3.1 Laplace s Equation 3.2 The Method of Images 3.3 Separation of Variables

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1 Lecture Note #3B Chpter 3. Potetis 3. Lpce s Equtio 3. The Method of Imges 3.3 Seprtio of ribes 3.3. Crtesi Coordites 3.3. Spheric coordites 3.4 Mutipoe Expsio Boudry coditios re very importt to sove the poteti probems Dimesio (D, D, 3D) ectgur, Spheric, The seprtio of vribes c be used to sove the poteti distributios from fixed poteti cofigurtio (rectgur, spheric,...) X Ih Uiversity The Lpce s equtio x y z x, y, z X xy yz z X x Y Y y Z Z z i) whe y ii) whe y iii) whe z iv) whe z b v) s x vi) y,z whe x. r r r r r si si r si The, the fi form of the geer soutio of the equtio for zimuth symmetric probems becomes r, A r P B cos r

2 3.3 Seprtio of ribes ( 변수분리 ) 3.3. Crtesi Coordites [Exmpe 3.3] Let Ih Uiversity x, y X xy y The Lpce s equtio c be expressed s (Soutio) The cofigurtio is idepedet of z, so this is -D probem. The Lpce s equtio becomes x y The boudry coditios : i) whe y X Y Y X x y X Y x Y y Divide the bove equtio by XY. ii) iii) iv) whe y whe x whe x f x gy X k k (3.) (3.) (3.) (3.3)

3 3.3 Seprtio of ribes [Exmpe 3.3] (Soutio) - cotiued () X X x Y Y y k k X x Y y k k X x kx kx x, y X xy y Ae Be C si ky Dcos ky X Y Y y Ae kx Be kx Csi ky Dcos ky (3.7) Now, et us ppy the boudry coditios to the soutio of (x,y). iv) whe x k k, y Ae Be C si ky Dcos ky A iii) whe x kx x, Be D D i) whe y kx ii) whe y x, Be C si k k,, 3, Thus, the geer soutio becomes kx y x, y C' e si where C = BC : proportio costt y, y C' si y (3.9) (3.3) (3.3) Ih Uiversity Fourier sie series 3

4 3.3 Seprtio of ribes [Exmpe 3.3] (Soutio) - cotiued () Eq. (3.3) x si( y/), d the itegrte from to : y ' y C' si si dy ysi ' y dy (3.3) If (y) = t < y < [() = () = ], Thus, the fi form of the geer soutio becomes C' y ' y, si si dy, y ysi dy kx x, y C' e si if if y ' ' C' 4,3,5 (3.33) y ysi dy e x (3.34), if y y C' si cos cos 4 y dy, if y o y si is eve Pge 7 is odd. (3.36) Ih Uiversity 4

5 3.3 Seprtio of ribes [Exmpe 3.3] Eq. (3.36) (Soutio) - cotiued (3) x, y 4,3,5 e x y si = = 5 = = x, y 4,3,5 e x si y t si y sih x (3.37) Thus, the fi form of the geer soutio stisfies the four boudry coditios. Ih Uiversity 5

6 3.3 Seprtio of ribes Two Mjor Properties of the Seprbe Soutios to the Prti Differeti Equtios Competeess A set of fuctios f (y) is compete if fuctio f(y) is ier combitio of them. f y C f y (3.38) Orthogo A set of fuctios is orthogo if the itegr of the product of y two differet members of the set is zero: y y f f' dy for ' (3.39) Ih Uiversity 6

7 f x Mthemtic Iformtio Fourier series For periodic odd fuctio f(x) with period L, the Fourier series of f(x) is expressed s with x x cos b si L L b L t f t cos dt, L L L,,, 3, L t f t si dt, L L L,,, 3, [ef.] G. B. Arfke d H. J. Weber, Mthemtic Methods for Physicists, 5 th Ed., Acdemic Press () pge 873. Trigoometric fuctio six y si x cos y si y cos x cosx y cos x cos y si x si y If x = y, cos x cos x - si x si x si x cos x If x y, x si x dx 4 si bx si b si x si bx dx b b cosx dx si x Ih Uiversity [ef.] I. S. Grdshtey d I. M. yzhik, Tbe of Itegrs, Series, d Products, Acdemic Press (98) pge 39. x Bck 7

The Lpce s equtio becomes The boudry coditios : X Y Let x, y X xy y Y X x y Divide the bove

