, surfce chrge density
25
26 R 2 g С R g d x d y 0 R 2 gd x d y0 x, y : test function x C, y C 0 Green's 1st identity on the 1st term ij x,yxx i yy j Rh ij i j x, y 1x x i h1y y j h for x x i h,y y j h 0 otherwise i, j N 0 i j x, y1 x, y i, j i j x, y N i j 0 number of lttice sites including the boundry
27 ij x i,y j h i,j i, j Ngx,y N 0 k R x, y x, y d x d ygx, y i j k i j R i j k, x, y d x d y 1 NN k k x i,y j i j R i j x, y d x d y h 2 R i j k d x d y for ki, j j forki1, ki, j1 ki1, j1
28 K : N N mtrix KG ndg: N -column vectors x, y e x, y A B xc y. 1, 2, 3 ABC N e j xy N e j x x j, y y j 1 N 1 e 1 b 1 xc 1 y 1 b 1 x 1 c 1 y b 1 x 2 c 1 y b 1 x 3 c 1 y 3 0
29 D1 x 1 y 1 1 x 2 y 2 1 x 3 y 3 x 2 x 1 y 3 y 1 x 3 x 1 y 2 y D D2 S e S e : re of the tringle 1 x 2 y 3 x 3 y 2 2 S e, b 1 y 2 y 3 2 S e c 1 x x S e N e j 3 i1 N i e x, y1, 3 i 1 i 1, 3 i1 b i 0, j b j x e c j y e 1 3 j1, 2, 3 where x x x x 1 2 3, y e e y y y f x,y x, y j N f j x, y f, j f 3 i1 c i 0
30 x,yn j e x,yei 3 j1 j e e N i e N j e d x d y e g N i e d x d y g x e, y e e N i e d x d y 2 e N i e d x d y S e i b i x e c 1 y e S e 3 e e N i x b, N i i y c k e i i j e N e i N e j d x d ys e b i b j c i c j K e i j 3 2 j1 K e e G e k e i j j e S e g e 3 in mtrix form i1,2,3 g e g x e, y e Kk i j, G i 1 3 T k i i T N 0 S e g e j N 1 T KG for i j Eij k i i e k i j e, k i j E j e e k i j boundry terms
31
i r-s THE MEMPHIS, TENN., SATURDAY. DEGfMBER
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