8 3.3 Seprtio of ribes ( 변수분리법 ) 3.3. Crtesi Coordites [Exmpe 3.4] Ih Uiversity (Soutio) The cofigurtio is idepedet of z, so this is -D probem. The Lpce s equtio becomes The boudry coditios : X Y Let x, y X xy y Y X x y Divide the bove equtio by XY. The Lpce s equtio c be expressed s X X x k i ) ii ) Y Y y iii ) iv ) k x whe y y whe y whe x b whe x b (3.4) (3.4) (3.4) (3.43) 8

9 3.3 Seprtio of ribes [Exmpe 3.4] Now, et us ppy the boudry coditios to the soutio of (x,y). Ih Uiversity (Soutio) - cotiued () X X x Y Y y k k k X x Ae kx kx x, y X xy y Ae Be C si ky Dcos ky The poteti (x,y) is symmetric with respect to x. x,y x,y i) whe y iii) whe x b X x Y y k b, y C' cosh b si y x, y X Y C' Y A B y where C = AC d D =AD : proportio costts. x y cosh si kx Be kx Csi ky Dcos ky kx kx x, y Ae e C si ky Dcos ky cosh kxc'si ky D'cos ky x, cosh kxc'si D'cos D' x, C'coshkxsik k (3.44) (3.45) (3.46) ii) whe y The poteti (x,y) is symmetric with respect to x, (x,y) stisfies the 3 rd d 4th boudry coditios simuteousy. (3.47) (3.48) 9

10 3.3 Seprtio of ribes [Exmpe 3.4] (Soutio) - cotiued () Eq. (3.47) x si( y/), d the itegrte the equtio over y from to : b y ' y C' cosh si si dy ysi C' b cosh y ' y, si si dy, If (y) = t < y < & () = () =, C y ysi dy ' y if ' if ' C' dy b cosh (3.49) (3.5) y ysi dy (3.5), if is eve. 4, if is odd. b y y ' cosh y si dy cos cos yo Thus, the fi form of the geer soutio becomes, from Eq. (3.48), x, y C' x y cosh si 4,3,5 cosh cosh x b y si (3.5) Ih Uiversity

11 3.3 Seprtio of ribes [Exmpe 3.4] (Soutio) - cotiued (3) x, y 4,3,5 cosh cosh x b y si (3.5) Ih Uiversity x cosh b cosh e x e x cosh x sech cosh x cosh x sech e e x b e e x x e e x x x b

12 3.3. Crtesi Coordites [Exmpe 3.5] 3.3 Seprtio of ribes Let Ih Uiversity (Soutio) x, y, z X xy yz z This is 3-dimesio probem. The Lpce s equtio becomes x y The boudry coditios : i) whe y ii) whe y iii) whe z iv) whe z b v) vi) s x z y,z whe x. Substitutig this ito Eq. (3.53) d divide it by XYZ., the Lpce s equtio c be expressed s X X x Y Y y Z Z z (3.53) (3.54) (3.55) (3.56)

13 [Exmpe 3.5] 3.3 Seprtio of ribes (Soutio) - cotiued () X X X x C Y C k k x k x x Ae Be, k x k x x, y, z X xy yz z Ae Be C si ky Dcos kye si z F cos z Now, et us ppy the boudry coditios to the soutio of (x, y, z). i) whe y, Y y C C C C C C k 3 X x k X, Y y Y, Z C 3 Z 3 k Y, z Z z Z. y Csi ky Dcos ky, v) s x k x, y, z Ae x, y, z D Z A D, (3.57) z Esi z F cos z. (3.58) (3.59) (3.6) iii) ii) iv) whe z x, y, z F whe whe Ih Uiversity y z b k x x, y, z C' e siksiz k x x, y, z b C' e sikysib where C = BCE : proportio costt. m b x, y, z C' e m F k m b x y mz, m si si b (3.6) (3.6) (3.63) (3.64) 3

14 [Exmpe 3.5] 3.3 Seprtio of ribes (Soutio) - cotiued () Eq. (3.64) Let us ppy the st boudry coditios to the soutio of (x, y, z). y,z whe x. vi) / m b x, y, z C' e m x y mz, m si si b y mz b,, m, m x y, z C' si si x y (3.65) (3.66) Eq. (3.66) x si( y/) si(m z/b), d the itegrte the equtio over y from to d over z from to b : b b C' si si dy si si dz x, ysi si m y ' y y ' y, si si dy, if ' if ' mz b b m' z b mz m' z, si si dz b b b, ' y m' z b dy dz (3.67) if m' m if m' m C', m 4 b b x, y y mz si si dy dz b (3.68) Ih Uiversity 4

15 3.3 Seprtio of ribes [Exmpe 3.5] (Soutio) - cotiued (3) If the (x, y) = ( costt poteti) t x =, the soutio becomes Eq. (3.68) C', m 4 b y mz b b si dy si dz (3.68) y y si dy cos, if is eve. cos, if is odd. yo C' m 4 b b, 6 m, m if or m is eve, if d, m re odd. (3.7) (3.7) 6 x y mz e si si m m b, m,3,5 / m b Eq. (3.65) x, y, z (3.7) Ih Uiversity 5

16 3.3. Spheric Coordites 3.3 Seprtio of ribes I the spheric coordites, Lpce s equtio is writte s si r r r r r si r si (3.73) For zimuth symmetric probems, is idepedet of. r r r si si (3.74) Let Ih Uiversity r r, The Lpce s equtio c be expressed s Divide the bove equtio by. r r r r r r r d dr r r r si r si si si d dr (3.75) (3.76) (3.77) 6

17 3.3. Spheric Coordites 3.3 Seprtio of ribes The geer soutio of the equtio B r Ar r - cotiued () d d r dr dr (3.78) [Check whether this soutio stisfies Eq. (3.77) or ot]. Eq. (3.76) si si is writte s d d d si d The soutio of the gur equtio is cos P Legedre poyomis is defied by the odrigues formu: d P!! dx x x si (3.79) : Legedre poyomis i the vribe cos. (3.8) (3.8) Ih Uiversity P 7

18 3.3. Spheric Coordites For =, Eq. (3.79) 3.3 Seprtio of ribes i) Oe soutio of the bove equtio is P cos d d d d si d d d d si d d P cos P cos (3.83) - cotiued () (3.8) ii) Aother possibe soutio of the bove equtio is t (3.84) u t du sec d d u du u cos d d du sec d du d u si Eq. (3.8) si si cos d d d si si d d d si x y si xcos y si ycos x si si si cos si cos d d Ih Uiversity t - t This mes tht this type soutio is ot pproprite. (3.85) 8

19 3.3 Seprtio of ribes 3.3. Spheric Coordites - cotiued (3) Thus, the geer soutio of the equtio for zimuth symmetric probems becomes B r r r Ar P cos, (3.86) The, the fi form of the geer soutio of the equtio for zimuth symmetric probems becomes r, A r P B cos r (3.87) P cos : Legedre poyomis i the vribe cos. Ih Uiversity 9

20 3.3. Spheric Coordites [Exmpe 3.6] 3.3 Seprtio of ribes (Soutio) Eq. (3.87) B r, A r P cos r B r, A P cos ( ) r, A r P cos The boudry coditios :, A P cos Eq. (3.9) x P (cos) si d itegrtig Ih Uiversity A P ' P cos P cos si d P cos si d xp xdx P cos P cos ' ' ', si d, B if ' if ' (3.87) (3.88) (3.89) (3.9) (3.9) (3.9) : orthogo property of the Legedre poyomis

21 3.3 Seprtio of ribes [Exmpe 3.6] (Soutio) - cotiued () Eq. (3.9) ' A ' Thus, A ' P ' cos si d P cos si d (3.93) (3.94) ( ) r, A r P cos : Poteti iside the sphere (3.89) Ih Uiversity

22 Ih Uiversity 3.3 Seprtio of ribes [Exmpe 3.6] (Soutio) - cotiued () If k si, Eq. (3.95) Eq. (3.9) : Eq. (3.96) Eq. (3.89) : k cos P cos cos A r, A r P cos A k k si P P P P cos P cos k r k cos k cos A P cos A P cos P cos P cos P k A, k x y cos xcos y si xsi y cos cos cos cos si cos A cos r P cos cos k si k r si (3.95) (3.96) (3.97) : Poteti iside the sphere

23 3.3. Spheric Coordites [Exmpe 3.7] 3.3 Seprtio of ribes (Soutio) Eq. (3.87) B r, A r P (3.98) cos r A (3.99) B r, A P cos ( ) B (3.) r, P cos r The boudry coditios : B, (3.) P cos Eq. (3.9) x P (cos) si d itegrtig B P P d P d (3.) cos cos si ' cos si ', if ' (3.9) P xp ' xdx P cos P ' cos si d, if ' Ih Uiversity : orthogo property of the Legedre poyomis 3

24 3.3 Seprtio of ribes [Exmpe 3.7] (Soutio) - cotiued () Eq. (3.) B ' ' ' P cos si d ' (3.3) ( ) B P cos si d r, P B cos r : Poteti outside the sphere (3.4) (3.) Ih Uiversity 4

25 3.3. Spheric Coordites [Exmpe 3.8] 3.3 Seprtio of ribes r E E zˆ Ih Uiversity (Soutio) The metic sphere is equipoteti zero poteti. The etire xy pe withi the sphere is t zero poteti. Fr from the sphere, the fied is E E zˆ, d the poteti is The boudry coditios : E z C, B r A P cos i), whe r B A Eq. (3.87) Sice = i the equtori xy pe, C i), whe r ii) E r cos θ, for r B A (3.) (3.) (3.3) (3.4) (3.5) 5

26 3.3 Seprtio of ribes [Exmpe 3.8] (Soutio) - cotiued () Eq. (3.87) B r, A r P cos r Eq. (3.5) Eq. (3.87) : ) E r cos θ, The iduced chrge desity : E E r (3.87) r, A r P cos for r r for r (3.6) ii r, A r P cos Er cos cos E cos r A E, A r P 3 E r cos r Eq. (3.8) Eq. (3.6) : r, r r E r cos E cos, exter fied effect Cotributio from the iduced chrge 3 cos 3 3 r E r Ih Uiversity r 3 cos q E d S ecosed (3.7) (3.8) (3.9) d S E (3.) 6

3.3. Spheric Coordites [Exmpe 3.9] 3.3 Seprtio of ribes Ih Uiversity (Soutio) ) Method (Direct itegrtio method) : Eq. (.3) ( ) ) Method (Seprtio of vribes method, 변수분리법 ) : Poteti iside the sphere : Eq.

27 3.3. Spheric Coordites [Exmpe 3.9] 3.3 Seprtio of ribes Ih Uiversity (Soutio) ) Method (Direct itegrtio method) : Eq. (.3) ( ) ) Method (Seprtio of vribes method, 변수분리법 ) : Poteti iside the sphere : Eq. (3.89) Eq. (3.) r r, P cos r Sice the poteti is cotiuous t the she surfce, Eq. (3.) = Eq. (3.3) t r =. Poteti outside the sphere : 4 B r S r r' (3.3) d' (3.) r, A r P cos r A P cos P cos B (3.) (3.4) r 7

28 3.3 Seprtio of ribes [Exmpe 3.9] (Soutio) - cotiued () B A B A (3.5) ( ) Ih Uiversity The rdi derivtive of suffers discotiuity t the surfce: Eq. (.36) Eqs. (3.) & (3.3) Eq. (3.6): Eq. (3.5) Eq. (3.7): B P cos A P cos Eq. (3.8) x P (cos) si d itegrtig (3.6) P cos A P cos P cos si d P cos si d ' ' P r xp xdx P cos P cos ' out r i r ' A, if ' si d, if ' E : orthogo property of the Legedre poyomis E E bove // bove E beow // beow. (3.7) (3.8) (3.9) (3.) 8

29 3.3 Seprtio of ribes [Exmpe 3.9] (Soutio) - cotiued () ' Eq. (3.9) ' A P cos si d ' ' ' (3.) ( ) Eq. (3.) Thus, A P cos si d r, A r P cos r (3.) (3.3) Eqs. (3.3) & (3.5) r r, A P cos r (3.4) Ih Uiversity 9

30 3.3 Seprtio of ribes [Exmpe 3.9] (Soutio) - cotiued (3) If k kpcos, cos (3.5) Eq. (3.) Eq. (3.3) A k k (3.6) P cos si d 3 k 3 r, r cos r (3.7) ( ) Eq. (3.4) Ih Uiversity 3 k 3 r r cos r, (3.8) If () is the iduced chrge o met sphere i exter fied E zˆ so tht k E, The Poteti iside the sphere Eq. (3.7) r E r cos E z r, The Poteti outside the sphere r 3, r E cos r (3.9) Eq. (3.8) (3.3) Eq. (3.9) E from Eq. (3.) E 3 zˆ Eq. (3.9) 3

31 Chpter 3. Potetis Next Css 3. Lpce s Equtio 3. The Method of Imges 3.3 Seprtio of ribes 3.3. Crtesi Coordites 3.3. Spheric coordites 3.4 Mutipoe Expsio Ih Uiversity 3

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